Making Metric Temporal Logic Rational

TL;DR

This paper introduces an extension of Metric Temporal Logic with rational expressions, proving its decidability and establishing a connection to timed automata, advancing the theoretical understanding of temporal logic expressiveness.

Contribution

It defines and analyzes the decidability of the rational extension of MTL, introduces a fragment with elementary decidability, and links it to automata theory.

Findings

Decidable satisfiability for the extended logic via reduction to MTL.

Identification of a fragment with elementary decidability.

Establishment of a tight automaton-logic correspondence.

Abstract

We study an extension of $\mtl$ in pointwise time with rational expression guarded modality $\reg_I(\re)$ where $\re$ is a rational expression over subformulae. We study the decidability and expressiveness of this extension ($\mtl$+$\varphi \ureg_{I, \re} \varphi$+$\reg_{I,\re}\varphi$), called $\regmtl$, as well as its fragment $\sfmtl$ where only star-free rational expressions are allowed. Using the technique of temporal projections, we show that $\regmtl$ has decidable satisfiability by giving an equisatisfiable reduction to $\mtl$. We also identify a subclass $\mitl+\ureg$ of $\regmtl$ for which our equi-satisfiable reduction gives rise to formulae of $\mitl$, yielding elementary decidability. As our second main result, we show a tight automaton-logic connection between $\sfmtl$ and partially ordered (or very weak) 1-clock alternating timed automata.

Equations

$${◊}_{[l,l]}\{\neg (\bigvee \Sigma \vee c_{i\oplus u})\mathsf{U}[{\mathsf{Th}}_i(q_1)\wedge (\neg \mathsf{Mrg}(i)\mathsf{U}[\mathsf{merge}(i_1,i)\wedge$$

$$(\neg \mathsf{Mrg}(i_1)\mathsf{U}[\mathsf{merge}(i_2,i_1)\wedge \dots (\neg \mathsf{Mrg}(i_{k_1})\mathsf{U}{c}_{i\oplus u})])])]\}$$

$${\phi }_{k,k_1}=\underset{k\ge k_1\ge 1}{\bigvee }[\mathsf{MergeseqPref}(k_1)\wedge \mathsf{MergeseqSuf}(k,k_1)]$$

$${◊}_{[l,l+1)}\{[({\mathsf{Th}}_{i_k}(q_f))\wedge (\neg {\mathsf{Mrg}}^{\prime }(i_k)\mathsf{S}[\mathsf{merge}(i_k,i_{k-1})\wedge (\neg {\mathsf{Mrg}}^{\prime }(i_{k-1})$$

$$\mathsf{S}[\mathsf{merge}(i_{k-1},i_{k-2})\wedge \dots \mathsf{merge}(i_c,i_{k_1})\wedge (\neg {\mathsf{Mrg}}^{\prime }(i_{k_1})\mathsf{S}{c}_{i\oplus 1})])])]\}$$

$$\{\neg (\bigvee \Sigma \vee c_{i\oplus l\oplus 1})\mathsf{U}[{\mathsf{Th}}_i(q_1)\wedge (\neg \mathsf{Mrg}(i)\mathsf{U}[\mathsf{merge}(i_1,i)\wedge$$

$$(\neg \mathsf{Mrg}(i_1)\mathsf{U}[\mathsf{merge}(i_2,i_1)\wedge \dots (\neg \mathsf{Mrg}(i_{k_1})\mathsf{U}{c}_{i\oplus l\oplus 1})])])]\}$$

$${◊}_{[l+1,l+2)}\{[c_{i\oplus l\oplus 1}\wedge (\neg \mathsf{Mrg}(i_{k_1})\mathsf{U}[\mathsf{merge}(i_c,i_{k_1})\wedge (\neg \mathsf{Mrg}(i_c)$$

$$\mathsf{U}[\mathsf{merge}(i_c,i_{k_1})\wedge \dots \mathsf{merge}(i_{k-1},i_{k-2})\wedge (\neg \mathsf{Mrg}(i_{k-1})\mathsf{U}$$

$$(l,c_1,c_2\dots ,c_k)\to \{(l^{\prime },c_1^{\prime \prime },c_2^{\prime \prime }\dots ,c_k^{\prime \prime })|c_i^{\prime \prime }\ge c_i^{\prime },1\le i\le k\}$$

$$(\mathcal{S}(a_1{b}_1{)}^{\ast }(a_2{b}_2{)}^{\ast }\dots (a_k{b}_k{)}^{\ast }\mathcal{F}{)}^{\ast }$$

Full text

1 Introduction

Temporal logics provide constructs to specify qualitative ordering between events in time. Real time logics are quantitative extensions of temporal logics with the ability to specify real time constraints amongst events. Logics $\mathsf{MTL}$ and $\mathsf{TPTL}$ are amongst the prominent real time logics [2]. Two notions of $\mathsf{MTL}$ semantics have been studied in the literature : continuous and pointwise [4]. The expressiveness and decidability results vary considerably with the semantics used : while the satisfiability checking of $\mathsf{MTL}$ is undecidable in the continuous semantics even for finite timed words [1], it is decidable in pointwise semantics with non-primitive recursive complexity [15]. Due to limited expressive power of $\mathsf{MTL}$, several additional modalities have been proposed : the $\mathsf{threshold\leavevmode\nobreak\ counting}$ modality [17] $\mathsf{C}_{I}^{\geq n}\phi$ states that in time interval $I$ relative to current point, $\phi$ occurs at least $n$ times. The $\mathsf{Pnueli}$ modality [17] $\mathsf{Pn}_{I}(\phi_{1},\ldots,\phi_{n})$ states that there is a subsequence of $n$ time points inside interval $I$ where at $i$th point the formula $\phi_{i}$ holds. In a recent result, Hunter [9] showed that, in continuous time semantics, $\mathsf{MTL}$ enriched with $\mathsf{C}$ modality (denoted $\mathsf{MTL}+\mathsf{C}$) is as expressive as $\mathsf{FO}$ with distance $\mathsf{FO}[<,+1]$, which is as expressive as $\mathsf{TPTL}$. Unfortunately, satisfiability and model checking of all these logics are undecidable. This has led us to focus on the pointwise case with only the future modality, i.e. logic $\mathsf{MTL}[\>\mathsf{U}_{I}]$, which we abbreviate as $\mathsf{MTL}$ in rest of the paper. Also, $\mathsf{MTL}+op$ means $\mathsf{MTL}$ with modalities $\>\mathsf{U}_{I}$ as well as $op$.

In pointwise semantics, it can be shown that $\mathsf{MTL}\subset\mathsf{MTL}+\mathsf{C}\subset\mathsf{MTL}+\mathsf{Pn}$ (see [11]). In this paper, we propose a generalization of threshold counting and Pnueli modalities by a $\mathsf{rational\leavevmode\nobreak\ expression}$ modality $\mathsf{Rat}_{I}\mathsf{re}(\phi_{1},\ldots,\phi_{k})$, which specifies that the truth of the subformulae, $\phi_{1},\ldots,\phi_{k}$, at the set of points within interval $I$ is in accordance with the rational expression $\mathsf{re}(\phi_{1},\ldots,\phi_{k})$. The resulting logic is called $\mathsf{RatMTL}$ and is the subject of this paper. The expressive power of logic $\mathsf{RatMTL}$ raises several points of interest. It can be shown that $\mathsf{MTL}+\mathsf{Pn}\subset\mathsf{RatMTL}$, and it can express several new and interesting properties: (1) Formula $\mathsf{Rat}_{(1,2)}((aa)^{*})$ states that within time interval $(1,2)$ there is an even number of occurrences of $a$. We will define a derived modulo counting modality which states this directly as the formula $\mathsf{MC}_{(1,2)}^{0\%2}a$.

(2) An exercise regime lasting between 60 to 70 seconds consists of arbitrary many repetitions of three pushup cycles which must be completed within 2 seconds. There is no restriction on delay between two cycles to accomodate weak athletes. This is given by $\mathsf{Rat}_{[60,70]}((UPP.up.up)^{*})$ where $UPP=(up\leavevmode\nobreak\ \mathsf{URat}_{(0,2],up}up)$.

The inability to specify rational expression constraints has been an important lacuna of LTL and its practically useful extensions such as PSL sugar [7], [6] (based on Dymanic Logic [8]) which extend LTL with both counting and rational expressions. This indicates that our logic $\mathsf{RatMTL}$ is a natural and useful logic for specifying properties. However, to our knowledge, impact of rational expression constraints on metric temporal modalities have not been studied before. As we show in the paper, timing and regularity constraints interact in a fairly complex manner.

As our first main result, we show that satisfiability of $\mathsf{RatMTL}$ is decidable by giving an equisatisfiable reduction to $\mathsf{MTL}$. The reduction makes use of the technique of oversampled temporal projections which was previously proposed [10], [11] and used for proving the decidability of $\mathsf{MTL}+\mathsf{C}$. The reduction given here has several novel features such as an $\mathsf{MTL}$ encoding of the run tree of an alternating automaton which restarts the DFA of a given rational expression at each time point (section 3.1). We identify two syntactic subsets of $\mathsf{RatMTL}$ denoted $\mathsf{MITL}+\mathsf{URat}$ with 2$\mathsf{EXPSPACE}$ hard satisfiability, and its further subset $\mathsf{MITL}+\mathsf{UM}$ with $\mathsf{EXPSPACE}$-complete satisfiability. As our second main result, we show that the star-free fragment $\mathsf{SfrMTL}$ of $\mathsf{RatMTL}$ characterizes exactly the class of partially ordered 1-clock alternating timed automata, thereby giving a tight logic automaton connection. The most non-trivial part of this proof is the construction of $\mathsf{SfrMTL}$ formula equivalent to a given partially ordered 1-clock alternating timed automaton $\mathcal{A}$ (Lemma 4.5).

2 Timed Temporal Logics

This section describes the syntax and semantics of the timed temporal logics needed in this paper : $\mathsf{MTL}$ and $\mathsf{TPTL}$. Let $\Sigma$ be a finite set of propositions. A finite timed word over $\Sigma$ is a tuple $\rho=(\sigma,\tau)$. $\sigma$ and $\tau$ are sequences $\sigma_{1}\sigma_{2}\ldots\sigma_{n}$ and $\tau_{1}\tau_{2}\ldots\tau_{n}$ respectively, with $\sigma_{i}\in\mathcal{P}(\Sigma)-\emptyset$, and $\tau_{i}\in\>\mathbb{R}_{\geq 0}$ for $1\leq i\leq n$ and $\forall i\in dom(\rho)$, $\tau_{i}\leq\tau_{i+1}$, where $dom(\rho)$ is the set of positions $\{1,2,\ldots,n\}$ in the timed word. For convenience, we assume $\tau_{1}=0$. The $\sigma_{i}$’s can be thought of as labeling positions $i$ in $dom(\rho)$. For example, given $\Sigma=\{a,b,c\}$, $\rho=(\{a,c\},0)(\{a\},0.7)(\{b\},1.1)$ is a timed word. $\rho$ is strictly monotonic iff $\tau_{i}<\tau_{i+1}$ for all $i,i+1\in dom(\rho)$. Otherwise, it is weakly monotonic. The set of finite timed words over $\Sigma$ is denoted $T\Sigma^{*}$. Given $\rho=(\sigma,\tau)$ with $\sigma=\sigma_{1}\dots\sigma_{n}$, $\sigma^{\mathsf{single}}$ denotes the set of words $\{w_{1}w_{2}\dots w_{n}\mid w_{i}\in\sigma_{i}\}$. For $\rho$ as above, $\sigma^{\mathsf{single}}$ consists of $(\{a\},0)(\{a\},0.7)(\{b\},1.1)$ and $(\{c\},0)(\{a\},0.7)(\{b\},1.1)$. Let $I\nu$ be a set of open, half-open or closed time intervals. The end points of these intervals are in $\mathbb{N}\cup\{0,\infty\}$. For example, $[1,3),[2,\infty)$. For $\tau\in\>\mathbb{R}_{\geq 0}$ and interval $\langle a,b\rangle$, with $<\in\{(,[\}$ and $>\in\{],)\}$, $\tau+\langle a,b\rangle$ stands for the interval $\langle\tau+a,\tau+b\rangle$.

Metric Temporal Logic($\mathsf{MTL}$). Given a finite alphabet $\Sigma$, the formulae of $\mathsf{MTL}$ are built from $\Sigma$ using boolean connectives and time constrained version of the modality $\>\mathsf{U}$ as follows:

$\varphi::=a(\in\Sigma)\leavevmode\nobreak\ |true\leavevmode\nobreak\ |\varphi\wedge\varphi\leavevmode\nobreak\ |\leavevmode\nobreak\ \neg\varphi\leavevmode\nobreak\ |\leavevmode\nobreak\ \varphi\>\mathsf{U}_{I}\varphi$, where $I\in I\nu$.

For a timed word $\rho=(\sigma,\tau)\in T\Sigma^{*}$, a position $i\in dom(\rho)$, and an $\mathsf{MTL}$ formula $\varphi$, the satisfaction of $\varphi$ at a position $i$ of $\rho$ is denoted $(\rho,i)\models\varphi$, and is defined as follows: (i) $\rho,i\models a$ $\leftrightarrow$ $a\in\sigma_{i}$, (ii) $\rho,i\models\neg\varphi$ $\leftrightarrow$ $\rho,i\nvDash\varphi$, (iii) $\rho,i\models\varphi_{1}\wedge\varphi_{2}$ $\leftrightarrow$ $\rho,i\models\varphi_{1}$ and $\rho,i\ \models\ \varphi_{2}$, (iv) $\rho,i\ \models\ \varphi_{1}\>\mathsf{U}_{I}\varphi_{2}$ $\leftrightarrow$ $\exists j>i$, $\rho,j\ \models\ \varphi_{2},\tau_{j}-\tau_{i}\in I$, and $\rho,k\ \models\ \varphi_{1}$ $\forall$ $i<k<j$.

The language of a $\mathsf{MTL}$ formula $\varphi$ is $L(\varphi)=\{\rho\mid\rho,1\models\varphi\}$. Two formulae $\varphi$ and $\phi$ are said to be equivalent denoted as $\varphi\equiv\phi$ iff $L(\varphi)=L(\phi)$. Additional temporal connectives are defined in the standard way: we have the constrained future eventuality operator $\Diamond_{I}a\equiv true\>\mathsf{U}_{I}a$ and its dual $\Box_{I}a\equiv\neg\Diamond_{I}\neg a$. We also define the next operator as $\mathsf{O}_{I}\phi\equiv\bot\>\mathsf{U}_{I}\phi$. Non strict versions of operators are defined as $\Diamond^{\mathsf{ns}}_{I}a=a\vee\Diamond_{I}a,\Box^{\mathsf{ns}}_{I}a\equiv a\wedge\Box_{I}a$, $a\>\mathsf{U}^{\mathsf{ns}}_{I}b\equiv b\vee[a\wedge(a\>\mathsf{U}_{I}b)]$ if $0\in I$, and $[a\wedge(a\>\mathsf{U}_{I}b)]$ if $0\notin I$. Also, $a\>\mathsf{W}b$ is a shorthand for $\Box a\vee(a\>\mathsf{U}b)$. The subclass of $\mathsf{MTL}$ obtained by restricting the intervals $I$ in the until modality to non-punctual intervals is denoted $\mathsf{MITL}$.

Timed Propositional Temporal Logic ($\mathsf{TPTL}$). $\mathsf{TPTL}$ is a prominent real time extension of $\mathsf{LTL}$, where timing constraints are specified with the help of freeze clocks. The set of $\mathsf{TPTL}$ formulas are defined inductively as $\varphi::=a(\in\Sigma)\leavevmode\nobreak\ |true\leavevmode\nobreak\ |\varphi\wedge\varphi\leavevmode\nobreak\ |\leavevmode\nobreak\ \neg\varphi\leavevmode\nobreak\ |\leavevmode\nobreak\ \varphi\>\mathsf{U}\varphi\leavevmode\nobreak\ |\leavevmode\nobreak\ y.\varphi\leavevmode\nobreak\ |\leavevmode\nobreak\ y\in I$. $\mathcal{C}$ is a set of clock variables progressing at the same rate, $y\in\mathcal{C}$, and $I$ is an interval as above. For a timed word $\rho=(\sigma_{1},\tau_{1})\dots(\sigma_{n},\tau_{n})$, we define the satisfiability relation, $\rho,i,\nu\models\phi$ saying that the formula $\phi$ is true at position $i$ of the timed word $\rho$ with valuation $\nu$ of all the clock variables as follows: (1) $\rho,i,\nu\models a$ $\leftrightarrow$ $a\in\sigma_{i}$, (2) $\rho,i,\nu\models\neg\varphi$ $\leftrightarrow$ $\rho,i,\nu\nvDash\varphi$, (3) $\rho,i,\nu\models\varphi_{1}\wedge\varphi_{2}$ $\leftrightarrow$ $\rho,i,\nu\models\varphi_{1}$ and $\rho,i,\nu\ \models\ \varphi_{2}$, (4) $\rho,i,\nu\models x.\varphi$ $\leftrightarrow$ $\rho,i,\nu[x\leftarrow\tau_{i}]\models\varphi$, (5) $\rho,i,\nu\models x\in I$ $\leftrightarrow$ $\tau_{i}-\nu(x)\in I$, (6) $\rho,i,\nu\ \models\ \varphi_{1}\>\mathsf{U}\varphi_{2}$ $\leftrightarrow$ $\exists j>i$, $\rho,j,\nu\ \models\ \varphi_{2}$, and $\rho,k,\nu\ \models\ \varphi_{1}$ $\forall$ $i<k<j$. $\rho$ satisfies $\phi$ denoted $\rho\models\phi$ iff $\rho,1,\bar{0}\models\phi$. Here $\bar{0}$ is the valuation obtained by setting all clock variables to 0. We denote by $k{-}\mathsf{TPTL}$ the fragment of $\mathsf{TPTL}$ using at most $k$ clock variables.

Theorem 2.1** ([15]).**

$\mathsf{MTL}$* satisfiability is decidable over finite timed words and is non-primitive recursive.*

$\mathsf{MTL}$ with Rational Expressions($\mathsf{RatMTL}$)

We propose an extension of $\mathsf{MTL}$ with rational expressions, that forms the core of the paper. These modalities can assert the truth of a rational expression (over subformulae) within a particular time interval with respect to the present point. For example, $\mathsf{Rat}_{(0,1)}(\varphi_{1}.\varphi_{2})^{+}$ when evaluated at a point $i$, asserts the existence of $2k$ points $\tau_{i}<\tau_{i+1}<\tau_{i+2}<\dots<\tau_{i+2k}<\tau_{i}+1$, $k>0$, such that $\varphi_{1}$ evaluates to true at $\tau_{i+2j+1}$, and $\varphi_{2}$ evaluates to true at $\tau_{i+2j+2}$, for all $0\leq j<k$.

$\mathsf{RatMTL}$ Syntax: Formulae of $\mathsf{RatMTL}$ are built from $\Sigma$ (atomic propositions) as follows:

$\varphi::=a(\in\Sigma)\leavevmode\nobreak\ |true\leavevmode\nobreak\ |\varphi\wedge\varphi\leavevmode\nobreak\ |\leavevmode\nobreak\ \neg\varphi\leavevmode\nobreak\ |\leavevmode\nobreak\ \mathsf{Rat}_{I}\mathsf{re}(\mathsf{S})\leavevmode\nobreak\ |\leavevmode\nobreak\ \varphi\mathsf{URat}_{I,\mathsf{re}(\mathsf{S})}\varphi$, where $I\in I\nu$ and $\mathsf{S}$ is a finite set of formulae of interest, and $\mathsf{re}(\mathsf{S})$ is defined as a rational expression over $\mathsf{S}$. $\mathsf{re}(\mathsf{S})::=\varphi(\in\mathsf{S})\leavevmode\nobreak\ |\leavevmode\nobreak\ \mathsf{re}(\mathsf{S}).\mathsf{re}(\mathsf{S})\leavevmode\nobreak\ |\leavevmode\nobreak\ \mathsf{re}(\mathsf{S})+\mathsf{re}(\mathsf{S})\leavevmode\nobreak\ |\leavevmode\nobreak\ [\mathsf{re}(\mathsf{S})]^{*}$. Thus, $\mathsf{RatMTL}$ is $\mathsf{MTL}+\mathsf{URat}+\mathsf{Rat}$. An atomic rational expression $\mathsf{re}$ is any well-formed formula $\varphi\in\mathsf{RatMTL}$.

$\mathsf{RatMTL}$ Semantics: For a timed word $\rho=(\sigma,\tau)\in T\Sigma^{*}$, a position $i\in dom(\rho)$, and a $\mathsf{RatMTL}$ formula $\varphi$, a finite set $\mathsf{S}$ of formulae, we define the satisfaction of $\varphi$ at a position $i$ as follows. For positions $i<j\in dom(\rho)$, let $\mathsf{Seg}(\mathsf{S},i,j)$ denote the untimed word over $\mathcal{P}(\mathsf{S})$ obtained by marking the positions $k\in\{i+1,\dots,j-1\}$ of $\rho$ with $\psi\in\mathsf{S}$ iff $\rho,k\models\psi$. For a position $i{\in}dom(\rho)$ and an interval $I$, let $\mathsf{TSeg}(S,I,i)$ denote the untimed word over $\mathcal{P}(\mathsf{S})$ obtained by marking all the positions $k$ such that $\tau_{k}-\tau_{i}\in I$ of $\rho$ with $\psi\in\mathsf{S}$ iff $\rho,k\models\psi$.

$\rho,i\models\varphi_{1}\mathsf{URat}_{I,\mathsf{re}(\mathsf{S})}\varphi_{2}$ $\leftrightarrow$ $\exists j{>}i$, $\rho,j{\models}\ \varphi_{2},\tau_{j}-\tau_{i}{\in}I$, $\rho,k\ {\models}\ \varphi_{1}$ ${\forall}i{<}k{<}j$ and, $[\mathsf{Seg}(\mathsf{S},i,j)]^{\mathsf{single}}\cap L(\mathsf{re}(S))\neq\emptyset$, where $L(\mathsf{re}(\mathsf{S}))$ is the language of the rational expression $\mathsf{re}$ formed over the set $\mathsf{S}$. The subclass of $\mathsf{RatMTL}$ using only the $\mathsf{URat}$ modality is denoted $\mathsf{RatMTL}[\mathsf{URat}]$ or $\mathsf{MTL}+\mathsf{URat}$ and if only non-punctual intervals are used, then it is denoted $\mathsf{RatMITL}[\mathsf{URat}]$ or $\mathsf{MITL}+\mathsf{URat}$. 2. 2.

$\rho,i\models\mathsf{Rat}_{I}\mathsf{re}$ $\leftrightarrow$ $[\mathsf{TSeg}(S,I,i)]^{\mathsf{single}}\cap L(\mathsf{re}(S))\neq\emptyset$.

The language accepted by a $\mathsf{RatMTL}$ formula $\varphi$ is given by $L(\varphi)=\{\rho\mid\rho,0\models\varphi\}$.

Example 1. Consider the formula $\varphi=a\mathsf{URat}_{(0,1),ab^{*}}b$. Then $\mathsf{re}{=}ab^{*}$, and the subformulae of interest are $a,b$. For $\rho{=}(\{a\},0)(\{a,b\},0.3)(\{a,b\},0.99)$, $\rho,1\models\varphi$, since $a{\in}\sigma_{2},b{\in}\sigma_{3}$, $\tau_{3}{-}\tau_{1}{\in}(0,1)$ and $a\in[\mathsf{Seg}(\{a,b\},1,3)]^{\mathsf{single}}\cap L(ab^{*})$. On the other hand, for the word $\rho=(\{a\},0)(\{a\},0.3)(\{a\},0.5)(\{a\},0.9)(\{b\},0.99)$, we know that $\rho,1\nvDash\varphi$, since even though $b\in\sigma_{5},a\in\sigma_{i}$ for $i<5$, $[\mathsf{Seg}(\{a,b\},1,5)]^{\mathsf{single}}=aaa$ and $aaa\notin L(ab^{*})$.

Example 2. Consider the formula $\varphi=\mathsf{Rat}_{(0,1)}[\neg\mathsf{Rat}_{(0,1)}a]$. For the word $\rho=(\{a,b\},0)$ $(\{a,b\},0.91)(\{a\},1.2)$, to check $\varphi$ at position 1, we check position 2 of the word, since $\tau_{2}-\tau_{1}\in(0,1)$. The formulae of interest for marking is $\{\neg\mathsf{Rat}_{(0,1)}a\}$. Position 2 is not marked, since $\rho,2\models\mathsf{Rat}_{(0,1)}a$. Then $[\mathsf{TSeg}(\mathsf{S},(0,1),1)]^{\mathsf{single}}=\emptyset\notin L(\neg\mathsf{Rat}_{(0,1)}a)$. However, for the word $\rho=(\{a,b\},0)$ $(\{a,b\},0.91)(\{b\},1.1)$, $\rho,1\models\varphi$, since position 2 is marked with $\neg\mathsf{Rat}_{(0,1)}a$, and $\neg\mathsf{Rat}_{(0,1)}a\in L(\neg\mathsf{Rat}_{(0,1)}a)\cap[\mathsf{TSeg}(S,(0,1),1)]^{\mathsf{single}}$.

Example 3. Consider the formula $\varphi=\mathsf{Rat}_{(0,1)}[\mathsf{Rat}_{(0,1)}a]^{*}$.

For \rho=(\{a,b\},0)$$(\{a,b\},0.7)(\{b\},0.98)(\{a,b\},1.4), we have $\rho,1{\nvDash}\mathsf{Rat}_{(0,1)}[\mathsf{Rat}_{(0,1)}a]^{*}$, since point 2 is not marked $\mathsf{Rat}_{(0,1)}a$, even though point 3 is.

Generalizing Counting, Pnueli & Mod Counting Modalities The following reductions show that $\mathsf{RatMTL}$ subsumes most of the extensions of $\mathsf{MTL}$ studied in the literature.

(1) Threshold Counting constraints [17], [12], [11] specify the number of times a property holds within some time region is at least (or at most) $n$. These can be expressed in $\mathsf{RatMTL}$: (i) $\mathsf{C}^{\geq n}_{I}\varphi\equiv\mathsf{Rat}_{I}(\mathsf{re}_{th})$, (ii) $\phi_{1}\mathsf{UT}_{I,\varphi\geq n}\phi_{2}\equiv\phi_{1}\mathsf{URat}_{I,\mathsf{re}_{th}}\phi_{2}$, where \mathsf{re}_{th}={true^{*}\!\underbrace{\varphi.true^{*}.\ldots.\varphi.true^{*}}_{\text{n times}}}.

(2) Pnueli Modalities [17], which enhance the expressiveness of $\mathsf{MITL}$ in continuous semantics preserving the complexity, can be written in $\mathsf{RatMTL}$: $\mathsf{Pn}(\phi_{1},\phi_{2},\ldots,\phi_{k})$ can be written as $\mathsf{Rat}_{I}(true^{*}.\phi_{1}.true^{*}\phi_{2}.\ldots.true^{*}.\phi_{k}.true^{*})$.

(3) Modulo Counting constraints [3], [13] specify the number of times a property holds modulo $n\in\mathbb{N}$, in some region. We extend these to the timed setting by proposing two modalities $\mathsf{MC}^{k\%n}_{I}$ and $\mathsf{UM}_{I,\varphi=k\%n}$. $\mathsf{MC}^{k\%n}_{I}\varphi$ checks if the number of times $\varphi$ is true in interval $I$ is $M(n)+k$, where $M(n)$ denotes a non-negative integer multiple of $n$, and $0\leq k\leq n-1$, while $\varphi_{1}\mathsf{UM}_{I,\#{\psi}=k\%n}\varphi_{2}$ when asserted at a point $i$, checks the existence of $j>i$ such that $\tau_{j}-\tau_{i}\in I$, $\varphi_{2}$ is true at $j$, $\varphi_{1}$ holds between $i,j$, and the number of times $\psi$ is true between $i,j$ is $M(n)+k$, $0\leq k\leq n-1$. As an example, $\psi=true\mathsf{UM}_{(0,1),\#b=1\%2}(a\vee b)$, when asserted at a point $i$, checks the existence of a point $j>i$ such that $a$ or $b\in\sigma_{j}$, $\tau_{j}-\tau_{i}\in(0,1)$, and the number of points between $i,j$ where $b$ is true is odd. Both these modalities can be rewritten equivalently in $\mathsf{RatMTL}$ as follows: $\mathsf{MC}^{k\%n}_{I}\varphi\equiv\mathsf{Rat}_{I}(\mathsf{re}_{mod})$ and $\phi_{1}\mathsf{UM}_{I,\varphi=k\%n}\phi_{2}\equiv\phi_{1}\mathsf{URat}_{I,\mathsf{re}_{mod}}\phi_{2}$ where \mathsf{re}_{mod}=([\underbrace{(\neg\varphi)^{*}.\varphi.\ldots.(\neg\varphi)^{*}.\varphi}_{\text{n times}}]^{*}.[\underbrace{(\neg\varphi)^{*}.\varphi.\ldots.(\neg\varphi)^{*}.\varphi}_{\text{k times}}]. The extension of $\mathsf{MTL}$ ($\mathsf{MITL}$) with only $\mathsf{UM}$ is denoted $\mathsf{MTL}+\mathsf{UM}$ ($\mathsf{MITL}+\mathsf{UM}$) while $\mathsf{MTL}+\mathsf{MC}$ ($\mathsf{MITL}+\mathsf{MC}$) denotes the extension using $\mathsf{MC}$.

3 Satisfiability of $\mathsf{RatMTL}$ and Complexity

The main results of this section are as follows.

Theorem 3.1**.**

*(1) Satisfiability of $\mathsf{RatMTL}$ is decidable. (2) Satisfiability of $\mathsf{MITL}+\mathsf{UM}$ is $\mathsf{EXPSPACE}$-complete. (3) Satisfiability of $\mathsf{MITL}+\mathsf{URat}$ is in $\mathsf{2EXPSPACE}$.

(4) Satisfiability of $\mathsf{MITL}+\mathsf{MC}$ is $\bf{F}_{\omega^{\omega}}$-hard.*

Details of Theorems 3.1.2, 3.1.3, 3.1.4 can be found in Appendices E.2, E.3 and E.4.

Theorem 3.2**.**

$\mathsf{MTL}+\mathsf{URat}\subseteq\mathsf{MTL}+\mathsf{Rat}$, $\mathsf{MTL}+\mathsf{UM}\subseteq\mathsf{MTL}+\mathsf{MC}$.

Theorem 3.2 shows that the $\mathsf{Rat}$ modality can capture $\mathsf{URat}$ (and likewise, $\mathsf{MC}$ captures $\mathsf{UM}$). Thus, $\mathsf{RatMTL}\equiv\mathsf{MTL}+\mathsf{Rat}$. Observe that any $\mathsf{re}$ can be decomposed into finitely many factors, i.e. $\mathsf{re}=\sum\limits_{i=1}^{n}R^{i}_{1}.R^{i}_{2}$. Given $true\mathsf{URat}_{[l,u),\mathsf{re}}\phi_{2}$, we assert $R^{i}_{1}$ within interval $(0,l]$ and $R^{i}_{2}$ in the prefix of the latter part within $[l,u)$, followed by $\phi_{2}$. $true\mathsf{URat}_{[l,u),\mathsf{re}}\phi_{2}\equiv\bigvee\limits_{i\in\{1,2\ldots,n\}}\mathsf{Rat}_{(0,l)}R^{i}_{1}\wedge\mathsf{Rat}_{[l,u)}R^{i}_{2}.\phi_{2}.\Sigma^{*}$. The proofs can be seen in Appendix G.

3.1 Proof of Theorem 3.1.1

Equisatisfiability We will use the technique of equisatisfiability modulo oversampling [10] in the proof of Theorem 3.1. Using this technique, formulae $\varphi$ in one logic (say $\mathsf{RatMTL}$) can be transformed into formulae $\psi$ over a simpler logic (say $\mathsf{MTL}$) such that whenever $\rho\models\varphi$ for a timed word $\rho$ over alphabet $\Sigma$, one can construct a timed word $\rho^{\prime}$ over an extended set of positions and an extended alphabet $\Sigma^{\prime}$ such that $\rho^{\prime}\models\psi$ and vice-versa [10], [11]. In oversampling, (i) $dom(\rho^{\prime})$ is extended by adding some extra positions between the first and last point of $\rho$, (ii) the labeling of a position $i\in dom(\rho)$ is over the extended alphabet $\Sigma^{\prime}\supset\Sigma$ and can be a superset of the previous labeling over $\Sigma$, while the new positions are labeled using only the new symbols $\Sigma^{\prime}-\Sigma$. We can recover $\rho$ from $\rho^{\prime}$ by erasing the new points and the new symbols. A restricted use of oversampling, when one only extends the alphabet and not the set of positions of a timed word $\rho$ is called simple extension. In this case, if $\rho^{\prime}$ is a simple extension of $\rho$, then $dom(\rho)=dom(\rho^{\prime})$, and by erasing the new symbols from $\rho^{\prime}$, we obtain $\rho$. See Figure 1 for an illustration. The formula $\psi$ over the larger alphabet $\Sigma^{\prime}\supset\Sigma$ such that $\rho^{\prime}\models\psi$ iff $\rho\models\varphi$ is said to be equisatisfiable modulo temporal projections to $\varphi$. In particular, $\psi$ is equisatisfiable to $\varphi$ modulo simple extensions or modulo oversampling, depending on how the word $\rho^{\prime}$ is constructed from the word $\rho$.

The oversampling technique is used in the proofs of parts 3.1.1, 3.1.3 and 3.1.4.

Equisatisfiable Reduction : $\mathsf{RatMTL}$ to $\mathsf{MTL}$

Let $\varphi$ be a $\mathsf{RatMTL}$ formula. To obtain equisatisfiable $\mathsf{MTL}$ formula $\psi$, we do the following.

We “flatten” the reg modalities to simplify the formulae, eliminating nested reg modalities. Flattening results in extending the alphabet. Each of the modalities $\mathsf{Rat}_{I},\mathsf{URat}$ that appear in the formula $\varphi$ are replaced with fresh witness propositions to obtain a flattened formula. For example, if $\varphi=\mathsf{Rat}_{(0,1)}[a\mathsf{URat}_{(1,2),\mathsf{Rat}_{(0,1)}(a+b)^{*}}b]$, then flattening yields the formula $\varphi_{flat}=w_{1}\wedge\Box^{\mathsf{ns}}[w_{1}\leftrightarrow\mathsf{Rat}_{(0,1)}w_{2}]\wedge\Box^{\mathsf{ns}}[w_{2}\leftrightarrow a\mathsf{URat}_{(1,2),w_{3}}b]\wedge\Box^{\mathsf{ns}}[w_{3}\leftrightarrow\mathsf{Rat}_{(0,1)}(a+b)^{*}]$, where $w_{1},w_{2},w_{3}$ are fresh witness propositions. Let $W$ be the set of fresh witness propositions such that $\Sigma\cap W=\emptyset$. After flattening, the modalities $\mathsf{Rat}_{I},\mathsf{URat}$ appear only in this simplified form as $\Box^{\mathsf{ns}}[w\leftrightarrow\mathsf{Rat}_{I},\mathsf{URat}]$. This simplified appearance of reg modalities are called temporal definitions and have the form $\Box^{\mathsf{ns}}[w\leftrightarrow\mathsf{Rat}_{I}\and]$ or $\Box^{\mathsf{ns}}[w\leftrightarrow x\mathsf{URat}_{I^{\prime},\and}y]$, where $\and$ is a rational expression over $\Sigma\cup W_{i}$, $W_{i}$ being the set of fresh witness propositions used in the flattening, and $I^{\prime}$ is either a unit length interval or an unbounded interval. 2. 2.

The elimination of reg modalities is achieved by obtaining equisatisfiable $\mathsf{MTL}$ formulae $\psi_{i}$ over $X_{i}$, possibly a larger set of propositions than $\Sigma\cup W_{i}$ corresponding to each temporal definition $T_{i}$ of $\varphi_{flat}$. Relativizing these $\mathsf{MTL}$ formulae and conjuncting them, we obtain an $\mathsf{MTL}$ formula $\bigwedge_{i}Rel(\psi_{i},\Sigma)$ that is equisatisfiable to $\varphi$ (see Figure 1 for relativization).

The above steps are routine [10], [11]. What remains is to handle the temporal definitions.

Embedding the Runs of the DFA

For any given $\rho$ over $\Sigma\cup W$, where $W$ is the set of witness propositions used in the temporal definitions $T$ of the forms $\Box^{\mathsf{ns}}[w\leftrightarrow\mathsf{Rat}_{I}\and]$ or $\Box^{\mathsf{ns}}[w\leftrightarrow x\mathsf{URat}_{I^{\prime},\and}y]$, the rational expression $\and$ has a corresponding minimal DFA recognizing it. We define an LTL formula $\mathsf{GOODRUN}(\phi_{e})$ which takes a formula $\phi_{e}$ as a parameter with the following behaviour. $\rho,i\models\mathsf{GOODRUN}(\phi_{e})$ iff for all $k>i$, $(\rho,k\models\phi_{e})\rightarrow(\rho[i,k]\in L(\and))$. To achieve this, we use two new sets of symbols $\mathsf{Threads}$ and $\mathsf{Merge}$ for this information. This results in the extended alphabet $\Sigma\cup W\cup\mathsf{Threads}\cup\mathsf{Merge}$ for the simple extension $\rho^{\prime}$ of $\rho$. The behaviour of $\mathsf{Threads}$ and $\mathsf{Merge}$ are explained below.

Consider $\and=\mathsf{re}(\mathsf{S})$. Let $\mathcal{A}_{\and}=(Q,2^{\mathsf{S}},\delta,q_{1},Q_{F})$ be the minimal DFA for $\and$ and let $Q=\{q_{1},q_{2},\ldots,q_{m}\}$. Let $\mathsf{In}=\{1,2,\dots,m\}$ be the indices of the states. Conceptually, we consider multiple runs of $\mathcal{A}_{\and}$ with a new run (new thread) started at each point in $\rho$. $\mathsf{Threads}$ records the state of each previously started run. At each step, each thread is updated from it previous value according to the transition function $\delta$ of $\mathcal{A}_{\and}$ and also augmented with a new run in initial state. Potentially, the number of threads would grow unboundedly in size but notice that once two runs are the same state at position $i$ they remain identical in future. Hence they can be merged into single thread (see Figure2). As a result, $m$ threads suffice. We record whether threads are merged in the current state using variables $\mathsf{Merge}$. An LTL formula records the evolution of $\mathsf{Threads}$ and $\mathsf{Merge}$ over any behaviour $\rho$. We can define formula $\mathsf{GOODRUN}(\phi_{e})$ in LTL over $\mathsf{Threads}$ and $\mathsf{Merge}$.

At each position, let $\mathsf{Th}_{i}(q_{x})$ be a proposition that denotes that the $i$th thread is active and is in state $q_{x}$, while $\mathsf{Th}_{i}(\bot)$ be a proposition that denotes that the $i$th thread is not active. The set $\mathsf{Threads}$ consists of propositions $\mathsf{Th}_{i}(q_{x}),\mathsf{Th}_{i}(\bot)$ for $1\leq i,x\leq m$. 2. 2.

If at a position $e$, we have $\mathsf{Th}_{i}(q_{x})$ and $\mathsf{Th}_{j}(q_{y})$ for $i<j$, and if $\delta(q_{x},\sigma_{e})=\delta(q_{y},\sigma_{e})$, then we can merge the threads $i,j$ at position $e+1$. Let $\mathsf{merge}(i,j)$ be a proposition that signifies that threads $i,j$ have been merged. In this case, $\mathsf{merge}(i,j)$ is true at position $e+1$. Let $\mathsf{Merge}$ be the set of all propositions $\mathsf{merge}(i,j)$ for $1\leq i<j\leq m$.

We now describe the conditions to be checked in $\rho^{\prime}$.

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Initial condition($\varphi_{init}$)- At the first point of the word, we start the first thread and initialize all other threads as $\bot$ : $\varphi_{init}=((\mathsf{Th}_{1}(q_{1}))\wedge\bigwedge\limits_{1<i\leq m}\mathsf{Th}_{i}(\bot))$.

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Initiating runs at all points($\varphi_{start}$)- To check the rational expression within an arbitrary interval, we need to start a new run from every point. $\varphi_{start}=\Box^{\mathsf{ns}}(\bigvee\limits_{i\leq m}\mathsf{Th}_{i}(q_{1}))$

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Disallowing Redundancy($\varphi_{no-red}$)- At any point of the word, if $i<j$ and $\mathsf{Th}_{i}(q_{x})$ and $\mathsf{Th}_{j}(q_{x})$ are both true, $q_{x}\neq q_{y}$. $\varphi_{no-red}=\bigwedge\limits_{x\in\mathsf{In}}\Box^{\mathsf{ns}}[\neg\bigvee\limits_{1\leq i<j\leq m}(\mathsf{Th}_{i}(q_{x})\wedge\mathsf{Th}_{j}(q_{x}))]$

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Merging Runs($\varphi_{\mathsf{merge}}$)- If two different threads $\mathsf{Th}_{i},\mathsf{Th}_{j}(i<j)$ reach the same state $q_{x}$ on reading the input at the present point, then we merge thread $\mathsf{Th}_{j}$ with $\mathsf{Th}_{i}$. We remember the merge with the proposition $\mathsf{merge}(i,j)$. We define a macro $\mathsf{Nxt}(\mathsf{Th}_{i}(q_{x}))$ which is true at a point $e$ if and only if $\mathsf{Th}_{i}(q_{y})$ is true at $e$ and $\delta(q_{y},\sigma_{e})=q_{x}$, where $\sigma_{e}\subseteq AP$ is the maximal set of propositions true at $e$: ${\bigvee\limits_{\{(q_{y},prop)\in(Q,2^{AP})|\delta(q_{y},prop){=}q_{x}\}}}[prop{\wedge}{\mathsf{Th}_{i}(q_{y})}]$.

Let $\psi(i,j,k,q_{x})$ be a formula that says that at the next position, $\mathsf{Th}_{i}(q_{x})$ and $\mathsf{Th}_{k}(q_{x})$ are true for $k>i$, but for all $j<i$, $\mathsf{Th}_{j}(q_{x})$ is not. $\psi(i,j,k,q_{x})$ is given by

$\mathsf{Nxt}(\mathsf{Th}_{i}(q_{x})){\wedge}{\bigwedge\limits_{j<i}}{\neg\mathsf{Nxt}(\mathsf{Th}_{j}(q_{x}))}{\wedge}{\mathsf{Nxt}(\mathsf{Th}_{k}(q_{x}))}$. In this case, we merge threads $\mathsf{Th}_{i},\mathsf{Th}_{k}$, and either restart $\mathsf{Th}_{k}$ in the initial state, or deactivate the $k$th thread at the next position. This is given by the formula $\mathsf{NextMerge(i,k)}=\mathsf{O}[\mathsf{merge}(i,k)\wedge(\mathsf{Th}_{k}(\bot)\vee\mathsf{Th}_{k}(q_{1}))\wedge\mathsf{Th}_{i}(q_{x})]$. $\varphi_{\mathsf{merge}}=\bigwedge\limits_{x,i,k\in\mathsf{In}\wedge k>i}\Box^{\mathsf{ns}}[\psi(i,j,k,q_{x})\rightarrow\mathsf{NextMerge(i,k)}]$.

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Propagating runs($\varphi_{pro},\varphi_{NO-pro}$)- If $\mathsf{Nxt}(\mathsf{Th}_{i}(q_{x}))$ is true at a point, and if for all $j<i$, $\neg\mathsf{Nxt}(\mathsf{Th}_{j}(q_{x}))$ is true, then at the next point, we have $\mathsf{Th}_{i}(q_{x})$. Let $\mathsf{NextTh}(i,j,q_{x})$ denote the formula $\mathsf{Nxt}(\mathsf{Th}_{i}(q_{x}))\wedge\neg\mathsf{Nxt}(\mathsf{Th}_{j}(q_{x}))$. The formula $\varphi_{pro}$ is given by

$\bigwedge\limits_{i,j\in\mathsf{In}\wedge i<j}\Box^{\mathsf{ns}}[\mathsf{NextTh}(i,j,q_{x}){\rightarrow}{\mathsf{O}[\mathsf{Th}_{i}(q_{x}){\wedge}{\neg\mathsf{merge}(i,j)}]]}$. If $\mathsf{Th}_{i}(\bot)$ is true at the current point, then at the next point, either $\mathsf{Th}_{i}(\bot)$ or $\mathsf{Th}_{i}(q_{1})$. The latter condition corresponds to starting a new run on thread $\mathsf{Th}_{i}$. $\varphi_{NO-pro}{=}\bigwedge\limits_{i\in\mathsf{In}}\Box^{\mathsf{ns}}\{\mathsf{Th}_{i}(\bot){\rightarrow}{\mathsf{O}(\mathsf{Th}_{i}(\bot)\vee\mathsf{Th}_{i}(q_{1}))}\}$

Let $\mathsf{Run}$ be the formula obtained by conjuncting all formulae explained above. Once we construct the simple extension $\rho^{\prime}$, checking whether the rational expression $\and$ holds in some interval $I$ in the timed word $\rho$, is equivalent to checking that if $u$ is the first action point within $I$, and if $\mathsf{Th}_{i}(q_{1})$ holds at $u$, then after a series of merges of the form $\mathsf{merge}(i_{1},i)$,$\mathsf{merge}(i_{2},i_{1})$, $\ldots\mathsf{merge}(j,i_{n})$, at the last point $v$ in the interval $I$, $\mathsf{Th}_{j}(q_{f})$ is true, for some final state $q_{f}$. This is encoded as $\mathsf{GOODRUN}(q_{f})$. It can be seen that the number of possible sequences of merges are bounded. Figure 2 illustrates the threads and merging. We can easily write a 1- $\mathsf{TPTL}$ formula that will check the truth of $\mathsf{Rat}_{[l,u)}\and$ at a point $v$ on the simple extension $\rho^{\prime}$ (see Appendix C). However, to write an $\mathsf{MTL}$ formula that checks the truth of $\mathsf{Rat}_{[l,u)}\and$ at a point $v$, we need to oversample $\rho^{\prime}$ as shown below.

Lemma 3.3**.**

Let $T=\Box^{\mathsf{ns}}[w\leftrightarrow\mathsf{Rat}_{I}\and]$ be a temporal definition built from $\Sigma\cup W$. Then we synthesize a formula $\psi\in\mathsf{MTL}$ over $\Sigma\cup W\cup X$ such that $T$ is equivalent to $\psi$ modulo oversampling.

Proof 3.4**.**

Lets first consider the case when the interval $I$ is bounded of the form $[l,u)$. Consider a point in $\rho^{\prime}$ with time stamp $\tau_{v}$. To assert $w$ at $\tau_{v}$, we look at the first action point after time point $\tau_{v}+l$, and check that $\mathsf{GOODRUN}(last(q_{f}))$ holds, where $last(q_{f})$ identifies the last action point just before $\tau_{v}+u$. The first difficulty is the possible absence of time points $\tau_{v}+l$ and $\tau_{v}+u$. To overcome this difficulty, we oversample $\rho^{\prime}$ by introducing points at times $t+l,t+u$, whenever $t$ is a time point in $\rho^{\prime}$. These new points are labelled with a new proposition $\mathsf{ovs}$. Sadly, $last(q_{f})$ cannot be written in $\mathsf{MTL}$.

To address this, we introduce new time points at every integer point of $\rho^{\prime}$. The starting point 0 is labelled $c_{0}$. Consecutive integer time points are marked $c_{i},c_{i\oplus 1}$, where $\oplus$ is addition modulo the maximum constant used in the time interval in the $\mathsf{RatMTL}$ formula. This helps in measuring the time elapse since the first action point after $\tau_{v}+l$, till the last action point before $\tau_{v}+u$ as follows: if $\tau_{v}+l$ lies between points marked $c_{j},c_{j\oplus 1}$, then the last integer point before $\tau_{v}+u$ is uniquely marked $c_{j\oplus u}$.

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Anchoring at $\tau_{v}$, we assert the following at distance $l$: no action points are seen until the first action point where $\mathsf{Th}_{i}(q_{1})$ is true for some thread $\mathsf{Th}_{i}$. Consider the next point where $c_{j\oplus u}$ is seen. Let $\mathsf{Th}_{i_{k_{1}}}$ be the thread to which $\mathsf{Th}_{i}$ has merged at the last action point just before $c_{j\oplus u}$. Let us call $\mathsf{Th}_{i_{k_{1}}}$ the “last merged thread” before $c_{j\oplus u}$. The sequence of merges from $\mathsf{Th}_{i}$ till $\mathsf{Th}_{i_{k_{1}}}$ asserts a prefix of the run that we are looking for between $\tau_{v}+l$ and $\tau_{v}+u$. To complete the run we mention the sequence of merges from $\mathsf{Th}_{i_{k_{1}}}$ which culminates in some $\mathsf{Th}_{i_{k}}(q_{f})$ at the last action point before $\tau_{v}+u$.

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Anchoring at $\tau_{v}$, we assert the following at distance $u$: we see no action points since $\mathsf{Th}_{i_{k}}(q_{f})$ at the action point before $\tau_{v}+u$ for some thread $\mathsf{Th}_{i_{k}}$, and there is a path linking thread $\mathsf{Th}_{i_{k_{1}}}$ to $\mathsf{Th}_{i_{k}}$ since the point $c_{j\oplus u}$. We assert that the “last merged thread”, $\mathsf{Th}_{i_{k_{1}}}$ is active at $c_{j\oplus u}$ : this is the linking thread which is last merged into before $c_{j\oplus u}$, and which is the first thread which merges into another thread after $c_{j\oplus u}$.

These two formulae thus “stitch” the actual run observed between points $\tau_{v}+l$ and $\tau_{v}+u$. The formal technical details can be seen in Appendix D. If $I$ was an unbounded interval of the form $[l,\infty)$, then we will go all the way till the end of the word, and assert $\mathsf{Th}_{i_{k}}(q_{f})$ at the last action point of the word. Thus, for unbounded intervals, we do not need any oversampling at integer points.

In a similar manner, we can eliminate the $\mathsf{URat}$ modality, the proof of which can be found in Appendix E. If we choose to work on logic $\mathsf{MITL}+\mathsf{URat}$, we obtain a $\mathsf{2EXPSPACE}$ upper bound for satisfiability checking, since elimination of $\mathsf{URat}$ results in an equisatisfiable $\mathsf{MITL}$ formula. This is an interesting consequence of the oversampling technique; without oversampling, we can eliminate $\mathsf{URat}$ obtaining 1-$\mathsf{TPTL}$ (Appendix C). However, 1-$\mathsf{TPTL}$ does not enjoy the benefits of non-punctuality, and is non-primitive recursive (Appendix F).

4 Automaton-Metric Temporal Logic-Freeze Logic Equivalences

The focus of this section is to obtain equivalences between automata, temporal and freeze logics. First of all, we identify a fragment of $\mathsf{RatMTL}$ denoted $\mathsf{SfrMTL}$, where the rational expressions in the formulae are all star-free. We then show the equivalence between $\mathsf{po}$-1-clock ATA, 1${-}\mathsf{TPTL}$, and $\mathsf{SfrMTL}$ ($\mathsf{po}$-1-clock ATA $\subseteq\mathsf{SfrMTL}\subseteq 1{-}\mathsf{TPTL}\equiv\mathsf{po}$-1-clock ATA). The main result of this section gives a tight automaton-logic connection in Theorem 4.1, and is proved using Lemmas 4.3, 4.4 and 4.5.

Theorem 4.1**.**

1${-}\mathsf{TPTL}$, $\mathsf{SfrMTL}$ and $\mathsf{po}$-1-clock ATA are all equivalent.

We first show that partially ordered 1-clock alternating timed automata ($\mathsf{po}$-1-clock ATA) capture exactly the same class of languages as $1{-}\mathsf{TPTL}$. We also show that $1{-}\mathsf{TPTL}$ is equivalent to the subclass $\mathsf{SfrMTL}$ of $\mathsf{RatMTL}$ where the rational expressions $\mathsf{re}$ involved in the formulae are such that $L(\mathsf{re})$ is star-free.

A 1-clock ATA [15] is a tuple $\mathcal{A}=(\Sigma,S,s_{0},F,\delta)$, where $\Sigma$ is a finite alphabet, $S$ is a finite set of locations, $s_{0}\in S$ is the initial location and $F\subseteq S$ is the set of final locations. Let $x$ denote the clock variable in the 1-clock ATA, and $x\bowtie c$ denote a clock constraint where $c\in\mathbb{N}$ and $\bowtie\in\{<,\leq,>,\geq\}$. Let $X$ denote a finite set of clock constraints of the form $x\bowtie c$. The transition function is defined as $\delta:S\times\Sigma\rightarrow\Phi(S\cup\Sigma\cup X)$ where $\Phi(S\cup\Sigma\cup X)$ is a set of formulae defined by the grammar $\varphi::=\top|\bot|\varphi_{1}\wedge\varphi_{2}|\varphi_{1}\vee\varphi_{2}|s|x\bowtie c|x.\varphi$ where $s\in S$, and $x.\varphi$ is a binding construct corresponding to resetting the clock $x$ to 0.

The notation $\Phi(S\cup\Sigma\cup X)$ thus allows boolean combinations as defined above of locations, symbols of $\Sigma$, clock constraints and $\top,\bot$, with or without the binding construct $(x.)$. A configuration of a 1-clock ATA is a set consisting of locations along with their clock valuation. Given a configuration $C$, we denote by $\delta(C,a)$ the configuration $D$ obtained by applying $\delta(s,a)$ to each location $s$ such that $(s,\nu)\in C$. A run of the 1-clock ATA starts from the initial configuration $\{(s_{0},0)\}$, and proceeds with alternating time elapse transitions and discrete transitions obtained on reading a symbol from $\Sigma$. A configuration is accepting iff it is either empty, or is of the form $\{(s,\nu)\mid s\in F\}$. The language accepted by a 1-clock ATA $\mathcal{A}$, denoted $L(\mathcal{A})$ is the set of all timed words $\rho$ such that starting from $\{(s_{0},0)\}$, reading $\rho$ leads to an accepting configuration. A $\mathsf{po}$-1-clock ATA is one in which (i) there is a partial order denoted $\prec$ on the locations, such that whenever $s_{j}$ appears in $\Phi(s_{i})$, $s_{j}\prec s_{i}$, or $s_{j}=s_{i}$. Let $\downarrow s_{i}=\{s_{j}\mid s_{j}\prec s_{i}\}$, (ii) $x.s$ does not appear in $\delta(s,a)$ for all $s\in S,a\in\Sigma$.

Example 4.2**.**

Consider the $\mathsf{po}$-1-clock ATA $\mathcal{A}=(\{a,b\},\{s_{0},s_{a},s_{\ell}\},s_{0},\{s_{0},s_{\ell}\},\delta)$ with transitions $\delta(s_{0},b)=s_{0},\delta(s_{0},a)=(s_{0}\wedge x.s_{a})\vee s_{\ell},$ $\delta(s_{a},a)=(s_{a}\wedge x<1)\vee(x>1)=\delta(s_{a},b),$ and $\delta(s_{\ell},b)=s_{\ell},\delta(s_{\ell},a)=\bot$. The automaton accepts all strings where every non-last $a$ has no symbols at distance 1 from it, and has some symbol at distance $>1$ from it.

Lemma 4.3**.**

$\mathsf{po}$-1-clock ATA and $1{-}\mathsf{TPTL}$ are equivalent in expressive power.

The translation from $1{-}\mathsf{TPTL}$ to $\mathsf{po}$-1-clock ATA is easy, as in the translation from $\mathsf{MTL}$ to $\mathsf{po}$-1-clock ATA. For the reverse direction, we start from the lowest location (say $s$) in the partial order, and replace the transitions of $s$ by a 1-$\mathsf{TPTL}$ formula that models timed words which are accepted, when started in $s$. The accepting behaviours of each location $s$, denoted $\mathsf{Beh}(s)$ is computed bottom up. The 1-$\mathsf{TPTL}$ formula that we are looking for is $\mathsf{Beh}(s_{0})$ where $s_{0}$ is the initial location. In example 4.2, $\mathsf{Beh}(s_{\ell})=\Box^{\mathsf{ns}}b$, $\mathsf{Beh}(s_{a}){=}(x<1)\>\mathsf{U}^{\mathsf{ns}}(x>1)$, $\mathsf{Beh}(s_{0})=[(a\wedge x.\mathsf{O}\mathsf{Beh}(s_{a}))\vee b]\>\mathsf{W}(a\wedge\mathsf{O}\mathsf{Beh}(s_{\ell}))$ =$((a\wedge(x.\mathsf{O}[(x<1)\>\mathsf{U}^{\mathsf{ns}}x>1]))\vee b)\>\mathsf{W}(a\wedge\mathsf{O}\Box^{\mathsf{ns}}b).$ Step by step details for Lemma 4.3 can be seen in Appendix H.

We next show that starting from a $\mathsf{SfrMTL}$ formula $\varphi$, we can construct an equivalent $1{-}\mathsf{TPTL}$ formula $\psi$. The proof of Lemma 4.4 can be found in Appendix I.

Lemma 4.4**.**

$\mathsf{SfrMTL}\subseteq 1-\mathsf{TPTL}$**

The idea is to iteratively keep replacing the $\mathsf{Rat}$ modality level by level, starting with the innermost one, until we have eliminated the topmost one. Consider the $\mathsf{SfrMTL}$ formula $\varphi=\mathsf{Rat}_{(0,1)}[\mathsf{Rat}_{(1,2)}(a+b)^{*}]$. To eliminate $\mathsf{Rat}_{(1,2)}(a+b)^{*}$ at a point, we freeze a clock $x$, and wait till $x\in(1,2)$, and check $(a+b)^{*}$ on this region. The LTL formula for $(a+b)^{*}$ is $\Box(a\vee b)$. Treating $x\in(1,2)$ as a proposition, we obtain $\zeta=x.(x\notin(1,2)\>\mathsf{U}[\psi_{1}\wedge\psi_{2}])$ where $\psi_{1}=x\in(1,2)\wedge[x\in(1,2)\>\mathsf{U}\mathsf{O}(\Box(x\notin(1,2)))]$ and $\psi_{2}=\Box[x\in(1,2)\rightarrow(a\vee b)]$. $\zeta$ asserts $\Box(a\vee b)$ exactly on the region (1,2), eliminating the modality $\mathsf{Rat}_{(1,2)}$. To eliminate the outer $\mathsf{Rat}_{(0,1)}$, we assert the existence of a point in (0,1) where $\mathsf{Rat}_{(1,2)}(a+b)^{*}$ is true by saying $x.(x\notin(0,1)\>\mathsf{U}([x\in(0,1)\wedge\zeta\wedge\mathsf{O}(\Box(x\notin(0,1)))]))$. This is 1-$\mathsf{TPTL}$ equivalent to $\varphi$.

Lemma 4.5**.**

*($\mathsf{po}$-1-clock ATA to $\mathsf{SfrMTL}$)

Given a $\mathsf{po}$-1-clock ATA $\mathcal{A}$, we can construct a $\mathsf{SfrMTL}$ formula $\varphi$ such that $L(\mathcal{A})=L(\varphi)$.*

Proof 4.6**.**

(Sketch) We give a proof sketch here, a detailed proof can be found in Appendix J. Let $\mathcal{A}$ be a $\mathsf{po}$-1-clock ATA with locations $S=\{s_{0},s_{1},\dots,s_{n}\}$. Let $K$ be the maximal constant used in the guards $x\sim c$ occurring in the transitions. Let $R_{2i}=[i,i],R_{2i+1}=(i,i+1),0\leq i<K$ and $R^{+}_{K}=(K,\infty)$ be the regions $\mathcal{R}$ of $x$. Let $R_{h}\prec R_{k}$ denote that region $R_{h}$ precedes region $R_{k}$. For each location $s$, $\mathsf{Beh}(s)$ as computed in Lemma 4.3 is a 1-$\mathsf{TPTL}$ formula that gives the timed behaviour starting at $s$, using constraints $x\sim c$ since the point where $x$ was frozen. In example 4.2, $\mathsf{Beh}(s_{a}){=}(x<1)\>\mathsf{U}^{\mathsf{ns}}(x>1)$, allows symbols $a,b$ as long as $x<1$ keeping the control in $s_{a}$, has no behaviour at $x=1$, and allows control to leave $s_{a}$ when $x>1$. For any $s$, we “distribute” $\mathsf{Beh}(s)$ across regions by untiming it. In example 4.2, $\mathsf{Beh}(s_{a})$ is $\Box^{\mathsf{ns}}(a\vee b)$ for regions $R_{0},R_{1}$, it is $\bot$ for $R_{2}$ and is $(a\vee b)$ for $R^{+}_{1}$. Given any $\mathsf{Beh}(s)$, and a pair of regions $R_{j}\preceq R_{k}$, such that $s$ has a non-empty behaviour in region $R_{j}$, and control leaves $s$ in $R_{k}$, the untimed behaviour of $s$ between regions $R_{j},\dots,R_{k}$ is written as LTL formulae $\varphi_{j},\dots,\varphi_{k}$. This results in a “behaviour description” (or $\mathsf{BD}$ for short) denoted $\mathsf{BD}(s,R_{j},R_{k})$ : this is a $2K+1$ tuple with $\mathsf{BD}[R_{l}]=\varphi_{l}$ for $j\leq l\leq k$, and $\mathsf{BD}[R]=\top$ denoting “dont care” for the other regions. Let $\mathsf{BDSet}(s)$ denote the set of all $\mathsf{BD}$s for a location $s$. For the initial location $s_{0}$, consider all $\mathsf{BD}(s_{0},R_{j},R_{k})\in\mathsf{BDSet}(s_{0})$ that have a behaviour starting in $R_{j}$, and ends in an accepting configuration in $R_{k}$. Each LTL formula $\mathsf{BD}(s_{0},R_{j},R_{k})[R_{i}]$ (or $\mathsf{BD}[R_{i}]$ when $s,R_{j},R_{k}$ are clear) is replaced with a star-free rational expression denoted $\mathsf{re}(\mathsf{BD}(s_{0},R_{j},R_{k})[R_{i}])$. Then $\mathsf{BD}(s_{0},R_{j},R_{k})$ is transformed into a $\mathsf{SfrMTL}$ formula $\varphi(s_{0},R_{j},R_{k})=\bigwedge_{j\leq g\leq k}\mathsf{Rat}_{R_{g}}\mathsf{re}(\mathsf{BD}(s_{0},R_{j},R_{k})[R_{g}])$. The language accepted by the $\mathsf{po}$-1-clock ATA $\mathcal{A}$ is then given by $\bigvee_{0\leq j\leq k\leq 2K}\varphi(s_{0},R_{j},R_{k})$.

Computing $\mathsf{BD}(s,R_{i},R_{j})$ for a location $s$ and pair of regions $R_{i}\preceq R_{j}$. We first compute $\mathsf{BD}(s,R_{i},R_{j})$ for locations $s$ which are lowest in the partial order, followed by computing $\mathsf{BD}(s^{\prime},R_{i},R_{j})$ for locations $s^{\prime}$ which are higher in the order. For any location $s$, $\mathsf{Beh}(s)$ has the form $\varphi$ or $\varphi_{1}\>\mathsf{W}\varphi_{2}$ or $\varphi_{1}\>\mathsf{U}^{\mathsf{ns}}\varphi_{2}$, where $\varphi,\varphi_{1},\varphi_{2}$ are disjunctions of conjunctions over $\Phi(S\cup\Sigma\cup X)$, where $S$ is the set of locations with or without the binding construct $x.$, and $X$ is a set of clock constraints of the form $x\sim c$. Each conjunct has the form $\psi\wedge x\in R$ where $\psi\in\Phi(\Sigma\cup S)$ and $R\in\mathcal{R}$. Let $\varphi_{1}=\bigvee(P_{i}\wedge C_{i}),\varphi_{2}=\bigvee(Q_{j}\wedge E_{j})$ where $P_{i},Q_{j}\in\Phi(\Sigma\cup S)$ and $C_{i},E_{j}\in\mathcal{R}$. Let $\mathcal{C}$ and $\mathcal{E}$ be shorthands for any $C_{k},E_{l}$.

If $\mathsf{Beh}(s)$ is an expression without $\>\mathsf{U},\mathsf{W}$ (the case of $\varphi$ above), then $\mathsf{BD}(s,R_{i},R_{i})$ is defined for a region $R_{i}$ if $\varphi=\bigvee(Q_{j}\wedge E_{j})$ and there is some $E_{l}$ with $x\in R_{i}$. It is a $2K+1$ tuple with $\mathsf{BD}(s,R_{i},R_{i})[R_{i}]=Q_{l}$, and the rest of the entries are $\top$ (for dont care). If $\mathsf{Beh}(s)$ has the form $\varphi_{1}\>\mathsf{W}\varphi_{2}$ or $\varphi_{1}\>\mathsf{U}^{\mathsf{ns}}\varphi_{2}$, then for $R_{i}\preceq R_{j}$, and a location $s$, $\mathsf{BD}(s,R_{i},R_{j})$ is a $2K+1$ tuple with (i) formula $\top$ in regions $R_{0},\dots,R_{i-1},R_{j+1},\dots,R_{K}^{+}$, (ii) If $C_{k}=E_{l}=(x\in R_{j})$ for some $C_{k},E_{l}$, then the LTL formula in region $R_{j}$ is $P_{k}\>\mathsf{U}Q_{l}$ if $s$ is not accepting, and is $P_{k}\>\mathsf{W}Q_{l}$ if $s$ is accepting, (iii) If no $C_{k}$ is equal to any $E_{l}$, and if $E_{l}=(x\in R_{j})$ for some $l$, then the formula in region $R_{j}$ is $Q_{l}$. If $C_{m}=(x\in R_{i})$ for some $m$, then the formula for region $R_{i}$ is $\Box^{\mathsf{ns}}P_{m}$. If there is some $C_{h}=(x\in R_{w})$ for $i<w<j$, then the formula in region $R_{w}$ is $\Box^{\mathsf{ns}}P_{h}\vee\epsilon$, where $\epsilon$ signifies that there may be no points in regions $R_{w}$. If there are no $C_{m}$’s such that $C_{m}=(x\in R_{w})$ for $R_{i}\prec R_{w}\prec R_{j}$, then the formula in region $R_{w}$ is $\epsilon$. $\epsilon$ is used as a special symbol in LTL whenever there is no behaviour in a region.

$\mathsf{BD}(s,R_{i},R_{j})$** for location $s$ lowest in po***. Let $s$ be a location that is lowest in the partial order. In general, if $s$ is the lowest in the partial order, then $\mathsf{Beh}(s)$ has the form $\varphi_{1}\>\mathsf{W}\varphi_{2}$ or $\varphi_{1}\>\mathsf{U}^{\mathsf{ns}}\varphi_{2}$ or $\varphi$ where $\varphi,\varphi_{1},\varphi_{2}$ are disjunctions of conjunctions over $\Phi(\Sigma\cup X)$. Each conjunct has the form $\psi\wedge x\in R$ where $\psi\in\Phi(\Sigma)$ and $R\in\mathcal{R}$. See Figure 4, with regions $R_{0},R_{1},R_{2},R^{+}_{1}$, and some example *$\mathsf{BD}$*s. In Figure 4, using the *$\mathsf{BD}$s of the lowest location $s_{3}$, we write the $\mathsf{SfrMTL}$ formula for $\mathsf{Beh}(s_{3})$ : $\psi(s_{3})=\varphi_{R_{0}}(s_{3})\wedge\varphi_{R_{1}}(s_{3})\wedge\varphi_{R_{2}}(s_{3})\wedge\varphi_{R^{+}_{1}}(s_{3})$, where each $\varphi_{R}$ describes the behaviour of $s_{3}$ starting from region $R$. For a fixed region $R_{i}$, $\varphi_{R_{i}}(s_{3})$ is $\bigwedge_{R_{g}\prec R_{i}}\mathsf{Rat}_{R_{g}}\epsilon\wedge\mathsf{Rat}_{R_{i}}\Sigma^{+}\rightarrow\{\bigvee_{R_{i}\prec R_{j}}\varphi(s_{3},R_{i},R_{j})\}$, where $\varphi(s_{3},R_{i},R_{j})$ is described above. $\mathsf{Rat}_{R_{g}}\epsilon$ means that there is no behaviour in $R_{g}$. $\varphi_{R_{0}}(s_{3})$ is given by $\mathsf{Rat}_{R_{0}}\Sigma^{+}\rightarrow\{(\mathsf{Rat}_{R_{0}}a^{*}\wedge\mathsf{Rat}_{R_{1}}[a^{*}+\epsilon]\wedge\mathsf{Rat}_{R_{2}}[a^{*}+\epsilon]\wedge\mathsf{Rat}_{R^{+}_{1}}[a^{*}+a^{*}b])\}$.

$\mathsf{BD}(s,R_{i},R_{j})$** for a location $s$ which is higher up***. If $s$ is not the lowest in the partial order, then $\mathsf{Beh}(s)$ can have locations $s^{\prime}\in\downarrow s$. $s^{\prime}$ occurs as $\mathsf{O}(s^{\prime})$ or $x.\mathsf{O}(s^{\prime})$ in $\mathsf{Beh}(s)$. For $x.\mathsf{O}\mathsf{Beh}(s_{3})$ in $\mathsf{BD}(s,R_{i},R_{j})$, since the clock is frozen, we plug-in the $\mathsf{SfrMTL}$ formula $\psi(s_{3})$ computed above for $x.\mathsf{O}\mathsf{Beh}(s_{3})$ in $\mathsf{BD}(s_{1},R_{i},R_{j})$. For instance, in figure 4, $x.\mathsf{O}\mathsf{Beh}(s_{3})$ appears in $\mathsf{BD}(s_{2},R_{2},R_{2})[R_{2}]$. We simply plug in the $\mathsf{SfrMTL}$ formula $\psi(s_{3})$ in its place. Likewise, for locations $s,t$, if $\mathsf{O}\mathsf{Beh}(t)$ occurs in $\mathsf{BD}(s,R_{i},R_{j})[R_{k}]$, we look up $\mathsf{BD}(t,R_{k},R_{l})\in\mathsf{BDSet}(t)$ for all $R_{k}\preceq R_{l}$ and combine $\mathsf{BD}(s,R_{i},R_{j}),\mathsf{BD}(t,R_{k},R_{l})$ in a manner described below. This is done to detect if the “next point” for $t$ has a behaviour in $R_{k}$ or later.*

• (a)

*If the next point for $t$ is in $R_{k}$ itself, then we combine $\mathsf{BD}_{1}=\mathsf{BD}(s,R_{i},R_{j})$ with $\mathsf{BD}_{2}\in\{\mathsf{BD}(t,R_{k},R_{l})\mid R_{k}\preceq R_{l}\}\subseteq\mathsf{BDSet}(t)$ as follows. $\mathsf{combine}(\mathsf{BD}_{1},\mathsf{BD}_{2})$ results in $\mathsf{BD}_{3}$ such that $\mathsf{BD}_{3}[R]=\mathsf{BD}_{1}[R]$ for $R\prec R_{k}$, $\mathsf{BD}_{3}[R]=\mathsf{BD}_{1}[R]\wedge\mathsf{BD}_{2}[R]$ for $R_{k}\prec R$, where $\wedge$ denotes component wise conjunction. $\mathsf{BD}_{3}[R_{k}]$ is obtained by replacing $\mathsf{O}\mathsf{Beh}(s_{2})$ in $\mathsf{BD}_{1}[R_{k}]$ with $\mathsf{BD}_{2}[R_{k}]$. Doing so enables the next point in $R_{k}$, emulating the behaviour of $t$ in $R_{k}$. *

• (b)

Assume the next point for $t$ lies in $R_{b}$, $R_{k}\prec R_{b}$. The difference with case (a) is that we combine $\mathsf{BD}_{1}=\mathsf{BD}(s,R_{i},R_{j})$ with $\mathsf{BD}_{2}{\in}\{\mathsf{BD}(t,R_{b},R_{l})\mid R_{k}\prec R_{b}\preceq R_{l}\}\subseteq\mathsf{BDSet}(t)$. Then $\mathsf{combine}(\mathsf{BD}_{1},\mathsf{BD}_{2})$ results in a $\mathsf{BD}$, say $\mathsf{BD}_{3}$ such that $\mathsf{BD}_{3}[R]=\mathsf{BD}_{1}[R]$ for $R\prec R_{k}$, $\mathsf{BD}_{3}[R]=\mathsf{BD}_{1}[R]\wedge\mathsf{BD}_{2}[R]$ for all $R_{b}\preceq R$, and $\mathsf{BD}_{3}[R]=\epsilon$ for $R_{k}\prec R\prec R_{b}$. The $\mathsf{O}\mathsf{Beh}(t)$ in $\mathsf{BD}_{1}[R_{k}]$ is replaced with $\Box\bot$ to signify that the next point is not enabled for $t$. See Figure 5 where $R_{b}=R_{2}$. The conjunction with $\Box\bot$ in $R_{0}$ signifies that the next point for $s_{2}$ is not in $R_{0}$; the $\epsilon$ in $R_{1}$ signifies that there are no points in $R_{1}$ for $s_{2}$. Conjuncting $\Box\bot$ in a region signifies that the next point does not lie in this region.

*We look at the “accepting” *$\mathsf{BD}$*s in $\mathsf{BDSet}(s_{0})$, viz., all $\mathsf{BD}(s_{0},R_{j},R_{k})$, such that acceptance happens in $R_{k}$, and $s_{0}$ has a behaviour starting in $R_{j}$. The LTL formulae $\mathsf{BD}(s_{0},R_{j},R_{k})[R]$ are replaced with star-free expressions $\mathsf{re}(\mathsf{BD}(s_{0},R_{j},R_{k})[R])$. Each accepting $\mathsf{BD}(s_{0},R_{j},R_{k})$ gives an $\mathsf{SfrMTL}$ formula $\bigwedge_{R_{j}\preceq R\preceq R_{k}}\mathsf{Rat}_{R}\mathsf{re}(\mathsf{BD}(s_{0},R_{j},R_{k})[R])$. The disjunction of these across all accepting *$\mathsf{BD}$s is the $\mathsf{SfrMTL}$ formula equivalent to $L(\mathcal{A})$.

5 Discussion

We propose $\mathsf{RatMTL}$ which significantly increases the expressive power of $\mathsf{MTL}$ and yet retains decidability over pointwise finite words. The $\mathsf{Rat}$ operator added to $\mathsf{MTL}$ syntactically subsumes several other modalities in literature including threshold counting, modulo counting and the pnueli modality. Decidability of $\mathsf{RatMTL}$ is proved by giving an equisatisfiable reduction to $\mathsf{MTL}$ using oversampled temporal projections. This reduction has elementary complexity and allows us to identify two fragments of $\mathsf{RatMTL}$ with $\mathsf{2EXPSPACE}$ and $\mathsf{EXPSPACE}$ satisfibility. In previous work [10], oversampled temporal projections were used to reduce $\mathsf{MTL}$ with punctual future and non-punctual past to $\mathsf{MTL}$. Our reduction can be combined with the one in [10] to obtain decidability of $\mathsf{RatMTL}$ and elementary decidability of $\mathsf{MITL}+\mathsf{URat}$ + non-punctual past. These are amongst the most expressive decidable extensions of $\mathsf{MTL}$ known so far. We also show an exact logic-automaton correspondence between the fragment $\mathsf{SfrMTL}$ and $\mathsf{po}$-a-clock ATA. Ouaknine and Worrell reduced $\mathsf{MTL}$ to $\mathsf{po}$-1 clock ATA. Our $\mathsf{SfrMTL}$ achieves the converse too. It is not difficult to see that full $\mathsf{RatMTL}$ can be reduced to equivalent 1 clock alternating timed automata. This provides an alternative proof of decidability of $\mathsf{RatMTL}$ but the proof will not extend to decidability of $\mathsf{RatMTL}+$ non-punctual past, nor prove elementary decidability of $\mathsf{MITL}+\mathsf{URat}+$non-punctual past. Hence, we believe that our proof technique has some advantages. An interesting related formalism of timed regular expressions was defined by Asarin, Maler, Caspi, and shown to be expressively equivalent to timed automata. Our $\mathsf{RatMTL}$ has orthogonal expressivity, and it is boolean closed. The exact expressive power of $\mathsf{RatMTL}$ which is between 1-clock ATA and $\mathsf{po}$-1-clock ATA is open.

Appendix A Rational Expressions and Star-Free Expressions

We briefly introduce rational expressions and star-free expressions over an alphabet $\Sigma$. A rational expression over $\Sigma$ is constructed inductively using the atomic expressions $a\in\Sigma,\epsilon,\emptyset$ and combining them using concatenation, Kleene-star and union.

A star-free expression also has the same atomic expressions, and allows combination using union, concatenation and complementation. For instance, $\Sigma^{*}$ is star-free since it can be written as $\neg\emptyset$.

Appendix B Exclusive Normal Form

We eliminate $\mathsf{Rat}_{I}\and$ and $x\mathsf{URat}_{I^{\prime},\and}y$ respectively from temporal definitions $\Box^{\mathsf{ns}}[w\leftrightarrow\mathsf{Rat}_{I}\and]$ or $\Box^{\mathsf{ns}}[w\leftrightarrow x\mathsf{URat}_{I^{\prime},\and}y]$. The idea is to first mark each point of the timed word $\rho$ over $\Sigma\cup W$ with the information whether $\and$ is true or not at that point, obtaining a simple extension $\rho^{\prime}$ of $\rho$, and then to refine this information by checking if $\and$ is true within an interval $I$.

Assume $\and=\mathsf{re}(\mathsf{S})$. To say that $\mathsf{re}(\mathsf{S})$ is true starting at a point in the timed word, we have to look at the truth of subformulae in $\mathsf{S}$. The alphabet of the minimal DFA to check $\mathsf{re}(\mathsf{S})$ is hence $2^{\mathsf{S}}=\mathsf{S}^{\prime}$. This results in the minimal DFA accepting an expression $\mathsf{re}^{\prime}(\mathsf{S}^{\prime})$, and not $\mathsf{re}(\mathsf{S})$. In the following, we show that $\mathsf{re}^{\prime}(\mathsf{S}^{\prime})$ is equivalent to $\mathsf{re}(\mathsf{S})$.

The first thing we do to avoid dealing with sets of formulae of $\mathsf{S}$ being true at each point is to assume that the sets $\mathsf{S}$ are exclusive: that is, at any point, exactly one formula from $\mathsf{S}$ can be true. If the sets $\mathsf{S}$ are all exclusive, then the formula is said to be in Exclusive Normal Form. If $\mathsf{S}$ is exclusive, then we will be marking positions in the word over $\mathsf{S}$ and not $\mathcal{P}(\mathsf{S})$. This way, the untimed words $\mathsf{Seg}(\mathsf{S},i,j)$ as well as $\mathsf{TSeg}(\mathsf{S},i,I)$ that were used in the semantics of $\varphi_{1}\mathsf{URat}_{I,\mathsf{re}(\mathsf{S})}\varphi_{2}$, $\mathsf{Rat}_{I,\mathsf{re}(\mathsf{S})}$ respectively will be words over $\mathsf{S}$. The satisfaction of $\varphi_{1}\mathsf{URat}_{I,\mathsf{re}(\mathsf{S})}\varphi_{2}$, $\mathsf{Rat}_{I,\mathsf{re}(\mathsf{S})}$ at any point $i$ will then amount to simply checking if $\mathsf{Seg}(\mathsf{S},i,j),\mathsf{TSeg}(\mathsf{S},i,I)\in L(\mathsf{re}(\mathsf{S}))$.

We now show that the exclusiveness of $\mathsf{S}$ can be achieved by a simple translation.

Lemma B.1**.**

Given any $\mathsf{RatMTL}$ formula $\varphi$ of the form $\mathsf{Rat}^{\mathsf{S}}_{I,\mathsf{re}(\mathsf{S})}$ or $\varphi_{1}\mathsf{URat}^{\mathsf{S}}_{I,\mathsf{re}(\mathsf{S})}\varphi_{2}$, there exists an equivalent formula $\psi\in\mathsf{RatMTL}$ in exclusive normal form.

Proof B.2**.**

Let $\mathsf{S}=\{\phi_{1},\dots,\phi_{n}\}$. Construct set $\mathsf{S}^{\prime}$ consisting of all formulae of the form $\bigwedge_{i\in K}\phi_{i}\wedge\bigwedge_{i\notin K}\neg\phi_{i}$ for all possible subsets $K\subseteq\{1,2,\dots,n\}$.

Consider any formula of the form $\mathsf{Rat}_{I}(\mathsf{re}(S))$. Let $W_{i}$ denote the set consisting of all subsets of $\{1,2,\ldots,K\}$ which contains $i$. The satisfaction of $\varphi_{i}$ is then equivalent to that of $\sum\limits_{W\in W_{i}}\phi_{W}$. We can thus replace any $\varphi_{i}$ occurring in $\mathsf{re}(\mathsf{S})$ with $\sum\limits_{W\in W_{i}}\phi_{W}$. This results in obtaining a rational expression $\mathsf{re}^{\prime}$ over $\mathsf{S}^{\prime}$.

It can be shown that $\mathsf{re}(\mathsf{S})$ is equivalent to $\mathsf{re}^{\prime}(\mathsf{S}^{\prime})$ by inducting on the structure of $\mathsf{re}$.

Thus, the minimal DFA we construct for $\mathsf{re}(\mathsf{S})$ in the temporal definition will end up accepting $\mathsf{re}^{\prime}(\mathsf{S}^{\prime})$, equivalent to $\mathsf{re}(\mathsf{S})$.

Appendix C 1-$\mathsf{TPTL}$ for $\mathsf{Rat}_{[l,u)}\and$

We encode in 1-$\mathsf{TPTL}$ an accepting run going through a sequence of merges capturing $\mathsf{Rat}_{[l,u)}\and$ at a point $e$. To encode an accepting run going through a sequence of merges capturing $\mathsf{Rat}_{[l,u)}\and$ at a point $e$, we assert $\varphi_{chk1}\vee\varphi_{chk2}$ at $e$, assuming $l\neq 0$. If $l=0$, we assert $\varphi_{chk3}$. Recall that $m$ is the number of states in the minimal DFA accepting $\and$.

• •

Let $\mathsf{cond1}=0\leq n<m$, and

• •

Let $\mathsf{cond2}=1\leq i_{1}<i_{2}<\ldots<i_{n}<i\leq m$.

• •

$\varphi_{chk1}=\bigvee\limits_{\mathsf{cond1}}\bigvee\limits_{\mathsf{cond2}}x.\Diamond(x<l\wedge\mathsf{O}[(x\geq l)\wedge\mathsf{GoodRun}])$

• •

$\varphi_{chk2}=\bigvee\limits_{\mathsf{cond1}}\bigvee\limits_{\mathsf{cond2}}x.(\mathsf{O}[(x\geq l)\wedge\mathsf{GoodRun}])$

• •

$\varphi_{chk3}=\bigvee\limits_{\mathsf{cond1}}\bigvee\limits_{\mathsf{cond2}}x.\mathsf{GoodRun}$

where $\mathsf{GoodRun}$ is the formula which describes the run starting in $q_{1}$ in thread $\mathsf{Th}_{i}$, going through a sequence of merges, and witnesses $q_{f}$ in a merged thread $\mathsf{Th}_{i_{1}}$ at a point when $x\in[l,u)$, and is the maximal point in $[l,u)$.

$\mathsf{GoodRun}$ is given by $\mathsf{Th}_{i}(q_{1})\wedge[\{\neg\mathsf{Mrg}(i)\}\>\mathsf{U}[\mathsf{merge}(i_{n},i)\wedge\{\neg\mathsf{Mrg}(i_{n})\}\>\mathsf{U}[\mathsf{merge}(i_{n-1},i_{n})\ldots\\ \{\neg\mathsf{Mrg}(i_{2})\}\>\mathsf{U}[\mathsf{merge}(i_{1},i_{2})\wedge\bigvee\limits_{q\in Q_{F}}\mathsf{Nxt}(\mathsf{Th}_{i_{1}}(q))\wedge x\in[l,u)\wedge\mathsf{O}(x>u)]\ldots]]]]$ where $\mathsf{Mrg}(i)$ is the formula $\bigvee\limits_{j<i}\mathsf{merge}(j,i)$.

The idea is to freeze the clock at the current point $e$, and start checking a good run from the first point in the interval $[l,u)$. $\varphi_{chk1}$ is the case when the next point after point $e$ is not at distance $[l,u)$ from $e$, while $\varphi_{chk2}$ handles the case when the next point after $e$ is at distance $[l,u)$ from $e$. In both cases, $l>0$. (If $l=0$, we assert $\varphi_{chk3}$). Let $\mathsf{Th}_{i}$ be the thread having the initial state $q_{1}$ in the start of the interval $I$. Let $i_{1}$ be the index of the thread to which $\mathsf{Th}_{i}$ eventually merged (at the last point in the interval $[l,u)$ from $e$). The next expected state of thread $\mathsf{Th}_{i_{1}}$ is one of the final states if and only if the sub-string within the interval $[l,u)$ from the point $e$ satisfies the regular expression $\and$. Note that when the frozen clock is $\geq l$, we start the run with $\mathsf{Th}_{i}(q_{1})$, go through the merges, and check that $x\in I$ when we encounter a thread $\mathsf{Th}_{i_{1}}(q_{f})$, with $q_{f}$ being a final state. To ensure that we have covered checking all points in $\tau_{e}+I$, we ensure that at the next point after $\mathsf{Th}_{i_{1}}(q_{f})$, $x>u$. The decidability of $1{-}\mathsf{TPTL}$ gives the decidability of $\mathsf{RatMTL}$.

Appendix D Proof of Lemma 3.3

Proof D.1**.**

Starting with the simple extension $\rho^{\prime}$ having the information about the runs of $\mathcal{A}_{\and}$, we explain the construction of the oversampled extension $\rho^{\prime\prime}$ as follows:

• •

*We first oversample $\rho^{\prime}$ at all the integer timestamps and mark them with propositions in $C=\{c_{0},\ldots,c_{max-1}\}$ where $max$ is the maximum constant used in timing constraints of the input formulae. An integer timestamp $k$ is marked $c_{i}$ if and only if $k=M(max)+i$ where $M(max)$ denotes a non-negative integral multiple of $max$ and $0\leq i\leq max-1$. This can be done easily by the formula *

$c_{0}{\wedge}{\bigwedge\limits_{i\in\{0,\ldots max-1\}}}\Box^{\mathsf{ns}}(c_{i}\rightarrow\neg\Diamond_{(0,1)}(\bigvee C)\wedge\Diamond_{(0,1]}c_{i\oplus 1})$* where $x\oplus y$ is addition of $x,y$ modulo $max$.*

• •

Next, a new point marked $\mathsf{ovs}$ is introduced at all time points $\tau$ whenever $\tau-l$ or $\tau-u$ is marked with $\bigvee\Sigma$. This ensures that for any time point $t$ in $\rho^{\prime\prime}$, the points $t+l,t+u$ are also available in $\rho^{\prime\prime}$.

*After the addition of integer time points, and points marked $\mathsf{ovs}$, we obtain the oversampled extension $(\Sigma\cup W\cup\mathsf{Threads}\cup\mathsf{Merge},C\cup\{\mathsf{ovs}\})$ $\rho^{\prime\prime}$ of $\rho^{\prime}$.

To check the truth of $\mathsf{Rat}_{[l,u)}\and$ at a point $v$, we need to assert the following: starting from the time point $\tau_{v}+l$, we have to check the existence of an accepting run $R$ in $\mathcal{A}_{\and}$ such that the run starts from the first action point in the interval $[\tau_{v}+l,\tau_{v}+u)$, is a valid run which goes through some possible sequence of merging of threads, and witnesses a final state at the last action point in $[\tau_{v}+l,\tau_{v}+u)$. To capture this, we start at the first action point in $[\tau_{v}+l,\tau_{v}+u)$ with initial state $q_{1}$ in some thread $\mathsf{Th}_{i}$, and proceed for some time with $\mathsf{Th}_{i}$ active, until we reach a point where $\mathsf{Th}_{i}$ is merged with some $\mathsf{Th}_{i_{1}}$. This is followed by $\mathsf{Th}_{i_{1}}$ remaining active until we reach a point where $\mathsf{Th}_{i_{1}}$ is merged with some other thread $\mathsf{Th}_{i_{2}}$ and so on, until we reach the last such merge where some thread say $\mathsf{Th}_{n}$ witnesses a final state at the last action point in $[\tau_{v}+l,\tau_{v}+u)$. A nesting of until formulae captures this sequence of merges of the threads, starting with $\mathsf{Th}_{i}$ in the initial state $q_{1}$. Starting at $v$, we have the point marked $\mathsf{ovs}$ at $\tau_{v}+l$, which helps us to anchor there and start asserting the existence of the run.*

The issue is that the nested until can not keep track of the time elapse since $\tau_{v}+l$. However, note that the greatest integer point in $[\tau_{v}+l,\tau_{v}+u)$ is uniquely marked with $c_{j\oplus u}$ whenever $c_{j}\leq\tau_{v}\leq c_{j\oplus 1}$ are the closest integer points to $\tau_{v}$. We make use of this by (i) asserting the run of $\mathcal{A}_{\and}$ until we reach $c_{j\oplus u}$ from $\tau_{v}+l$. Let the part of the run $R$ that has been witnessed until $c_{j\oplus u}$ be $R_{pref}$. Let $R=R_{pref}.R_{suf}$ be the accepting run. (ii) From $\tau_{v}+l$, we jump to $\tau_{v}+u$, and assert the reverse of $R_{suf}$ till we reach $c_{j\oplus u}$. This ensures that $R=R_{pref}.R_{suf}$ is a valid run in the interval $[\tau_{v}+l,\tau_{v}+u)$.

Let $\mathsf{Mrg}(j)=[\bigvee\limits_{k<j}\mathsf{merge}(k,j)\vee c_{j\oplus u}]$.

We first write a formula that captures $R_{pref}$. Given a point $v$, the formula captures a sequence of merges through threads $i>i_{1}>\dots>i_{k_{1}}$, and $m$ is the number of states of $\mathcal{A}_{\and}$.

Let $\varphi_{Pref,k_{1}}=\bigvee_{m\geq i>i_{1}>\dots>i_{k_{1}}}\mathsf{MergeseqPref}(k_{1})$ where $\mathsf{MergeseqPref}(k_{1})$ is the formula

[TABLE]

[TABLE]

Note that this asserts the existence of a run till $c_{i\oplus u}$ going through a sequence of merges starting at $\tau_{v}+l$. Also, $\mathsf{Th}_{i_{k_{1}}}$ is the guessed last active thread till we reach $c_{i\oplus u}$ which will be merged in the continuation of the run from $c_{i\oplus u}$.

Now we start at $\tau_{v}+u$ and assert that we witness a final state sometime as part of some thread $\mathsf{Th}_{i_{k}}$, and walk backwards such that some thread $i_{t}$ got merged to $i_{k}$, and so on, we reach a thread $\mathsf{Th}_{i_{c}}$ to which thread $\mathsf{Th}_{i_{k_{1}}}$ merges with. Note that $\mathsf{Th}_{i_{k_{1}}}$ was active when we reached $c_{i\oplus u}$. This thread $\mathsf{Th}_{i_{k_{1}}}$ is thus the “linking point” of the forward and reverse runs. See Figure 6.

*Let $\varphi_{Suf,k,k_{1}}=\bigvee_{1\leq i_{k}<\dots<i_{k_{1}}\leq m}\mathsf{MergeseqSuf}(k,k_{1})$ where $\mathsf{MergeseqSuf}(k,k_{1})$ is the formula

$\Diamond_{[u,u]}\{\neg(\bigvee\Sigma\vee c_{i\oplus u})\>\mathsf{S}[(\mathsf{Th}_{i_{k}}(q_{f}))\wedge(\neg\mathsf{Mrg}(i_{k})\>\mathsf{S}$* $[\mathsf{merge}(i_{k},i_{k-1})\wedge(\neg\mathsf{Mrg}(i_{k-1})\>\mathsf{S}$

$[\mathsf{merge}(i_{k-1},i_{k-2})\wedge\dots\mathsf{merge}(i_{c},i_{k_{1}})\wedge(\neg\mathsf{Mrg}(i_{k_{1}})\>\mathsf{S}c_{i\oplus u})])])]\}$.*

For a fixed sequence of merges, the formula

[TABLE]

captures an accepting run using the merge sequence. Disjuncting over all possible sequences for a starting thread $\mathsf{Th}_{i}$, and disjuncting over all possible starting threads gives the required formula capturing an accepting run. Note that this resultant formulae is also relativized with respect to $\Sigma$ and also conjuncted with $Rel(\mathsf{Run},\Sigma)$ (where $\mathsf{Run}$ is the formula capturing the run information in $\rho^{\prime}$ as seen in section 3.1) to obtain the equisatisfiable $\mathsf{MTL}$ formula. The relativization of $\mathsf{Run}$ with respect to $\Sigma$ can be done as illustrated in Figure 1.) Note that $\mathsf{S}$ can be eliminated obtaining an equisatisfiable $\mathsf{MTL}[\>\mathsf{U}_{I}]$ formula modulo simple projections [16].

If $I$ was an unbounded interval of the form $[l,\infty)$, then in formula $\varphi_{k,k_{1}}$, we do not require $\mathsf{MergeseqSuf}(k,k_{1})$; instead, we will go all the way till the end of the word, and assert $\mathsf{Th}_{i_{k}}(q_{f})$ at the last action point of the word. Thus, for unbounded intervals, we do not need any oversampling at integer points.

Appendix E Elimination of $\mathsf{URat}_{I,\mathsf{re}}$

Lemma E.1**.**

Let $T=\Box^{\mathsf{ns}}[a\leftrightarrow x\mathsf{URat}_{I,\mathsf{re}}y]$ be a temporal definition built from $\Sigma\cup W$. Then we synthesize a formula $\psi\in\mathsf{MTL}$ over $\Sigma\cup W\cup X$ such that $T$ is equivalent to $\psi$ modulo oversampling.

Proof E.2**.**

We discuss first the case of bounded intervals. The proof technique is very similar to Lemma 3.3. The differences that arise are as below.

Checking $\mathsf{re}$ in $\mathsf{Rat}_{I}\mathsf{re}$ at point $v$ is done at all points $j$ such that $\tau_{j}-\tau_{v}\in I$. To ensure this, we needed the punctual modalities $\Diamond_{[u,u]},\Diamond_{[l,l]}$. On the other hand, to check $\mathsf{URat}_{I,\mathsf{re}}$ from a point $v$, the check on $\mathsf{re}$ is done from the first point after $\tau_{v}$, and ends at some point within $[\tau_{v}+l,\tau_{v}+u)$. Assuming $\tau_{v}$ lies between integer points $c_{i},c_{i\oplus 1}$, we can witness the forward run in $\mathsf{MergeseqPref}$ from the next point after $\tau_{v}$ till $c_{i\oplus 1}$, and for the reverse run, go to some point in $\tau_{v}+I$ where the final state is witnessed in a merged thread, and walk back till $c_{i\oplus 1}$, ensuring the continuity of the threads merged across $c_{i\oplus 1}$. The punctual modalities are hence not required and we do not need points marked $\mathsf{ovs}$. 2. 2.

The formulae $\mathsf{MergeseqPref}(k_{1})$, $\mathsf{MergeseqSuf}(k,k_{1})$ of the lemma 3.3 are replaced as follows:

• •

$\mathsf{MergeseqPref}(k_{1}):\{\neg(\bigvee\Sigma\vee c_{i\oplus 1})\>\mathsf{U}[\mathsf{Th}_{i}(q_{1})\wedge(\neg\mathsf{Mrg}(i)\>\mathsf{U}[\mathsf{merge}(i_{1},i)\wedge(\neg\mathsf{Mrg}(i_{1})\>\mathsf{U}$

$[\mathsf{merge}(i_{2},i_{1})\wedge\dots(\neg\mathsf{Mrg}(i_{k_{1}})\>\mathsf{U}c_{i\oplus 1})])])]\}$.

• •

$\mathsf{MergeseqSuf}(k,k_{1}):\Diamond_{I}\{[(\mathsf{Th}_{i_{k}}(q_{f})){\wedge}(\neg\mathsf{Mrg}(i_{k})\>\mathsf{S}$* *$[\mathsf{merge}(i_{k},i_{k-1})\wedge(\neg\mathsf{Mrg}(i_{k-1})\>\mathsf{S}$

$[\mathsf{merge}(i_{k-1},i_{k-2})\wedge$* $\dots\mathsf{merge}(i_{c},i_{k_{1}})\wedge(\neg\mathsf{Mrg}(i_{k_{1}})\>\mathsf{S}c_{i\oplus 1})])])]\}$*

The above takes care of $\mathsf{re}$ in $x\mathsf{URat}_{I,\mathsf{re}}y$ : we also need to say that $x$ holds continously from the current point to some point in $I$. This is done by pushing $x$ into $\mathsf{re}$ (see the translation of $\varphi_{1}\mathsf{URat}_{I,\mathsf{re}}\varphi_{2}$ to $\mathsf{Rat}_{I}\mathsf{re}^{\prime}$ in Appendix G). The resultant formulae is relativized with respect to $\Sigma$ and also conjuncted with $Rel(\mathsf{Run},\Sigma)$ to obtain the equisatisfiable $\mathsf{MTL}$ formula.

*Now we consider unbounded intervals. The major challenge for the unbounded case is that the point where we asserting $\mathsf{Th}_{i_{k}}(q_{f})$ (call this point $w$) may be far away from the point $v$ where we begin : that is, if $\tau_{v}$ is flanked between integer points marked $c_{i}$ and $c_{i\oplus 1}$, it is possible to see multiple occurrences of $c_{i\oplus 1}$ between $\tau_{v}$ and and the point in $\tau_{v}+I$ which witnesses $\mathsf{Th}_{i_{k}}(q_{f})$. In this case, when walking back reading the reverse of the suffix, it is not easy to stitch it back to the first $c_{i\oplus 1}$ seen after $\tau_{v}$. The possible non-uniqueness of $c_{i\oplus 1}$ thus poses a problem in reusing our technique in the bounded interval case. Thus we consider two cases:

Case 1: In this case, we assume that our point $w$ lies within $[\tau_{v}+l,\lceil\tau_{v}+l\rceil)$. Note that $\lceil\tau_{v}+l\rceil$ is the nearest point from $v$ marked with $c_{i\oplus l\oplus 1}$. This can be checked by asserting $\neg c_{i\oplus l\oplus 1}$ all the way till $c_{i\oplus 1}$ while walking backward from $w$, where $\mathsf{Th}_{i_{k}}(q_{f})$ is witnessed. The formula $\mathsf{MergeseqPref}(k_{1})$ does not change. $\mathsf{MergeseqSuf}(k,k_{1})$ is as follows:

[TABLE]

[TABLE]

*where $\mathsf{Mrg}^{\prime}(i)=[\bigvee\limits_{j<i}\mathsf{merge}(j,i)\vee c_{i\oplus l\oplus 1}]$

Case 2: In this case, we assume the complement. That is the point $w$ occurs after $\lceil\tau_{v}+l\rceil$. In this case, we assert the prefix till $c_{i\oplus l\oplus 1}$ and then continue asserting the suffix from this point in the forward fashion unlike other cases. The changed $\mathsf{MergeseqPref}$ and $\mathsf{MergeseqSuf}$ are as follows:*

• •

$\mathsf{MergeseqPref}(k_{1})$:

[TABLE]

[TABLE]

• •

$\mathsf{MergeseqSuf}(k,k_{1})$:

[TABLE]

[TABLE]

$(\mathsf{Th}_{i_{k}}(q_{f}))])])]\}$* where $\mathsf{Mrg}(i)=[\bigvee\limits_{j<i}\mathsf{merge}(j,i)]$*

E.1 Complexity of $\mathsf{RatMTL}$ Fragments

Given a formula $\varphi$ in ($\mathsf{MITL}$ or $\mathsf{MTL}$ or $\mathsf{RatMTL}$), the size of $\varphi$ denoted $|\varphi|$ is defined by taking into consideration, the number of temporal modalities $\>\mathsf{U}_{I},\mathsf{O}_{I}$, the number of boolean connectives, as well as the maximal constant occurring in the formulae (encoded in binary). The size is defined as $logK\times$(the number of temporal modalities in $\varphi$ + number of boolean connectives in $\varphi$), where $K$ is the max constant appearing in the formulae. For example, $|a\>\mathsf{U}_{(0,2)}(\neg b\wedge c\>\mathsf{U}_{(0,1)}d)|=log2\times 4$. In all our complexity results, we assume a binary encoding of all constants involved in the formulae.

To prove the complexity results we first need the following lemma.

Lemma E.3**.**

Given any $\mathsf{MITL}$ formula $\varphi$ with $|\varphi|=\mathcal{O}(2^{n})$ (for some $n\in\mathbb{N}$) with maximum constant $K$ used in timing intervals, the satisfiability checking for $\varphi$ is $\mathsf{EXPSPACE}$ in $n$ and $log(K)$.

Proof E.4**.**

Given any $\mathsf{MITL}$ formula $\varphi$ with $|\varphi|=\mathcal{O}(2^{n})$, there are at most $\mathsf{expn}=\mathcal{O}(2^{n})$ number of temporal modalities and boolean connectives. Let $K$ be the maximal constant used in $\varphi$. We give a satisfiability preserving reduction from $\varphi$ to $\psi\in\mathsf{MITL[\>\mathsf{U}_{0,\infty},\>\mathsf{S}]}$. $\mathsf{MITL[\>\mathsf{U}_{0,\infty},\>\mathsf{S}]}$ is the fragment of $\mathsf{MITL}$ with untimed past and the intervals in future modalities are only of the form $\langle 0,u\rangle$ or $\langle l,\infty)$. The satisfiability checking for $\mathsf{MITL}[\>\mathsf{U}_{0,\infty},\>\mathsf{S}]$ is in $\mathsf{PSPACE}$ [1]. Hence, the reduction from a $\mathsf{MITL}$ formula with $\mathsf{expn}=\mathcal{O}(2^{n})$ number of modalities to an $\mathsf{MITL}[\>\mathsf{U}_{0,\infty},\>\mathsf{S}]$ formula with $\mathcal{O}(poly(K.\mathsf{expn}))$ modalities preserving the max constant $K$, gives an $\mathsf{EXPSPACE}$ upper bound in $n,logK$. The $\mathsf{EXPSPACE}$ hardness of $\mathsf{MITL}$ can be found in [1]. The reduction from $\mathsf{MITL}$ to $\mathsf{MITL[\>\mathsf{U}_{0,\infty},\>\mathsf{S}]}$ is achieved as follows:

1. (a)

Break each $\>\mathsf{U}_{I}$ formulae in $\mathsf{MITL}$ where $I$ is a bounded interval, into disjunctions of $\>\mathsf{U}_{I_{i}}$ modality, where each $I_{i}$ is a unit length interval and union of all $I_{i}$ is equal to $I$. That is, $\phi_{1}\>\mathsf{U}_{\langle l,u\rangle}\phi_{2}\equiv\phi_{1}\>\mathsf{U}_{\langle l,l+1)}\phi_{2}\vee\phi_{1}\>\mathsf{U}_{[l+1,l+2)}\phi_{2}\ldots\vee\phi_{1}\>\mathsf{U}_{[u-1,u\rangle}\phi_{2}$. This increases the size from $\mathsf{expn}$ to $\mathsf{expn}\times K$. 2. (b)

Next, we flatten all the modalities containing bounded intervals. This results in replacing subformulae of the form $\phi_{1}\>\mathsf{U}_{[l,l+1)}\phi_{2}$ with new witness variables. This results in the conjunction of temporal definitions of the form $\Box^{\mathsf{ns}}[a\leftrightarrow\phi_{1}\>\mathsf{U}_{[l,l+1)}\phi_{2}]$ to the formula, and creates only a linear blow up in the size.

Now consider any temporal definition $\Box^{\mathsf{ns}}[a\leftrightarrow\phi_{1}\>\mathsf{U}_{[l,l+1)}\phi_{2}]$. We show a reduction to an equisatisfiable $\mathsf{MITL[\>\mathsf{U}_{0,\infty},\>\mathsf{S}]}$ formula by eliminating each $\Box^{\mathsf{ns}}[a\leftrightarrow\phi_{1}\>\mathsf{U}_{[l,l+1)}\phi_{2}]$ and replacing it with untimed $\>\mathsf{S}$ and $\>\mathsf{U}$ modalities with intervals $<0,u>$ and $<l,\infty)$.

• •

First we oversample the words at integer points $C=\{c_{0},c_{1},c_{2},\dots,c_{K-1}\}$. An integer timestamp $k$ is marked $c_{i}$ if and only if $k=M(K)+i$, where $M(K)$ denotes a non-negative integer multiple of $K$, and $0\leq i\leq K-1$. This can be done easily by the formula

$c_{0}{\wedge}{\bigwedge\limits_{i\in\{0,\ldots K-1\}}}\Box^{\mathsf{ns}}(c_{i}{\rightarrow}\neg\Diamond_{(0,1)}(\bigvee C)\wedge\Diamond_{(0,1]}c_{i\oplus 1})$**

where $x\oplus y$ is $(x+y)\%K$ (recall that $(x+y)\%K=M(K)+(x+y),0\leq x+y\leq K-1$).

• •

Consider any point $p$ within a unit integer interval whose end points are marked $c_{i-1},c_{i}$. Then $\phi_{1}\>\mathsf{U}_{[l,l+1)}\phi_{2}$ is true at that point $p$ if and only if, $\phi_{1}$ is true on all the action points till a point $j$ in the future, such that

• –

either $j$ occurs within $[l,\infty)$ from $p$ and there is no $c_{i\oplus l}$ between $p$ and $j$ ($\tau_{j}\in[\tau_{p}+l,\lceil\tau_{p}+l\rceil]$* ) (see figure 7)*

$\phi_{C1,p}=(\phi_{1}\wedge\neg c_{i\oplus l})\>\mathsf{U}_{[l,\infty)}\phi_{2}$**

• –

or, $j$ occurs within $[0,l+1)$ from $p$, and $j$ is within a unit interval whose end points are marked $c_{i\oplus l}$ and $c_{i\oplus l\oplus 1}$ ($\tau_{j}\in[\lceil\tau_{p}+l\rceil,\tau_{p}+l+1)$* ) (see figure 8)*

$\phi_{C2,p}=\phi_{1}\>\mathsf{U}_{[0,l+1)}(\phi_{2}\wedge(\neg(\bigvee C))\>\mathsf{S}c_{i\oplus l})$**

*The temporal definition $\Box^{\mathsf{ns}}[a\leftrightarrow\phi_{1}\>\mathsf{U}_{[l,l+1)}\phi_{2}]$ is then captured by *

$\bigvee\limits_{i=1}^{K-1}\Box^{\mathsf{ns}}[\{a\wedge(\neg(\bigvee C)\>\mathsf{U}c_{i})\}\leftrightarrow\phi_{C1,i}\vee\phi_{C2,i}]$**

To eliminate each bounded interval modality as seen in (a),(b) above, we need an $\mathcal{O}(K)$ increase in size. Each temporal definition is replaced with a formula with of size $\mathcal{O}(K)$. Thus the size of the new formula is $\mathcal{O}(2^{n})\times\mathcal{O}(K)\times\mathcal{O}(K)$, and the total number of propositions needed is $2^{\Sigma}\cup\{c_{0},\ldots,c_{K-1}\}$. Assuming binary encoding for $K$, we get a $\mathsf{MITL}[\>\mathsf{U}_{0,\infty},\>\mathsf{S}]$ formulae whose size is exponential in $n$ and $logK$. As the satisfiability checking for $\mathsf{MITL}[\>\mathsf{U}_{0,\infty},\>\mathsf{S}]$ is in $\mathsf{PSPACE}$ [1], we get an $\mathsf{EXPSPACE}$ upper bound in $n,logK$. The $\mathsf{EXPSPACE}$ hardness of $\mathsf{MITL}$ can be found in [1].

E.2 Proof of Theorem 3.1.2 : $\mathsf{MITL}+\mathsf{UM}$ is $\mathsf{EXPSPACE}$-complete

Starting from an $\mathsf{MITL}+\mathsf{UM}$ formula, we first show how to obtain an equivalent $\mathsf{MITL}$ formula modulo simple projections. The constants appearing in a $\mathsf{MITL}+\mathsf{UM}$ formula come from those which are part of the time intervals $I$ decorating the temporal modalities, as well as those from counting constraints $k\%n$. If we consider some $\>\mathsf{U}$ modality, say $\>\mathsf{U}_{(l,u),\#b=k\%n}$, then the number of bits needed to encode this modality is $(logl+logu+logk+logn)=\mathcal{O}(log(u)+log(n))$. Let $n_{max}$ and $u_{max}$ be the maximal constants appearing in the counting constraints as well as time intervals of a $\mathsf{MITL}+\mathsf{UM}$ formula $\phi$. Then $|\phi|=(log(n_{max})+log(u_{max}))\times$(the number of temporal modalities in $\phi$+ number of boolean connectives in $\phi$).

Elimination of $\mathsf{UM}$

In this section, we show how to eliminate $\mathsf{UM}$ from $\mathsf{MTL}+\mathsf{UM}$ over strictly monotonic timed words. This can be extended to weakly monotonic timed words. Given any $\mathsf{MTL}+\mathsf{UM}$ formula $\varphi$ over $\Sigma$, we first “flatten" the $\mathsf{UM}$ modalities of $\varphi$ and obtain a flattened formula.

Example. The formula $\varphi=[a\>\mathsf{U}(e\wedge(f\>\mathsf{U}_{(2,3),\#b=2\%5}y))]$ can be flattened by replacing the $\mathsf{UM}$ with a fresh witness proposition $w$ to obtain

$\varphi_{flat}=[a\>\mathsf{U}(e\wedge w)]{\wedge}\Box^{\mathsf{ns}}\{w\leftrightarrow(f\>\mathsf{U}_{(2,3),\#b=2\%5}y)\}$.

Starting from $\chi\in\mathsf{MTL}+\mathsf{UM}$, in the following, we now show how to obtain equisatisfiable $\mathsf{MTL}$ formulae corresponding to each temporal projection containing a $\mathsf{UM}$ modality.

Flattening : Flatten $\chi$ obtaining $\chi_{flat}$ over $\Sigma\cup W$, where $W$ is the set of witness propositions used, $\Sigma\cap W=\emptyset$. 2. 2.

Eliminate Counting : Consider, one by one, each temporal definition $T_{i}$ of $\chi_{flat}$. Let $\Sigma_{i}=\Sigma\cup W\cup X_{i}$, where $X_{i}$ is a set of fresh propositions, $X_{i}\cap X_{j}=\emptyset$ for $i\neq j$.

• •

For each temporal projection $T_{i}$ containing a $\mathsf{UM}$ modality of the form $x\>\mathsf{U}_{I,\#b=k\%n}y$, Lemma E.5 gives $\zeta_{i}\in\mathsf{MTL}$ over $\Sigma_{i}$ such that $\zeta_{i}$ is equisatisfiable to $T_{i}$ modulo simple extensions. 3. 3.

Putting it all together : The formula $\zeta{=}\bigwedge_{i=1}^{k}\zeta_{i}\in\mathsf{MTL}$ is such that it is equisatisfiable to modulo simple extensions, over the extra propositions $X=\bigcup_{i=1}^{k}X_{i}$.

For elimination of $\mathsf{UM}$, marking witnesses correctly is ensured using an extra set of symbols $B=\{b_{0},...,b_{n}\}$ which act as counters incremented in a circular fashion. Each time a witness of the formula which is being counted is encountered, the counter increments, else it remains same. The evaluation of the mod counting formulae can be reduced to checking the difference between indices between the first and the last symbol in the time region where the counting constraint is checked.

Construction of Simple Extension

Consider a temporal definition $T=\Box^{\mathsf{ns}}[a\leftrightarrow x\mathsf{UM}_{I,\#b=k\%n}y]$, built from $\Sigma\cup W$. Let $\oplus$ denote addition modulo $n+1$.

Construction of a ($\Sigma\cup W,B)$- simple extension. We introduce a fresh set of propositions $B=\{b_{0},b_{1},\ldots,b_{n-1}\}$ and construct a family of simple extensions $\rho^{\prime}=(\sigma^{\prime},\tau)$ from $\rho=(\sigma,\tau)$ as follows:

• •

$C1$: $\sigma^{\prime}_{1}=\sigma_{1}\cup\{b_{0}\}$. If $b_{k}\in\sigma^{\prime}_{i}$ and if $b\in\sigma_{i+1}$, $\sigma^{\prime}_{i+1}=\sigma_{i+1}\cup\{b_{k\oplus 1}\}$.

• •

$C2$: If $b_{k}\in\sigma^{\prime}_{i}$ and $b\notin\sigma_{i+1}$, then $\sigma^{\prime}_{i+1}=\sigma_{i+1}\cup\{b_{k}\}$.

• •

$C3$: $\sigma^{\prime}_{i}$ has exactly one symbol from $B$ for all $1\leq i\leq|dom(\rho)|$. 2. 2.

Formula specifying the above behaviour. The variables in $B$ help in counting the number of $b$’s in $\rho$. $C1,C2$ and $C3$ are written in $\mathsf{MTL}$ as follows:

• •

$\delta_{1}{=}\bigwedge\limits_{k=0}^{n}\Box^{\mathsf{ns}}[(\mathsf{O}b\wedge b_{k})\rightarrow\mathsf{O}b_{k\oplus 1}]$ and

• •

$\delta_{2}{=}\bigwedge\limits_{k=0}^{n}\Box^{\mathsf{ns}}[(\mathsf{O}\neg b\wedge b_{k})\rightarrow\mathsf{O}b_{k}]$

• •

$\delta_{3}{=}\bigwedge\limits_{k=0}^{n}\Box^{\mathsf{ns}}[b_{k}\rightarrow\bigwedge\limits_{j\neq k}\neg b_{j}]$

Lemma E.5**.**

Consider a temporal definition $T=\Box^{\mathsf{ns}}[a\leftrightarrow x\>\mathsf{U}_{I,\#b=k\%n}y]$, built from $\Sigma\cup W$. Then we synthesize a formula $\psi\in\mathsf{MTL}$ over $\Sigma\cup W\cup X$ which is equivalent to $T$ modulo simple extensions.

Proof E.6**.**

Construct a simple extension $\rho^{\prime}$ as shown in section E.2. 2. 2.

Now checking whether at point $i$ in $\rho$, $x\>\mathsf{U}_{I,\#b=k\%n}y$ is true, is equivalent to checking that at point $i$ in $\rho^{\prime}$ there exist a point $j$ in the future where $y$ is true and for all the points between $j$ and $i$, $x$ is true and the difference between the index values of the symbols from $B$ at $i$ and $j$ is $k\%n$. $\phi_{mark,a}{=}\Box^{\mathsf{ns}}\bigwedge\limits_{i\in\{1,\ldots n-1\}}(a\wedge b_{i}{\leftrightarrow}[x\>\mathsf{U}_{I}(y\wedge b_{j})])$ where $j=k+i\%n$. 3. 3.

The formula $\delta_{1}\wedge\delta_{2}\wedge\delta_{3}\wedge\phi_{mark,a}$ is equivalent to $T$ modulo simple projections.

Notice that in the above reduction, if we start with an $\mathsf{MITL}+\mathsf{UM}$ formula, we will obtain an $\mathsf{MITL}$ formula since we do not introduce any new punctual intervals.

Lemma E.7**.**

Satisfiability of $\mathsf{MITL}+\mathsf{UM}$ is $\mathsf{EXPSPACE}$-complete.

Proof E.8**.**

Assume that we have a $\mathsf{MITL}+\mathsf{UM}$ formula $\phi$ with $|\phi|=m$, and hence $\leq m$ $\mathsf{UM}$ modalities. Let $\phi$ be over the alphabet $\Sigma$. The number of propositions used is hence $2^{\Sigma}$, and let $K$ be the maximal constant appearing in the intervals of $\phi$. Let $k_{1}\%n_{1},\dots,k_{m}\%n_{m}$ be the modulo counting entities in $\phi$. Let $n_{max}$ be the maximum of $n_{1},\dots,n_{m}$. Going by the construction above, we obtain $m$ temporal definitions $T_{1},\dots,T_{m}$, corresponding to the $m$ $\mathsf{UM}$ modalities.

To eliminate each $T_{i}$, we introduce $n_{max}$ formulae of the form $\phi_{mark,a}$, evaluated on timed words over $2^{\Sigma}\cup B_{1}\cup\dots\cup B_{m}$. This is enforced by $\delta_{1},\delta_{2},\delta_{3}$. The number of propositions in the obtained $\mathsf{MITL}$ formula is hence $|2^{\Sigma}|.|B_{1}+B_{2}+\dots+B_{m}|$. The size of the new formula is is $\mathcal{O}(m.n_{max})$, while the maximum constant appearing in the intervals is same as $K$. Thus we have an exponential size (the size now is $\mathcal{O}(m.2^{logn_{max}})$) $\mathsf{MITL}$ formulae with max constant as $K$. The $\mathsf{EXPSPACE}$-hardness of $\mathsf{MITL}+\mathsf{UM}$ follows from that of $\mathsf{MITL}$. Lemma E.3 now shows that satisfiability checking for $\mathsf{MITL}+\mathsf{UM}$ is $\mathsf{EXPSPACE}$-complete.

E.3 Proof of Theorem 3.1.3 : $\mathsf{MITL}+\mathsf{URat}$ is in $\mathsf{2EXPSPACE}$

Proof E.9**.**

Consider a $\mathsf{URat}$ modality $a\mathsf{URat}_{I,\mathsf{re}(\mathsf{S})}b$, where $\mathsf{re}$ is a rational expression over $\mathsf{S}$ and $a,b\in\Sigma$. The size of a $\mathsf{URat}$ modality is $size(\mathsf{re})+log(l)+log(u)$, where $l,u$ are the lower and upper bounds of the interval $I$, and $size(\mathsf{re})$ is the size of the rational expression $\mathsf{re}$. We first flatten $\varphi$ by introducing witness propositions for each $\mathsf{URat}$ modality obtaining temporal definitions of the form $\Box^{\mathsf{ns}}(w\leftrightarrow\mathsf{URat}_{I,\mathsf{re}(\mathsf{S})}\psi)$. Flattening only creates a linear blow up in formula size. Assume that $\varphi$ is flattened. Let $T_{i}=\Box^{\mathsf{ns}}(w_{i}\leftrightarrow a\mathsf{URat}^{\mathsf{S}_{i}}_{I_{i},\mathsf{re}_{i}(\mathsf{S}_{i})}b_{i})$ be a temporal definition, and let there be $t$ temporal definitions. Let $l_{i},u_{i}$ be the bounds of the interval $I_{i}$. The size of $\varphi$, $|\varphi|$ is then defined as $\mathcal{O}(\sum_{i=1}^{t}(n_{i}+log(l_{i})+log(u_{i}))$, where $n_{i}$ is the size of $\mathsf{re}_{i}$. Let $u$ be the maximum constant appearing in the intervals $I_{i}$.

Let us consider a temporal definition $T=\Box^{\mathsf{ns}}(w\leftrightarrow\mathsf{URat}_{I,\mathsf{re}(\mathsf{S})}\psi)$.

We look at the number of propositions needed in obtaining the equisatisfiable $\mathsf{MTL}$ formula.

1. (a)

The size of the rational expression $\mathsf{re}$ in $T$ is $n$. The DFA accepting $\mathsf{re}$ has $\leq 2^{n}$ states. The transitions of this DFA are over formulae from $S$. Since we convert the formulae into ExNF, we also convert this DFA into one whose transitions are over $2^{\mathsf{S}}$. Hence, the number of transitions in the DFA is $\leq 2^{n}\times 2^{|\mathsf{S}|}$. Let $\mathsf{S}^{\prime}=2^{\mathsf{S}}$. 2. (b)

This DFA is simulated using the symbols $\mathsf{Threads},\mathsf{Merge}$. There can be at most $2^{n}$ threads, and each thread be in one of the $2^{n}$ states. Thus, the number of propositions $\mathsf{Th}_{i}(q)$ is at most ${2^{n}}^{2^{n}}$. Given that there are $t$ temporal definitions, we need $t\times{2^{n}}^{2^{n}}$ extra symbols. 3. (c)

Each integer point in the timed word is marked with a symbol $c_{i}$, $0\leq i\leq u-1$ (see the proof of lemma 3.3 in Appendix D). 4. (d)

*The number of propositions *$\mathsf{merge}(i,j)$is $\leq 2^{n}\times 2^{n}$. 5. (e)

Thus, the number of symbols needed is $2^{|\Sigma|}\times u\times t\times{2^{n}}^{2^{n}}\times(2^{n}\times 2^{n})$. 2. 2.

Next, we count the size of the formulae needed while constructing the equisatisfiable $\mathsf{MTL}$ formula.

1. (a)

For each temporal definition, we define the formulae $\mathsf{Nxt}(\mathsf{Th}_{i}(q_{x}))$ for each thread $\mathsf{Th}_{i}$. The argument of $\mathsf{Nxt}$ can take at most $2^{n}$ possibilities ($2^{n}$* states of a DFA) on each of the $2^{n}$ threads. Thus, the total number of $\mathsf{Nxt}(\mathsf{Th}(q))$ formulae is $2^{n}\times 2^{n}=\mathcal{O}(poly(2^{n}))$. Note that each $\mathsf{Nxt}$ formulae simulates the transition function of the DFA. $\mathsf{Nxt}(\mathsf{Th}_{i}(q^{\prime}))$ is determined depending on the present state $q$ of the thread $\mathsf{Th}_{i}$, and the formulae (in $\mathcal{S^{\prime}}$) that are true at the present point. Thus, the size of each $\mathsf{Nxt}$ formulae is $2^{n}\times 2^{|S|}$. Thus, the total size of all the $\mathsf{Nxt}$ formulae is $\mathcal{O}(poly(2^{n}))\times\mathcal{O}(poly(2^{n}\times 2^{|S|}))=\mathcal{O}(poly(2^{n+|S|})$.* 2. (b)

Next, we look at formulae $\mathsf{NextMerge(i,k)}$. Note that both the arguments refer to threads, and hence can take at most $2^{n}$ values. Thus, the total number of formulae is $2^{n}\times 2^{n}=\mathcal{O}(poly(2^{n})$. Each $\mathsf{NextMerge}$ formulae checks whether the states at the 2 threads $\mathsf{Th}_{i},\mathsf{Th}_{k}$ are equal or not. Thus, the size of each formulae is $\mathcal{O}(2^{n})$. The total blow up due to $\mathsf{NextMerge}$ formulae is hence, $\mathcal{O}(2^{n}\times 2^{n}\times 2^{n})=\mathcal{O}poly(2^{n})$. 3. (c)

Next, we look at formulae $\mathsf{MergeseqPref}(k_{1})$. This formulae states all the possible merges from the present point to the integer point within the interval $(l-1,l)$. There are at most $2^{n}$ merges possible, as the merge always happens from a higher indexed thread to a lower one. The number of merges is equal to the nesting depth of the formula $\mathsf{MergeseqPref}(k_{1})$. Note that the nesting depth can be at most $2^{n}$. The number of propositions $\mathsf{merge}(i,j)$ is $2^{n}\times 2^{n}$. Let there be $k\leq 2^{n}$ merges until we see the integer point in $(l-1,l)$. At each of these $k$ merges, we have $2^{n}\times 2^{n}$ possibilities, the maximum possible number of propositions $\mathsf{merge}(i,j)$ ($i,j\leq 2^{n}$). Hence, the number of possible merge sequences we can generate is $(2^{n}\times 2^{n})^{k}\leq(2^{n}\times 2^{n})^{2^{n}}$. There are $2^{n}$ possible values of $k$ and the possible number of disjunctions of the formulae is at most $(2^{n}\times 2^{n})^{2^{n}}\times 2^{n}=\mathcal{O}(poly(2^{poly(2^{n})}))$. 4. (d)

The counting for $\mathsf{MergeseqSuf}(k_{1},k)$ is symmetric.

Adding all the blow ups due to various formulae $\mathsf{Nxt}(\mathsf{Th}(q))$, $\mathsf{NextMerge(i,k)}$, $\mathsf{MergeseqPref}(k_{1})$ and $\mathsf{MergeseqSuf}(k_{1},k)$, we see the number to be doubly exponential $\mathcal{O}(poly(2^{poly(2^{n})}))$. Thus, we obtain an $\mathsf{MITL}$ formula of doubly exponential size, with doubly exponential number of new propositions. By applying the reduction as in lemma E.3, we will obtain a formula in $\mathsf{MITL}[\>\mathsf{U}_{0,\infty},\>\mathsf{S}]$, which is still doubly exponential, and which preserves the max constant. The $\mathsf{PSPACE}$ procedure of $\mathsf{MITL}[\>\mathsf{U}_{0,\infty},\>\mathsf{S}]$ thus ensures that we have a 2$\mathsf{EXPSPACE}$ procedure for satisfiability checking for $\mathsf{MITL}+\mathsf{URat}$. Arriving at a tighter complexity for this class is an interesting problem and is open.

E.4 Proof of Theorem 3.1.4: $\mathsf{MITL}+\mathsf{MC}$ is $\bf{F}_{\omega^{\omega}}$-hard

In this section, we discuss the complexity of $\mathsf{MITL}+\mathsf{MC}$, proving Theorem 3.1.4. To prove this, we obtain a reduction from the reachability problem of Insertion Channel Machines with Emptiness Testing ($\mathsf{ICMET}$). We now show how to encode the reachability problem of $\mathsf{ICMET}$ in $\mathsf{MITL}+\mathsf{MC}$.

Recalling $\mathsf{ICMET}$

A channel machine $\mathcal{A}$ consists of a tuple having a finite set of states $S$, a finite alphabet $M$ used to write on the channels, a finite set $C$ of channels, and a transition relation $\Delta\subseteq S\times Op\times S$ where $Op$ is a finite set of operations on the channels. These operations have the forms $c!a$, $c?a$ and $c=\epsilon$ which respectively write a message $a$ to the tail of channel $c$, read the first message $a$ from a channel $c$, and test if channel $c$ is empty.

A configuration of the channel machine $\mathcal{A}$ is a pair $(s,U)$ where $s$ is a state and $U$ is a tuple of length $|C|$ which describes the contents of all the $|C|$ channels. Each entry in this tuple is hence a string over the alphabet $M$. We use $\mathsf{Conf}$ to denote the configurations of the channel machine. The configurations are connected to each other depending on the operations performed. In particular,

1. (a)

From a configuration $(q,U)$, the transition $(q,c!a,q^{\prime})$ results in a configuration $(q^{\prime},U^{\prime})$ where $U^{\prime}$ is the $|C|$-tuple which does not alter the contents of channels other than $c$, and appends $a$ to channel $c$. 2. (b)

From a configuration $(q,U)$, the transition $(q,c?a,q^{\prime})$ results in a configuration $(q^{\prime},U^{\prime})$ where $U^{\prime}$ is the $|C|$-tuple which does not alter the contents of channels other than $c$, and reads $a$ from the head of channel $c$. 3. (c)

From a configuration $(q,U)$, the transition $(q,c=\epsilon,q^{\prime})$ results in the configuration $(q^{\prime},U)$ if channel $c$ is empty. The contents of all the channels are unaltered. If channel $c$ is non-empty, then the machine is stuck.

If the only transitions allowed are as above, then we call $\mathcal{A}$ an error-free channel-machine. We now look at channel machines with insertion errors. These allow extra transitions between configurations as follows.

1. (d)

If a configuration $(q,U)$ can evolve into $(q^{\prime},V)$ using one transition as above, then we allow any configuration $(q,U^{\prime})$, where $U^{\prime}$ is a $|C|$-tuple of words obtained by deleting any number of letters from any word in $U$, to evolve into $(q^{\prime},V^{\prime})$ where $V^{\prime}$ is obtained by adding any number of letters to any word in $V$. Thus insertion errors are created by inserting arbitrarily many symbols into some word.

The channel machines as above are called $\mathsf{ICMET}$. A run of an $\mathsf{ICMET}$ is a sequence of transitions $\gamma_{0}\stackrel{{\scriptstyle op_{0}}}{{\rightarrow}}\gamma_{1}\dots\stackrel{{\scriptstyle op_{n-1}}}{{\rightarrow}}\gamma_{n}\dots$ that is consistent with the above operational semantics.

Reduction from $\mathsf{ICMET}$ reachability to satisfiability of $\mathsf{MITL}+\mathsf{MC}$

Consider any $\mathsf{ICMET}$ ${\cal C}=(S,M,\Delta,C)$, with set of states $S=\{s_{0},\ldots,s_{n}\}$ and channels $C=\{c_{1},\ldots,c_{k}\}$. Let $M$ be a finite set of messages used for communication in the channels.

We encode the set of all possible configurations of $\cal C$, with a timed language over the alphabet $\Sigma=M_{a}\cup M_{b}\cup\Delta\cup S\cup\{H\}$, where $M_{a}=\{m_{a}|m\in M\}$ $M_{b}=\{m_{b}|m\in M\}$, and $H$ is a new symbol.

The $j$th configuration for $j\geq 0$ is encoded in the interval $[(2k+2)j,(2k+2)(j+1)-1)$ where $k$ refers to number of channels. The $j$th configuration begins at the time point $(2k+2)j$. At a distance $[2i-1,2i]$ from this point, $1\leq i\leq k$, the contents of the $i^{th}$ channel are encoded as shown in the point 7. The intervals of the form $(2i,2i+1)$, $0\leq i\leq k+1$ from the start of any configuration are time intervals within which no action takes place. The current state at the $j$th configuration is encoded at $(2k+2)j$, and the transition that connects configurations $j,j+1$ is encoded at $(2k+2)j+(2k+1)$.

Lets look at the encoding of the contents of channel $i$ in the $j$th configuration. Let $m_{h_{i}}$ be the message at the head of the channel $i$. Each message $m_{i}$ is encoded using consecutive occurrences of symbols $m_{i,a}$ and $m_{i,b}$. In our encoding of channel $i$, the first point marked $m_{h_{i},a}$ in the interval $(2k+2)j+[2i-1,2i]$ is the head of the channel $i$ and denotes that $m_{h_{i}}$ is the message at the head of the channel. The last point marked $m_{t_{i},b}$ in the interval is the tail of the channel, and denotes that message $m_{t_{i}}$ is the message stored at the tail of the channel.

Exactly at $2k+1$ time units after the start of the $j^{th}$ configuration, we encode the transition from the state at the $j^{th}$ configuration to the $(j+1)^{st}$ configuration (starts at $(2k+2)(j+1)$). Note that the transition has the form $(s,c!m,s^{\prime})$ or $(s,c?m,s^{\prime})$ or $(s,c=\epsilon,s^{\prime})$. 4. 4.

We introduce a special symbol $H$, which acts as separator between the head of the message and the remaining contents, for each channel. 5. 5.

A sequence of messages $w_{1}w_{2}w_{3}\ldots w_{z}$ in any channel is encoded as a sequence

$w_{1,a}w_{1,b}Hw_{2,a}w_{2,b}w_{3,a}w_{3,b}\ldots w_{z,a}w_{z,b}$.

Let $S=\bigvee_{i=0}^{n}s_{i}$ denote the states of the $\mathsf{ICMET}$, $\alpha=\bigvee_{i=0}^{m}\alpha_{i}$, denote the transitions $\alpha_{i}$ of the form $(s,c!m,s^{\prime})$ or $(s,c?m,s^{\prime})$ or $(s,c=\epsilon,s^{\prime})$. Let $action=true$ and let $M_{a}=\bigvee_{m_{x}\in M}(m_{x,a})$, $M_{b}=\bigvee_{m_{x}\in M}(m_{x,b})$, with $M=M_{a}\vee M_{b}$.

All the states must be at distance $2k+2$ from the previous state (first one being at 0) and all the propositions encoding transitions must be at the distance $2k+1$ from the start of the configuration.

$\varphi_{S}{=}\leavevmode\nobreak\ s_{0}\wedge\Box[S\Rightarrow\{\Diamond_{(0,2k+2]}(S)\wedge\Box_{(0,2k+2)}(\neg S)\wedge\Diamond_{(0,2k+1]}\alpha\wedge\Box_{[0,2k+1)}(\neg\alpha)\wedge\Diamond_{(2k+1,2k+2)}(\neg\alpha)\}]$ 2. 2.

All the messages are in the interval $[2i-1,2i]$ from the start of configuration. No action takes place at $(2i-2,2i-1)$ from the start of any configuration.

$\varphi_{m}{=}\leavevmode\nobreak\ \Box\{S{\Rightarrow}\bigwedge_{i=1}^{k}\Box_{[2i-1,2i]}(M{\vee}H){\wedge}$ $\Box_{(2i-2,2i-1)}(\neg action)\}$ 3. 3.

Consecutive source and target states must be in accordance with a transition $\alpha$. For example, $s_{j}$ appears consecutively after $s_{i}$ reading $\alpha_{i}$ iff $\alpha_{i}$ is of the form $(s_{i},y,s_{j})\in\Delta$, with $y\in\{c_{i}!m,c_{i}?m,c_{i}=\epsilon\}$.

$\varphi_{\Delta}{=}\bigwedge_{s,s^{\prime}\in S}\Box\{(s{\wedge}\Diamond_{(0,2k+2]}s^{\prime}){\Rightarrow}(\Diamond_{(0,2k+1]}{\bigvee\Delta_{s,s^{\prime}}})\}$ where $\Delta_{s,s^{\prime}}$ are possible $\alpha_{i}$ between $s,s^{\prime}$. 4. 4.

We introduce a special symbol $H$ along with other channel contents which acts as a separator between the head of the channel and rest of the contents. Thus $H$ has following properties

• •

There is one and only one time-stamp in the interval $(2i-1,2i)$ from the start of the configuration where $H$ is true. The following formula says that there is an occurrence of a $H$:

$\varphi_{H_{1}}{=}\Box[(S{\wedge}\Diamond_{(2i-1,2i)}M){\Rightarrow}(\bigwedge_{i=1}^{k}\Diamond_{(2i-1,2i)}(H))]$

The following formula says that there can be only one $H$: $\varphi_{H_{2}}{=}\Box(H{\Rightarrow}\neg\Diamond_{(0,1)}H)$

• •

Every message $m_{x}$ is encoded by truth of proposition $m_{x,a}$ immediately followed by $m_{x,b}$. Thus for any message $m_{x}$, the configuration encoding the channel contents has a sub-string of the form $(m_{x,a}m_{x,b})^{*}$ where $m_{x}$ is some message in $M$.

$\varphi_{m}{=}\Box[m_{x,a}{\Rightarrow}\mathsf{O}_{(0,1]}m_{x,b}]{\wedge}\Box[m_{x,b}{\Rightarrow}\mathsf{O}_{(0,1)}M_{a}$ ${\vee}\mathsf{O}(\bigvee\Delta\vee H)]{\wedge}(\neg M_{b}\>\mathsf{U}M_{a})$

• •

If the channel is not empty (there is at least one message $m_{a}m_{b}$ in the interval $(2i-1,2i)$ corresponding to channel $i$ contents) then there is one and only one $m_{b}$ before $H$. The following formula says that there can be at most one $m_{b}$ before $H$.

$\varphi_{H_{3}}{=}\Box[\neg\{M_{b}\wedge\Diamond_{(0,1)}(M_{b}\wedge\Diamond_{(0,1)}H)\}]$

The following formula says that there is one $M_{b}$ before $H$ in the channel, if the channel is non-empty.

$\varphi_{H_{4}}{=}\Box[S{\Rightarrow}\{\bigwedge_{j=1}^{k}(\Diamond_{[2j-1,2j]}(M_{b}){\Rightarrow}$ $\Diamond_{[2j-1,2j]}(M_{b}\wedge\Diamond_{(0,1)}H))\}]$

Let $\varphi_{H}{=}\varphi_{H_{1}}\wedge\varphi_{H_{2}}\wedge\varphi_{H_{3}}\wedge\varphi_{H_{4}}$. 5. 5.

Encoding transitions:

• (a)

We first define a macro for copying the contents of the $i^{th}$ channel to the next configuration with insertion errors. If there were some $m_{x,a},m_{x,b}$ at times $t,t^{\prime}$, $m_{x,b}$ is copied to $t^{\prime\prime}+2k+2$ (where $t^{\prime\prime}\in[t,t^{\prime})$), representing the channel contents in the next configuration. This is specified by means of an even count check. From any 3 consecutive points $u,v,w$ such that $m_{x,a}$ and $m_{x,b}$ are true at $v$ and $w$, respectively, we assert that there are even (or odd) number of $m_{x,b}$ within $(0,2k+2)$ from both $v$ and $w$. This implies that there must be an odd number of $m_{x,b}$’s within time interval $[\tau_{v}+2k+2,\tau_{w}+2k+2]$. Thus, there must be at least one $m_{x,b}$ copied from the point $w$ to some point in the interval $[\tau_{v}+2k+2,\tau_{w}+2k+2]$. The rest of the even number of erroneous $m_{x,b}$ in $[\tau_{v}+2k+2,\tau_{w}+2k+2]$, along with the arbitrary insertion errors within $[\tau_{u}+2k+2,\tau_{v}+2k+2]$ models the insertion error of the $\mathsf{ICMET}$ (see Figure 11). The formula $\mathsf{copy}_{g}$ is as follows. $\Box_{[2i-1,2i]}[\bigwedge_{m_{x}\in M}(m_{x,a}{\wedge}\mathsf{iseven}_{(0,2k+2)}(m_{x,b})){\Rightarrow}\mathsf{O}(\mathsf{iseven}_{(0,2k+2)}(m_{x,b}))]\\ {\wedge}\Box_{[2i-1,2i]}[\bigwedge_{m_{x}\in M}(m_{x,a}{\wedge}\neg\mathsf{iseven}_{(0,2k+2)}(m_{x,b})){\Rightarrow}\mathsf{O}(\neg\mathsf{iseven}_{(0,2k+2)}(m_{x,b}))]$

• (b)

If the transition is of the form $c_{i}=\epsilon$. The following formulae checks that there are no events in the interval $(2i-1,2i)$ corresponding to channel $i$, while all the other channel contents are copied.

$\varphi_{c_{i}=\epsilon}{=}S\wedge\Box_{(2i-1,2i)}(\neg action){\wedge}\bigwedge\limits_{g=1}^{k}\mathsf{copy}_{g}$

• (c)

If the transition is of the form $c_{i}!m_{x}$ where $m\in M$. An extra message is appended to the tail of channel $i$, and all the $m_{a}m_{b}$’s are copied to the next configuration. $M_{b}\wedge\Box_{(0,1)}(\neg M))$ denotes the last time point of channel $i$; if this occurs at time $t$, we know that this is copied at a timestamp strictly less than $2k+2+t$ (by 5(a)). Thus we assert that truth of $\Diamond_{(2k+2,2k+3)}m_{x,b}$ at $t$.

$\varphi_{c_{i}!m}{=}S{\wedge}\bigwedge\limits_{g=1}^{k}\mathsf{copy}_{g}{\wedge}\Diamond_{[2i-1,2i)}\{(M{\wedge}\Box_{(0,1)}(\neg M)){\Rightarrow}(\Diamond_{(2k+2,2k+3)}(m_{x,b}))\}$

• (d)

If the transition is of the form $c_{i}?m$ where $m\in M$. The contents of all channels other than $i$ are copied to the intervals encoding corresponding channel contents in the next configuration. We also check the existence of a first message in channel $i$; such a message has a $H$ at distance $(0,1)$ from it.

$\varphi_{c_{i}?m_{x}}{=}S{\wedge}{\bigwedge\limits_{j\neq i,g=1}^{k}}\mathsf{copy}_{g}{\wedge}\Diamond_{(2i-1,2i)}\{m_{x,b}{\wedge}\Diamond_{(0,1)}(H)\}{\wedge}\\ \Box_{[2i-1,2i]}[{\bigwedge_{m_{x}\in M}}(m_{x,a}{\wedge}\mathsf{iseven}_{(0,2k+2)}(m_{x,b}){\wedge}\neg\Diamond_{(0,1)}H){\Rightarrow}\mathsf{O}(\mathsf{iseven}_{(0,2k+2)}(m_{x,b}))]{\wedge}\\ \Box_{[2i-1,2i]}[{\bigwedge_{m_{x}\in M}}(m_{x,a}{\wedge}\neg\mathsf{iseven}_{(0,2k+2)}(m_{x,b}){\wedge}\neg\Diamond_{(0,1)}H){\Rightarrow}\mathsf{O}(\neg\mathsf{iseven}_{(0,2k+2)}(m_{x,b}))]$ 6. 6.

Channel contents must change in accordance to the relevant transition. Let $L$ be a set of labels (names) for the transitions. Let $l\in L$ and $\alpha_{l}$ be a transition labeled $l$.

$\varphi_{C}=\Box[S\Rightarrow\bigwedge_{l\in L}(\Diamond_{(0,2k+1]}(\bigvee\alpha_{l}\Rightarrow\varphi_{l}))]$

where $\varphi_{l}$ are the formulae as seen in 5 above ($\varphi_{c_{i}?m_{x}},\varphi_{c_{i}!m},\varphi_{c_{i}=\epsilon}$). 7. 7.

Let $s_{t}$ be a state of the $\mathsf{ICMET}$ whose reachability we are interested in. We check $s_{t}$ is reachable from $s_{0}$ : $\phi_{reach}=\Diamond(s_{t})$

Thus the formula encoding $\mathsf{ICMET}$ is: $\varphi_{S}\wedge\varphi_{\Delta}\wedge\varphi_{m}\wedge\varphi_{H}\wedge\varphi_{C}\wedge\varphi_{reach}$

This is a formula in $\mathsf{MITL}+\mathsf{UM}$, and we have reduced the reachability problem of $\mathsf{ICMET}$ with insertion errors to checking satisfiability of this formula.

Appendix F Non-punctual 1-$\mathsf{TPTL}$ is $\mathsf{NPR}$

In this section, we show that non-punctuality does not provide any benefits in terms of complexity of satisfiability for $\mathsf{TPTL}$ as in the case of $\mathsf{MITL}$. We show that satisfiability checking of non-punctual $\mathsf{TPTL}$ is itself non-primitive recursive. This highlights the importance of our oversampling reductions from $\mathsf{RatMTL}$ and $\mathsf{RatMITL}$ to $\mathsf{MTL}$ and $\mathsf{MITL}$ respectively, giving $\mathsf{RegMITL}$ an elementary complexity. It is easier to reduce $\mathsf{RatMITL}[\mathsf{URat}]$ to 1-variable, non-punctual, $\mathsf{TPTL}$ without using oversampling, but this gives a non-primitive recursive bound on complexity. Our reduction of $\mathsf{RatMITL}[\mathsf{URat}]$ to equisatisfiable $\mathsf{MITL}$ using oversampling, however has a $\mathsf{2EXPSPACE}$ upperbound.

Non-punctual $\mathsf{TPTL}$ with 1 Variable ($1-\mathsf{OpTPTL}$)

We study a subclass of $1-\mathsf{TPTL}$ called open $1-\mathsf{TPTL}$ and denoted as $1-\mathsf{OpTPTL}$. The restrictions are mainly on the form of the intervals used in comparing the clock $x$ as follows:

• •

Whenever the single clock $x$ lies in the scope of even number of negations, $x$ is compared only with open intervals, and

• •

Whenever the single clock $x$ lies in the scope of an odd number of negations, $x$ is compared to a closed interval.

Note that this is a stricter restriction than non-punctuality as it can assert a property only within an open timed region. Our complexity result hence applies to $\mathsf{TPTL}$ with non-punctual intervals. Our hardness result uses a reduction from counter machines.

Counter Machines

A deterministic $k$-counter machine is a $k+1$ tuple ${\cal M}=(P,C_{1},\ldots,C_{k})$, where

$C_{1},\ldots,C_{k}$ are counters taking values in $\mathbb{N}\cup\{0\}$ (their initial values are set to zero); 2. 2.

$P$ is a finite set of instructions with labels $p_{1},\dots,p_{n-1},p_{n}$. There is a unique instruction labelled HALT. For $E\in\{C_{1},\ldots,C_{k}\}$, the instructions $P$ are of the following forms:

1. (a)

$p_{i}$: $Inc(E)$, goto $p_{j}$, 2. (b)

$p_{i}$: If $E=0$, goto $p_{j}$, else go to $p_{k}$, 3. (c)

$p_{i}$: $Dec(E)$, goto $p_{j}$, 4. (d)

$p_{n}$: HALT.

A configuration $W=(i,c_{1},\ldots,c_{k})$ of ${\cal M}$ is given by the value of the current program counter $i$ and values $c_{1},c_{2},\ldots,c_{k}$ of the counters $C_{1},C_{2},\ldots,C_{k}$. A move of the counter machine $(l,c_{1},c_{2},\ldots,c_{k})\rightarrow(l^{\prime},c_{1}^{\prime},c_{2}^{\prime},\ldots,c_{k}^{\prime})$ denotes that configuration $(l^{\prime},c_{1}^{\prime},c_{2}^{\prime},\ldots,c_{k}^{\prime})$ is obtained from $(l,c_{1},c_{2},\ldots,c_{k})$ by executing the $l^{th}$ instruction $p_{l}$. If $p_{l}$ is an increment or decrement instruction, $c^{\prime}_{l}=c_{l}+1$ or $c_{l}-1$, while $c^{\prime}_{i}=c_{i}$ for $i\neq l$ and $p^{\prime}_{l}$ is the respective next instruction, while if $p_{l}$ is a zero check instruction, then $c^{\prime}_{i}=c_{i}$ for all $i$, and $p^{\prime}_{l}=p_{j}$ if $c_{l}=0$ and $p_{k}$ otherwise.

Incremental Error Counter Machine ($\mathsf{IECM}$)

An incremental error counter machine ($\mathsf{IECM}$) is a counter machine where a particular configuration can have counter values with arbitrary positive error. Formally, an incremental error $k$-counter machine is a $k+1$ tuple ${\cal M}=(P,C_{1},\ldots,C_{k})$ where $P$ is a set of instructions like above and $C_{1}$ to $C_{k}$ are the counters. The difference between a counter machine with and without incremental counter error is as follows:

Let $(l,c_{1},c_{2}\ldots,c_{k})\rightarrow(l^{\prime},c_{1}^{\prime},c_{2}^{\prime}\ldots,c_{k}^{\prime})$ be a move of a counter machine without error when executing $l^{th}$ instruction. 2. 2.

The corresponding move in the increment error counter machine is

[TABLE]

Thus the value of the counters are non deterministic. We use these machines for proving lower bound complexity in section F.1.

Theorem F.1**.**

[14]** The halting problem for deterministic $k$ counter machines is undecidable for $k\geq 2$.

Theorem F.2**.**

[5]** The halting problem for incremental error $k$-counter machines is non primitive recursive for $k\geq 5$.

F.1 Satisfiability Checking for $1-\mathsf{OpTPTL}$

Theorem F.3**.**

Satisfiability Checking of \mbox{1-\mathsf{OpTPTL}}[\Diamond,\mathsf{O}] is decidable with non primitive recursive lower bound over finite timed words.

Proof F.4**.**

We encode the runs of $k$ counter incremental error counter machine using $1-\mathsf{OpTPTL}$ formulae with $\Diamond,\mathsf{O}$ modalities. We will encode a particular computation of any counter machine using timed words. The main idea is to construct a \mbox{1-\mathsf{OpTPTL}}[\Diamond,\mathsf{O}] formula $\varphi_{\mathsf{IECM}}$ for any given $k$-incremental counter machine $\mathsf{IECM}$ such that $\varphi_{\mathsf{IECM}}$ is satisfied by only those timed words that encode the halting computation of $\mathsf{IECM}$. Moreover, for every halting computation $\mathcal{C}$ of the $\mathsf{IECM}$, at least one timed word $\rho_{C}$ encodes $\mathcal{C}$ and satisfies $\varphi_{\mathsf{IECM}}$.

We encode each computation of a $k$-incremental counter machine $(P,C)$ where $P=\{p_{1},\ldots,p_{n}\}$ is the set of instructions and $C=\{c_{1},\ldots,c_{k}\}$ is the set of counters using timed words over the alphabet $\Sigma_{\mathsf{IECM}}=\bigcup_{j\in\{1,\ldots,k\}}(S\cup F\cup\{a_{j},b_{j}\})$ where $S=\{s^{p}|p\in{1,\ldots,n}\}$ and $F=\{f^{p}|p\in{1,\ldots,n}\}$ as follows: The $i^{th}$ configuration, $(p,c_{1},\ldots,c_{k})$ is encoded in the timed region $[i,i+1)$ with the sequence

$s^{p}(a_{1}b_{1})^{c_{1}}(a_{2}b_{2})^{c_{2}}\ldots(a_{k}b_{k})^{c_{k}}f^{p}$.

The concatenation of these time segments of a timed word encodes the whole computation. Untiming our language yields the language

[TABLE]

where $\mathcal{S}=\bigvee\limits_{p\in\{1,2,\ldots,n\}}s^{p}$ and $\mathcal{F}=\bigvee\limits_{p\in\{1,2,\ldots,n\}}f^{p}$.

To construct a formula $\varphi_{\mathsf{IECM}}$, the main challenge is to propagate the behaviour from the time segment $[i,i+1)$ to the time segment $[i+1,i+2)$ such that the latter encodes the $i+1^{th}$ configuration of the $\mathsf{IECM}$ in accordance with the counter values of the $i^{th}$ configuration. The usual idea is to copy all the $a$’s from one configuration to another using punctuality. This is not possible in a non-punctual logic. We preserve the number of $a$s and $b$s using the following idea:

• •

Given any non last $(a_{j},t)(b_{j},t^{\prime})$ before $\mathcal{F}$(for some counter $c_{j}$), of a timed word encoding a computation. We assert that the last symbol in $(t,t+1)$ is $a_{j}$ and the last symbol in $(t^{\prime},t^{\prime}+1$) is $b_{j}$.

• •

We can easily assert that the untimed sequence of the timed word is of the form

[TABLE]

• •

The above two conditions imply that there is at least one $a_{j}$ within time $(t+1,t^{\prime}+1)$. Thus, all the non last $a_{j},b_{j}$ are copied to the segment encoding next configuration. Now appending one $a_{j}b_{j}$,two $a_{j}b_{j}$’s or no $a_{j}b_{j}$’s depends on whether the instruction was copy, increment or decrement operation.

$\varphi_{\mathsf{IECM}}$* is obtained as a conjunction of several formulae. Let $\mathcal{S},\mathcal{F}$ be a shorthand for $\bigwedge\limits_{p\in\{1,\ldots,n\}}s^{p}$ and $\bigwedge\limits_{p\in\{1,\ldots,n\}}f^{p}$, respectively. We also define macros $A_{j}=\bigvee\limits_{w\geq j}a_{w}$ and $A_{k+1}=\bot$ We now give formula for encoding the machine. Let $\mathsf{C}=\{1,\ldots,k\}$ and $\mathsf{P}=\{1,\ldots,n\}$ be the indices of the counters and the instructions.*

• •

Expressing untimed sequence*: The words should be of the form*

[TABLE]

This could be expressed in the formula below

$\varphi_{1}=\bigwedge\limits_{j\in\mathsf{C},p\in\mathsf{P}}\Box^{\mathsf{ns}}[s^{p}\rightarrow\mathsf{O}(A_{1}\vee f^{p})]\wedge\Box^{\mathsf{ns}}[a_{j}\rightarrow\mathsf{O}(b_{j})]\wedge\\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \Box^{\mathsf{ns}}[b_{j}\rightarrow\mathsf{O}(A_{j+1}\vee f^{p})]\wedge\Box^{\mathsf{ns}}[f^{p}\rightarrow\mathsf{O}(\mathcal{S}\vee\Box^{\mathsf{ns}}(false))]$**

• •

Initial Configuration*: There is no occurrence of $a_{j}b_{j}$ within $[0,1]$. The program counter value is $1$.*

$\varphi_{2}=x.\{s^{1}\wedge\mathsf{O}(f^{1}\wedge x\in(0,1))\}$**

• •

Copying $\mathcal{S},\mathcal{F}$: Any $(\mathcal{S},u)$ (read as any symbol from $\mathcal{S}$ at time stamp $u$) $(\mathcal{F},v)$ (read as (read as any symbol from $\mathcal{F}$ at time stamp $v$)) has a next occurrence $(\mathcal{S},u^{\prime})$, $(\mathcal{F},v^{\prime})$ in the future such that $u^{\prime}-u\in(k,k+1)$ and $v^{\prime}-v\in(k-1,k)$. Note that this condition along with $\varphi_{1}$ and $\varphi_{2}$ makes sure that $\mathcal{S}$ and $\mathcal{F}$ occur only within the intervals of the form $[i,i+1)$ where $i$ is the configuration number. Recall that $s^{n},f^{n}$ represents the last instruction (HALT).

$\varphi_{3}=[\Box^{\mathsf{ns}}x.\{(\mathcal{S}\wedge\neg s^{n})\rightarrow\neg\Diamond(x\in[0,1]\wedge\mathcal{S})\wedge\Diamond(\mathcal{S}\wedge x\in(1,2))\}\wedge\\ \Box^{\mathsf{ns}}x.\{(\mathcal{F}\wedge\neg f^{n})\rightarrow\Diamond(\mathcal{F}\wedge x\in(0,1))\}]$**

Note that the above formula ensures that subsequent configurations are encoded in smaller and smaller regions within their respective unit intervals, since consecutive symbols from $\mathcal{S}$ grow apart from each other (a distance $>1$), while consecutive symbols from $\mathcal{F}$ grow closer to each other (a distance $<1$). See Figure 13.

• •

Beyond $p_{n}$=HALT, there are no instructions

$\varphi_{4}\ =\ \Box^{\mathsf{ns}}[f^{n}\rightarrow\Box(false)]$**

• •

At any point of time, exactly one event takes place. Events have distinct time stamps.

$\varphi_{6}\ =\ [\bigwedge\limits_{y\in\Sigma_{\mathsf{IECM}}}\Box^{\mathsf{ns}}[y\rightarrow\neg(\bigwedge\limits_{\Sigma_{\mathsf{IECM}}\setminus\{y\}}(x))]\wedge\Box^{\mathsf{ns}}[\Box(false)\vee\mathsf{O}(x\in(0,\infty))]$**

• •

Eventually we reach the halting configuration $\langle p_{n},c_{1},\ldots,c_{k}\rangle$: $\varphi_{6}={\Diamond^{\mathsf{ns}}}s^{n}\\$

• •

Every non last $(a_{j},t)(b_{j},t^{\prime})$ occurring in the interval $(i,i+1)$ should be copied in the interval $(i+1,i+2)$. We specify this condition as follows:

• –

state that from every non last $a_{j}$ the last symbol within $(0,1)$ is $a_{j}$. Similarly from every non last $b_{j}$, the last symbol within $(0,1)$ is $b_{j}$. Thus $(a_{j},t)(b_{j},t^{\prime})$ will have a $(b_{j},t^{\prime}+1-\epsilon)$ where $\epsilon\in(0,t^{\prime}-t)$.

• –

Thus all the non last $a_{j}b_{j}$ will incur a $b_{j}$ in the next configuration. $\varphi_{1}$ makes sure that there is an $a_{j}$ between two $b_{j}$’s. Thus this condition along with $\varphi_{1}$ makes sure that the non last $a_{j}b_{j}$ sequence is conserved. Note that there can be some $a_{j}b_{j}$ which are arbitrarily inserted. These insertions errors model the incremental error of the machine. Any such inserted $(a_{j},t_{ins})(b_{j},t_{ins}^{\prime})$ in $(i+1,i+2)$ is such that there is a $(a_{j},t)(b_{j},t^{\prime})$ in $(i,i+1)$ with $t_{ins}^{\prime}\in(t+1,t^{\prime}+1)$. Just for the sake of simplicity we assume that $a_{k+1}=false$.

*Let $nl(a_{j})=a_{j}\wedge\neg last(a_{j})$, $nl(b_{j})=b_{j}\wedge\neg last(b_{j})$, $\psi_{nh}=\neg\Diamond(f^{n}\wedge x\in[0,1])$, *

$last(a_{j})=a_{j}\wedge\mathsf{O}(\mathsf{O}(\mathcal{F}\vee A_{j+1})))$* and $last(b_{j})=b_{j}\wedge\mathsf{O}(\mathcal{F}\vee A_{j+1})$.*

$\varphi_{7}=\bigwedge\limits_{j\in\mathsf{C}}\Box^{\mathsf{ns}}x.[(nl(a_{j})\wedge\psi_{nh})\rightarrow\Diamond(a_{j}\wedge x\in(0,1)\wedge\mathsf{O}(x\in(1,2)))]\wedge\\ \Box^{\mathsf{ns}}x.[(nl(b_{j})\wedge\psi_{nh})\rightarrow\Diamond(b_{j}\wedge x\in(0,1)\wedge\mathsf{O}(x\in(1,2)))]$**

We define a short macro $Copy_{\mathsf{C}\setminus W}$: Copies the content of all the intervals encoding counter values except counters in $W$. Just for the sake of simplicity we denote

$Copy_{\mathsf{C}\setminus W}=\bigwedge\limits_{j\in\mathsf{C}\setminus W}\Box^{\mathsf{ns}}x.\{last(a_{j})\rightarrow(a_{j}\wedge x\in(0,1)\wedge\mathsf{O}(b_{j}\wedge x\in(1,2)\wedge\mathsf{O}(\mathcal{F})))\}$**

Using this macro we define the increment,decrement and jump operation.

Consider the zero check instruction $p_{g}$: If $C_{j}=0$ goto $p_{h}$, else goto $p_{d}$. $\delta_{1}$ specifies the next configuration when the check for zero succeeds. $\delta_{2}$ specifies the else condition.

$\varphi^{g,j=0}_{8}\ =\ Copy_{\mathsf{C}\setminus\{\emptyset\}}\wedge\delta_{1}\wedge\delta_{2}$**

$\delta_{1}=\Box^{\mathsf{ns}}[\{s^{g}\wedge((\neg a_{j})\>\mathsf{U}\mathcal{F})\}\rightarrow(\neg\mathcal{S})\>\mathsf{U}s^{h})]$**

$\delta_{2}=\Box^{\mathsf{ns}}[\{s^{g}\wedge((\neg a_{j})\>\mathsf{U}a_{j})\}\rightarrow(\neg\mathcal{S})\>\mathsf{U}s^{d})]$. 2. 2.

$p_{g}$: $Inc(C_{j})$ goto $p_{h}$. The increment is modelled by appending exactly one $a_{j}b_{j}$ in the next interval just after the last copied $a_{j}b_{j}$

$\varphi^{g,inc_{j}}_{8}\ =Copy_{\mathsf{C}\setminus\emptyset}\wedge\Box^{\mathsf{ns}}(s^{g}\rightarrow(\neg\mathcal{S})\>\mathsf{U}s^{h})\wedge\psi^{inc}_{0}\wedge\psi^{inc}_{1}$**

• •

The formula $\psi^{inc}_{0}=\Box^{\mathsf{ns}}[(s^{g}\wedge(\neg a_{j}\>\mathsf{U}f^{g}))\rightarrow(\neg\mathcal{S}\>\mathsf{U}x.(s^{h}\wedge\Diamond(x\in(0,1)\wedge a_{j}))]$ specifies the increment of the counter $j$ when the value of $j$ is zero.

• •

The formula $\psi^{inc}_{1}=\Box^{\mathsf{ns}}[\{s^{g}\wedge((\neg\mathcal{F})\>\mathsf{U}(a_{j}))\}\rightarrow(\neg\mathcal{F})\>\mathsf{U}x.\{last(a_{j})\wedge\Diamond(x\in(0,1)\wedge(a_{j}\wedge\mathsf{O}\mathsf{O}(last(a_{j})\wedge x\in(1,2))))\}]$ specifies the increment of counter $j$ when $j$ value is non zero by appending exactly one pair of $a_{j}b_{j}$ after the last copied $a_{j}b_{j}$ in the next interval. 3. 3.

$p_{g}$: $Dec(C_{j})$ goto $p_{h}$. Let $second-last(a_{j})=a_{j}\wedge\mathsf{O}(\mathsf{O}(last(a_{j})))$. Decrement is modelled by avoiding copy of last $a_{j}b_{j}$ in the next interval.

$\varphi^{g,dec_{j}}_{8}\ =\ Copy_{\mathsf{C}\setminus j}\wedge\Box^{\mathsf{ns}}(s^{g}\rightarrow(\neg\mathcal{S})\>\mathsf{U}s^{h})\wedge\psi^{dec}_{0}\wedge\psi^{dec}_{1}$**

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The formula $\psi^{dec}_{0}=\Box^{\mathsf{ns}}[\{s^{g}\wedge(\neg a_{j})\>\mathsf{U}f^{g})\}\rightarrow\{(\neg\mathcal{S})\>\mathsf{U}\{s^{h}\wedge((\neg a_{j})\>\mathsf{U}(\mathcal{F})\}]$ specifies that the counter remains unchanged if decrement is applied to the $j$ when it is zero.

• •

The formula $\psi^{dec}_{1}=\Box^{\mathsf{ns}}[\{s^{g}\wedge((\neg\mathcal{F})\>\mathsf{U}(a_{j}))\}\rightarrow(\neg\mathcal{F})\>\mathsf{U}x.\{second-last(a_{j})\wedge\Diamond(x\in(0,1)\wedge(a_{j}\wedge\mathsf{O}\mathsf{O}([A_{j+1}\vee\mathcal{F}]\wedge x\in(1,2))))\}]$ decrements the counter $j$, if the present value of $j$ is non zero. It does that by disallowing copy of last $a_{j}b_{j}$ of the present interval to the next.

The formula $\varphi_{\mathsf{IECM}}=\bigwedge\limits_{i\in\{1,\ldots,7\}}\varphi_{i}\wedge\bigwedge\limits_{p\in\mathsf{P}}\varphi^{p}_{8}$.

Appendix G Details on Expressiveness

Theorem G.1**.**

$\mathsf{MTL}+\mathsf{URat}\subseteq\mathsf{MTL}+\mathsf{Rat}$** 2. 2.

$\mathsf{MTL}+\mathsf{UM}\subseteq\mathsf{MTL}+\mathsf{MC}$**

Proof G.2**.**

We first prove $\mathsf{MTL}+\mathsf{URat}\subseteq\mathsf{MTL}+\mathsf{Rat}$.

Note that $\phi_{1}\mathsf{URat}_{I,\mathsf{re}}\phi_{2}$ is equivalent to $true\mathsf{URat}^{\mathsf{S}^{\prime}}_{I,\mathsf{re}^{\prime}}\phi_{2}$, where $\mathsf{re}^{\prime}$ is a regular expression obtained by conjuncting $\phi_{1}$ to all formulae $\psi$ occurring in the top level subformulae of $\mathsf{re}$, and $\mathsf{S}^{\prime}=\mathsf{S}\cup\{\phi_{1}\}$. For example, if we had $a\mathsf{URat}_{(0,1),(\mathsf{Rat}_{(1,2)}[\mathsf{Rat}_{(2,3)}(b+c)^{*}])}d$, then we obtain $true\mathsf{URat}_{(0,1),(a\wedge\mathsf{Rat}_{(1,2)}[\mathsf{Rat}_{(2,3)}(b+c)^{*}])}d$. When evaluated at a point $i$, the conjunction ensures that $\phi_{1}$ holds good at all the points between $i$ and $j$, where $\tau_{j}-\tau_{i}\in I$. To reduce $true\mathsf{URat}^{\mathsf{S}^{\prime}}_{I,\mathsf{re}^{\prime}}\phi_{2}$ to a $\mathsf{Rat}_{I}$ formula, we need the following lemma.

Lemma G.3**.**

Given any regular expression $R$, there exist finitely many regular expressions $R^{1}_{1},R^{1}_{2},\dots,R^{n}_{1},R^{n}_{2}$ such that $R=\bigcup_{i=1}^{n}R^{i}_{1}.R^{i}_{2}$. That is, for any string $\sigma\in R$ and for any decomposition of $\sigma$ as $\sigma_{1}.\sigma_{2}$, there exists some $i\leq n$ such that $\sigma_{1}\in R^{i}_{1}$ and $\sigma_{2}\in R^{i}_{2}$.

Proof G.4**.**

Let $\mathcal{A}$ be the minimal DFA for $R$. Let the number of states in $\mathcal{A}$ be $n$. The set of strings that leads to some state $q_{i}$ from the initial state $q_{1}$ is definable by a regular expression $R^{i}_{1}$. Likewise, the set of strings that lead from $q_{i}$ to some final state of $\mathcal{A}$ is also definable by some regular expression $R^{i}_{2}$. Given that there are $n$ states in the DFA $\mathcal{A}$, we have $L(\mathcal{A})=\bigcup_{i=1}^{n}R^{i}_{1}.R^{i}_{2}$. Consider any string $\sigma\in L(\mathcal{A})$, and any arbitrary decomposition of $\sigma$ as $\sigma_{1}.\sigma_{2}$. If we run the word $\sigma_{1}$ over $\mathcal{A}$, we might reach at some state $q_{i}$. Thus $\sigma_{1}\in L(R^{i}_{1})$. If we read $\sigma_{2}$ from $q_{i}$, it should lead us to one of the final states (by assumption that $\sigma\in R$) . Thus $\sigma_{2}\in L(R^{i}_{2})$.

Lets consider $true\mathsf{URat}_{I,\mathsf{re}^{\prime}}\phi_{2}$ when $I=[l,u)$.111If $I=[l,l]$, then $true\mathsf{URat}_{I,\mathsf{re}^{\prime}}\phi_{2}=\mathsf{Rat}_{[0,l]}\mathsf{re}^{\prime}.\phi_{2}$ If $true\mathsf{URat}_{[l,u),\mathsf{re}^{\prime}}\phi_{2}$ evaluates to true at a point $i$, we know that $\phi_{2}$ holds good at some point $j$ such that $\tau_{j}-\tau_{i}\in[l,u)$, and that $[\mathsf{Seg}(\mathsf{S}^{\prime},i,j)]^{\mathsf{single}}\cap L(\mathsf{re}^{\prime})\neq\emptyset$. By the above lemma, for any word $\sigma\in L(\mathsf{re}^{\prime})$, and any decomposition $\sigma=\sigma_{1}.\sigma_{2}$, there exist an $i\in\{1,2,\ldots,n\}$ such that $\sigma_{1}\in L(R^{i}_{1})$ and $\sigma_{2}\in L(R^{i}_{2})$. Thus we decompose at a point $j^{\prime}$ with every possible $R^{k}_{1}.R^{k}_{2}$ pair such that

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$\tau_{j^{\prime}}\in\tau_{i}+[l,u)$, $[\mathsf{TSeg}(\mathsf{S},(0,l),i)]^{\mathsf{single}}\cap L(R^{k}_{1})\neq\emptyset$,

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$[\mathsf{TSeg}(\mathsf{S},[l,u),i)]^{\mathsf{single}}\cap L(R^{k}_{2}).\phi_{2}.\Sigma^{*}\neq\emptyset$, where $\phi_{2}\in\mathsf{S}$.

Note that (i)$\phi_{2}$* holds good at the point $j$ such that $\tau_{j}\in[\tau_{i}+l,\tau_{i}+u)$, and, (ii) the expression $R^{k}_{2}$ evaluates to true in $[l,\tau_{j})$. We simply assert $\Sigma^{*}$ on the remaining part $(\tau_{j},u)$ of the interval.*

Thus $true\mathsf{URat}_{[l,u),\mathsf{re}^{\prime}}\phi_{2}\equiv\bigvee\limits_{i\in\{1,2\ldots,n\}}\mathsf{Rat}_{(0,l)}R^{i}_{1}\wedge\mathsf{Rat}_{[l,u)}(R^{i}_{2}.\phi_{2}.\Sigma^{*})$. 2. 2.

We first show that the $\mathsf{UM}$ modality can be captured by $\mathsf{MC}$. Consider any formula $\phi_{1}\mathsf{UM}_{I,\#\phi_{3}=k\%n}\phi_{2}$. At any point $i$ this formulae is true if and only if there exists a point $j$ in future such that $\tau_{j}-\tau_{i}\in I$ and the number of points between $i$ and $j$ where $\phi_{3}$ is true is $k\%n$, and $\phi_{1}$ is true at all points between $i$ and $j$. To count between $i$ and $j$, we can first count the behaviour $\phi_{3}$ from $i$ to the last point of the word, followed by the counting from $j$ to the last point of the word. Then we check that the difference between these counts to be $k\%n$.

Let $cnt_{\phi}(x,\phi_{3})=\{\phi\wedge\mathsf{MC}^{x\%n}_{(0,\infty)}(\phi_{3})\}$. Using this macro, $\phi_{1}\mathsf{UM}_{I,\#\phi_{3}=k\%n}\phi_{2}$ is equivalent to $\bigvee_{k_{1}=0}^{n-1}[\psi_{1}\vee\psi_{2}]$ where

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$\psi_{1}{=}\{cnt_{true}(k_{1},\phi_{3})\wedge(\phi_{1}\>\mathsf{U}_{I}cnt_{\phi_{2}\wedge\neg\phi_{3}}(k_{2},\phi_{3}))\}$,

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$\psi_{2}{=}{\{cnt_{true}(k_{1},\phi_{3})\wedge(\phi_{1}\>\mathsf{U}_{I}cnt_{\phi_{2}\wedge\phi_{3}}(k_{2}{-}1,\phi_{3}))\}}$,

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$k_{1}{-}k_{2}{=}k$**

The only difference between $\psi_{1},\psi_{2}$ is that in one, $\phi_{3}$ holds at position $j$, while in the other, it does not. The $k_{2}-1$ is to avoid the double counting in the case $\phi_{3}$ holds at $j$.

Appendix H $\mathsf{po}$-1-clock ATA to $1{-}\mathsf{TPTL}$

In this section, we explain the algorithm which converts a $\mathsf{po}$-1-clock ATA $\mathcal{A}$ into a $1{-}\mathsf{TPTL}$ formula $\varphi$ such that $L(\mathcal{A})=L(\varphi)$.

Step 1. Rewrite the transitions of the automaton. Each $\delta(s,a)$ can be written in an equivalent form as (i) $C_{1}\vee C_{2}$ or (ii) $C_{1}$ or (iii) $C_{2}$ where

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$C_{1}$ has the form $s\wedge\varphi_{1}$, where $\varphi_{1}\in\Phi(\downarrow s\cup\{a\}\cup X)$,

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$C_{2}$ has the form $\varphi_{2}$, where $\varphi_{2}\in\Phi(\downarrow s\cup\{a\}\cup X)$

In particular, if $s$ is the lowest location in the partial order, then $\varphi_{1},\varphi_{2}\in\Phi(\{a\}\cup X)$. Denote this equivalent form by $\delta^{\prime}(s,a)$.

For the example above, we obtain $\delta^{\prime}(s_{0},a)=(s_{0}\wedge(a\wedge x.s_{a}))\vee(a\wedge s_{\ell}),\delta^{\prime}(s_{0},b)=s_{0}\wedge b$, $\delta^{\prime}(s_{a},a)=(s_{a}\wedge x<1)\vee(x>1)$ $\delta^{\prime}(s_{\ell})=(s_{\ell}\wedge b)$ 2. 2.

Step 2. For each location $s$, construct $\Delta(s)$ which combines $\delta^{\prime}(s,a)$ for all $a\in\Sigma$, by disjuncting them first, and again putting them in the form in step 1. Thus, we obtain $\Delta(s)=\bigvee_{a}\delta^{\prime}(s,a)$ which can be written as a disjunction $D_{1}\vee D_{2}$ or simply $D_{1}$ or simply $D_{2}$ where $D_{1},D_{2}$ have the forms $s\wedge\varphi_{1}$ and $\varphi_{2}$ respectively, where $\varphi_{1},\varphi_{2}\in\Phi(\downarrow s\cup\Sigma\cup X)$.

For the example above, we obtain $\Delta(s_{0})=(s_{0}\wedge[(a\wedge x.s_{a}))\vee b])\vee(a\wedge s_{\ell})$, $\Delta(s_{a})=(s_{a}\wedge x<1)\vee(x>1)$ $\Delta(s_{\ell})=s_{\ell}\wedge b$. 3. 3.

Step 3. We now convert each $\Delta(s)$ into a normal form $\mathcal{N}(s)$. $\mathcal{N}(s)$ is obtained from $\Delta(s)$ as follows.

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If $s$ occurs in $\Delta(s)$, replace it with $\mathsf{O}s$.

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Replace each $s^{\prime}$ occurring in each $\Phi_{i}(\downarrow s)$ with $\mathsf{O}s^{\prime}$.

Let $\mathcal{N}(s)=\mathcal{N}_{1}\vee\mathcal{N}_{2}$, where $\mathcal{N}_{1},\mathcal{N}_{2}$ are normal forms. Intuitively, the states appearing on the right side of each transition are those which are taken up in the next step. The normal form explicitely does this, and takes us a step closer to $1{-}\mathsf{TPTL}$.

Continuing with the example, we obtain $\mathcal{N}(s_{0})=(\mathsf{O}s_{0}\wedge[(a\wedge x.\mathsf{O}s_{a}))\vee b])\vee(a\wedge\mathsf{O}s_{\ell})$ $\mathcal{N}(s_{a})=(\mathsf{O}s_{a}\wedge x<1)\vee(x>1)$ $\mathcal{N}(s_{\ell})=\mathsf{O}s_{\ell}\wedge b$. 4. 4.

Step 4.

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Start with the state $s_{n}$ which is the lowest in the partial order.

Let $\mathcal{N}(s_{n})=(\mathsf{O}s_{n}\wedge\varphi_{1})\vee\varphi_{2}$, where $\varphi_{1},\varphi_{2}\in\Phi(\Sigma,X)$. Solving $\mathcal{N}(s_{n})$, one obtains the solution $\mathsf{Beh}(s_{n})$ as $\varphi_{1}\>\mathsf{W}\varphi_{2}$ if $s_{n}$ is an accepting location, and as $\varphi_{1}\>\mathsf{U}^{\mathsf{ns}}\varphi_{2}$ if $s_{n}$ is non-accepting. Intuitively, $\mathsf{Beh}(s_{n})$ is the behaviour of $s_{n}$: that is, it describes the timed words that are accepted when we start in $s_{n}$. In the running example, we obtain $\mathsf{Beh}(s_{\ell})=b\>\mathsf{W}\bot=\Box^{\mathsf{ns}}b$ and $\mathsf{Beh}(s_{a})=(x<1)\>\mathsf{U}^{\mathsf{ns}}x>1$.

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Consider now some $\mathcal{N}(s_{i})=(\mathsf{O}s_{i}\wedge\varphi_{1})\vee\varphi_{2}$. First replace each $s^{\prime}$ in $\varphi_{i}$ with $\mathsf{Beh}(s^{\prime})$. $\mathsf{Beh}(s_{i})$ is then obtained as $\mathsf{Beh}(\varphi_{1})\>\mathsf{W}\mathsf{Beh}(\varphi_{2})$ if $s_{i}$ is an accepting location, and as $\mathsf{Beh}(\varphi_{1})\>\mathsf{U}^{\mathsf{ns}}\mathsf{Beh}(\varphi_{2})$ if $s_{i}$ is non-accepting.

Substituting $\mathsf{Beh}(s_{a})$ and $\mathsf{Beh}(s_{\ell})$ in $\mathcal{N}(s_{0})$, we obtain

$(\mathsf{O}s_{0}\wedge[(a\wedge x.\mathsf{O}\mathsf{Beh}(s_{a}))\vee b])\vee(a\wedge\mathsf{O}\mathsf{Beh}(s_{\ell}))$, solving which, we get

$\mathsf{Beh}(s_{0})=[(a\wedge x.\mathsf{O}\mathsf{Beh}(s_{a}))\vee b]\>\mathsf{W}(a\wedge\mathsf{O}\mathsf{Beh}(s_{\ell}))$.

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Thus, $\mathsf{Beh}(s_{0})$ which represents all timed words which are accepted when we start at $s_{0}$ is given by $((a\wedge(x.\mathsf{O}[(x<1)\>\mathsf{U}^{\mathsf{ns}}x>1]))\vee b)\>\mathsf{W}(a\wedge\mathsf{O}\Box^{\mathsf{ns}}b)$. The $1{-}\mathsf{TPTL}$ formula equivalent to $L(\mathcal{A})$ is then given by $\mathsf{Beh}(s_{0})$.

H.1 Correctness of Construction

The above algorithm is correct; that is, the 1${-}\mathsf{TPTL}$ formula $\mathsf{Beh}(s_{0})$ indeed captures the language accepted by the $\mathsf{po}$-1-clock ATA.

For the proof of correctness, we define a 1-clock ATA with a $\mathsf{TPTL}$ look ahead. That is, $\delta:S\times\Sigma\rightarrow\Phi(S\cup X\cup\chi(\Sigma\cup\{x\}))$, where $\chi(\Sigma\cup\{x\})$ is a $\mathsf{TPTL}$ formula over alphabet $\Sigma$ and clock variable $x$. We allow open $\mathsf{TPTL}$ formulae for look ahead; that is, one which is not of the form $x.\varphi$. All the freeze quantifications $x.$ lie within $\varphi$. The extension now allows to take a transition $(s,\nu)\rightarrow[\kappa\wedge\psi(x)]$, where $\psi(x)$ is a $\mathsf{TPTL}$ formula, if and only if the suffix of the input word with value of $x$ being $\nu$ satisfies $\psi(x)$. We induct on the level of the partial order on the states.

Base Case: Let the level of the partial order be zero. Consider 1-clock ATA having only one location $s_{0}$. Let the transition function be $\delta(s_{0},a)=\mathcal{B}_{a}(\psi_{a}(x),X,s_{0})$ for every $a\in\Sigma$. By our construction, we reduce $s_{0}$ into $\Delta(s_{0})=\bigvee\limits_{a\in\Sigma}[\mathcal{B}_{a}(\psi_{a}(x),X,\mathsf{O}(s_{0}))]$. Let $\Delta(s_{0})=\bigvee(P_{i}\wedge\psi_{i}(x)\wedge X_{i}\wedge\mathsf{O}s_{0})\vee\bigvee(Q_{j}\wedge\psi_{j}(x)\wedge X_{j})$. $\delta(s_{0},a)=s_{0}\wedge X_{1}\wedge\psi_{1}(x)$ specifies that the clock constraints $X_{1}$ are satisfied and the suffix satisfies the formula $\psi_{1}(x)$ on reading an $a$. Thus for this $\delta(s_{0},a)$, we have $\mathsf{O}s_{0}\wedge X_{1}\wedge\psi_{1}(x)\wedge a$ as a corresponding disjunct in $\Delta$ which specifies the same constraints on the word from the current point onwards. Thus the solution to the above will be satisfied at a point with some $x=\nu$ if and only if there is an accepting run from $s_{0}$ to the final configuration with $x=\nu$.

If the $s_{0}$ is a final location, the solution to this is $\varphi=\bigvee(P_{i}\wedge\psi_{i}(x)\wedge X_{i}\wedge\mathsf{O}s_{0})\>\mathsf{W}\bigvee(Q_{j}\wedge\psi_{j}(x)\wedge X_{j})$. If it is non-final, then it would be $\>\mathsf{U}$ instead of $\>\mathsf{W}$. Note that this implies that whenever $s_{0}$ is invoked with value of $x$ being $\nu$, the above formula would be true with $x=\nu$ thus getting an equivalent $1{-}\mathsf{TPTL}$ formulae.

Assume that for automata with $n-1$ levels in the partial order, we can construct an equivalent $1-\mathsf{TPTL}$ formula as per our construction. Consider an automaton with $n$ levels. Consider all the locations at the lowest level (that is, those location that can only call itself), $s_{0},\ldots,s_{k}$. Apply the same construction. As explained above, the constructed formulae, while eliminating a location will be true at a point if and only if there is an accepting run starting from the corresponding location with the same clock value. Let the formula obtained for any $s_{i}$ be $\varphi_{i}$.

The occurrence of an $s_{i}$ in any $\Delta(s_{i<n})$ can be substituted with $\varphi_{i}$ as a look ahead. This gives us an $n-1$ level 1-clock ATA with $\mathsf{TPTL}$ look ahead. By induction, we obtain that every 1-clock $\mathsf{po}$-ATA can be reduced to $1{-}\mathsf{TPTL}$ formulae.

Appendix I Proof of Lemma 4.4

Proof I.1**.**

Let $\rho$ be a timed word such that $\rho,i\models\mathsf{Rat}_{I}\mathsf{re}$. $\mathsf{re}$ can be either a simple star-free expression over $\Sigma$, or can be of the form $\mathsf{Rat}_{I^{\prime}}\mathsf{re}^{\prime}$ or $\mathsf{Rat}_{I_{1}}\mathsf{re}_{1}+\mathsf{Rat}_{I_{2}}\mathsf{re}_{2}$ or $\mathsf{Rat}_{I_{1}}\mathsf{re}_{1}.\mathsf{Rat}_{I_{2}}\mathsf{re}_{2}$ or $(\mathsf{Rat}_{I^{\prime}}\mathsf{re}^{\prime})^{*}$. Recursively, each of $\mathsf{re}^{\prime}$, $\mathsf{re}_{1},\mathsf{re}_{2}$ also can be expanded out as above. The idea of the proof is to eliminate “all levels” of $\mathsf{Rat}_{I}$ starting from the inner most one, by replacing them with $1{-}\mathsf{TPTL}$ formulae using structural induction. We first explain the proof in the case of $\mathsf{Rat}_{I}\mathsf{re}$, where $\mathsf{re}$ is a star-free expression over $\Sigma$, and then look at general cases.

We first consider the case when all intervals are bounded.

Given $\mathsf{Rat}_{I}\mathsf{re}$, and a point $i$ in a word $\rho$, $\mathsf{Rat}_{I}$ checks $\mathsf{re}$ at all points $j$ in $\rho$ such that $\tau_{j}-\tau_{i}\in I$. We first eliminate the interval $I$ from $\mathsf{Rat}_{I}\mathsf{re}$ by imagining a witness variable $w_{I}$ that evaluates to true at all points $j$ of $\rho$ such that $\tau_{j}-\tau_{i}\in I$. $w_{I}$ is used to cover all points distant $I$ from $i$. 2. 2.

We eliminate the interval $I$ in $\mathsf{Rat}_{I}$ by rewriting $\mathsf{Rat}_{I}\mathsf{re}$ as $\mathsf{Rat}((\neg w_{I})^{*}.\mathsf{re}.(\neg w_{I})^{*})$, which, when asserted at a point $i$, checks the truth of the expression $(\neg w_{I})^{*}.\mathsf{re}.(\neg w_{I})^{*}$ in the suffix from $i$. If $w_{I}$ indeed captures $\tau_{j}-\tau_{i}\in I$, then indeed we are checking $\mathsf{re}$ in the interval $I$. Let $\varphi_{\mathsf{re}}$ be an LTL formula that is equivalent to $\mathsf{re}$. This is possible since $L(\mathsf{re})$ is a star-free language. 3. 3.

We next replace $w_{I}$ by using a freeze clock variable $x$ which checks $x\in I$ whenever we assert $w_{I}$.

1. (a)

We will look at the simplest case when $\mathsf{re}$ is a regular expression over $\Sigma$. Let $\varphi_{\mathsf{re}}$ be the LTL formula equivalent to $\mathsf{re}$.

1. i.

Let $\mathsf{re}=a$ for $a\in\Sigma$. We expand the alphabet by allowing proposition $x\in I$ (and its negation $x\notin I$). The formula $\psi=((x\notin I)\>\mathsf{U}\{\varphi_{\mathsf{re}}\wedge[(x\in I)\wedge(\mathsf{O}(\Box(x\notin I)))]\})$ is then an LTL formula that says that there is a single point in the region $I$, and $a$ holds at that point. Then $\psi_{\mathsf{re}}=x.\psi$ is a 1-$\mathsf{TPTL}$* formula that captures $\mathsf{Rat}_{I}a$.* 2. ii.

Let $\mathsf{re}=a.b$ for $b\in\Sigma$. Then we inductively assume LTL formulae $\varphi_{a}$ and $\varphi_{b}$ that capture $a$ and $b$. As above, we allow the proposition $x\in I$. Then the formula $\psi_{\mathsf{re}}=x.\{(x\notin I)\>\mathsf{U}[(x\in I\wedge\varphi_{a}\wedge\mathsf{O}(x\in I\wedge\varphi_{b}\wedge\mathsf{O}(\Box(x\notin I))))]\}$ asserts the existence of two points in the interval $I$ respectively satisfying in order, $a$ and $b$. This argument can be extended to work for any finite concatenation $\mathsf{re}=a_{1}.a_{2}.\dots.a_{n}$. 3. iii.

Let $\mathsf{re}=(\mathsf{re}_{1})^{*}$ be rational expression over $\Sigma$. Let $\varphi_{\mathsf{re}_{1}}$ be the LTL formula equivalent to $\mathsf{re}_{1}$. Then the formula $x.[(x\notin I)\>\mathsf{U}((x\in I)\wedge\Box[(x\in I)\rightarrow\varphi_{\mathsf{re}_{1}}])]$ is a 1-$\mathsf{TPTL}$* formula that asserts the formula $\varphi_{\mathsf{re}_{1}}$ at all points in $I$.* 4. iv.

Let $\mathsf{re}=\mathsf{re}_{1}+\mathsf{re}_{2}$ be a rational expression over $\Sigma$. Let $\varphi_{\mathsf{re}_{1}}$ and $\varphi_{\mathsf{re}_{2}}$ be LTL formulae equivalent to $\mathsf{re}_{1},\mathsf{re}_{2}$. Then $((x\notin I)\>\mathsf{U}\{(\varphi_{\mathsf{re}_{1}}\vee\varphi_{\mathsf{re}_{2}})\wedge[(x\in I)\wedge(\mathsf{O}(\Box(x\notin I)))]\})$ is then an LTL formula that says that there is a single point in the region $I$, and one of $\varphi_{\mathsf{re}_{1}},\varphi_{\mathsf{re}_{2}}$ holds at that point. 2. (b)

Finish the base case of the structural induction, where $\mathsf{re}$ was a rational expression over $\Sigma$, we now move on to general cases.

1. i.

Let us now consider the case when we have a formula $\mathsf{Rat}_{I_{1}}[\mathsf{Rat}_{I_{2}}\mathsf{re}]$.

Then we first obtain as seen above, $x.\zeta_{\mathsf{re}}$ equivalent to $\mathsf{Rat}_{I_{2}}\mathsf{re}$. Let $x.\zeta_{\mathsf{re}}=w$. Then, $\zeta_{out}=((x\notin I_{1})\>\mathsf{U}\{w\wedge(x\in I_{1})\wedge\mathsf{O}(\Box(x\notin I_{1}))\})$ is an LTL formula which asserts the existence of a single point lying in the interval $I_{1}$ where $w$ is true. Then $x.\zeta_{out}$ is a 1-$\mathsf{TPTL}$* formula over $\Sigma$ that is equivalent to $\mathsf{Rat}_{I_{1}}[\mathsf{Rat}_{I_{2}}\mathsf{re}]$.* 2. ii.

If we have $\mathsf{Rat}_{I}[\mathsf{Rat}_{I_{1}}\mathsf{re}_{1}.\mathsf{Rat}_{I_{2}}\mathsf{re}_{2}]$, then let $w_{1}=x.\zeta_{\mathsf{re}_{1}},w_{2}=x.\zeta_{\mathsf{re}_{2}}$. Then the formula $x.\{(x\notin I)\>\mathsf{U}[(x\in I\wedge w_{1}\wedge\mathsf{O}(x\in I\wedge w_{2}\wedge\mathsf{O}(\Box(x\notin I))))]\}$ asserts the existence of two points in the interval $I$ respectively satisfying in order, $\mathsf{Rat}_{I_{1}}\mathsf{re}_{1}$ and $\mathsf{Rat}_{I_{2}}\mathsf{re}_{2}$. 3. iii.

If we have $\mathsf{Rat}_{I}[(\mathsf{Rat}_{I_{1}}\mathsf{re}_{1})^{*}]$, then let $w_{1}=x.\zeta_{\mathsf{re}_{1}}$.

Let $w=(x\notin I)\>\mathsf{U}((x\in I)\wedge\Box[(x\in I)\rightarrow w_{1}])$. Then $x.w$ is a 1-$\mathsf{TPTL}$* formula that asserts that at all points in $I$, the formula $\mathsf{Rat}_{I_{1}}\mathsf{re}_{1}$ evaluates to true.* 4. iv.

If we have $\mathsf{Rat}_{I}(\mathsf{Rat}_{I_{1}}\mathsf{re}_{1}+\mathsf{Rat}_{I_{2}}\mathsf{re}_{2})$, then let $w_{1}=x.\zeta_{\mathsf{re}_{1}}$ and $w_{2}=x.\zeta_{\mathsf{re}_{2}}$. Then $x.\{(x\notin I)\>\mathsf{U}((w_{1}\vee w_{2})\wedge x\in I\wedge(\mathsf{O}(\Box(x\notin I))))\}$ is a 1-$\mathsf{TPTL}$* formula that checks that $\mathsf{Rat}_{I_{1}}\mathsf{re}_{1}$ or $\mathsf{Rat}_{I_{2}}\mathsf{re}_{2}$ evaluates to true at the single point in the interval $I$.*

Note that the boolean combinations like conjunction, disjunction and unary operations like negation can be handled in a straightforward way, once we are done with the above. While encountering boolean combinations, we simply combine the 1-$\mathsf{TPTL}$* formulae obtained so far.*

In case $I$ is an unbounded interval, then we need not concatenate $\mathsf{O}(\Box(x\notin I))$ at the end, since the time stamps of all points in the suffix lie in $I$. The rest of the proof is the same. ∎

Appendix J Proof of Lemma 4.5

The Main Idea: Let $\mathcal{A}$ be a $\mathsf{po}$-1-clock ATA with locations $S=\{s_{0},s_{1},\dots,s_{n}\}$. Let $K$ be the maximal constant used in the guards $x\sim c$ occurring in the transitions. Let $R_{2i}=[i,i],R_{2i+1}=(i,i+1),0\leq i<K$ and $R^{+}_{K}=(K,\infty)$ be the regions $\mathcal{R}$ of $x$. Let $R_{h}\prec R_{k}$ denote that region $R_{h}$ precedes region $R_{k}$. For each location $s$, $\mathsf{Beh}(s)$ as seen above (also Figure 4) gives the timed behaviour starting at $s$, using constraints $x\sim c$ since the point where $x$ was frozen. In example 4.2, $\mathsf{Beh}(s_{a}){=}(x<1)\>\mathsf{U}^{\mathsf{ns}}(x>1)$, allows symbols $a,b$ as long as $x<1$ keeping the control in $s_{a}$, has no behaviour at $x=1$, and allows control to leave $s_{a}$ when $x>1$. For any $s$, we “distribute” $\mathsf{Beh}(s)$ across regions by untiming it. In example 4.2, $\mathsf{Beh}(s_{a})$ is $\Box^{\mathsf{ns}}(a\vee b)$ for regions $R_{0},R_{1}$, it is $\bot$ for $R_{2}$ and is $(a\vee b)$ for $R^{+}_{1}$. Given any $\mathsf{Beh}(s)$, and a pair of regions $R_{j}\preceq R_{k}$, such that $s$ has a non-empty behaviour in region $R_{j}$, and control leaves $s$ in $R_{k}$, the untimed behaviour of $s$ between regions $R_{j},\dots,R_{k}$ is written as LTL formulae $\varphi_{j},\dots,\varphi_{k}$. This results in a “behaviour description” (or $\mathsf{BD}$ for short) denoted $\mathsf{BD}(s,R_{j},R_{k})$ : this is a $2K+1$ tuple with $\mathsf{BD}[R_{l}]=\varphi_{l}$ for $j\leq l\leq k$, and $\mathsf{BD}[R]=\top$ denoting “dont care” for the other regions. Each LTL formula $\mathsf{BD}(s,R_{j},R_{k})[R_{i}]$ (or $\mathsf{BD}[R_{i}]$ when $s,R_{j},R_{k}$ are clear) is replaced with a star-free rational expression denoted $\mathsf{re}(\mathsf{BD}(s,R_{j},R_{k})[R_{i}])$. Then $\mathsf{BD}(s,R_{j},R_{k})$ is transformed into a $\mathsf{SfrMTL}$ formula $\varphi(s,R_{j},R_{k})=\bigwedge_{j\leq g\leq k}\mathsf{Rat}_{R_{g}}\mathsf{re}(\mathsf{BD}(s,R_{j},R_{k})[R_{g}])$. The language accepted by the $\mathsf{po}$-1-clock ATA $\mathcal{A}$ is then given by $\bigvee_{0\leq j\leq k\leq 2K}\varphi(s_{0},R_{j},R_{k})$ where $s_{0}$ is the initial location, and the word is accepted while in region $R_{k}$. This disjunction allows all possible accepting behaviours from the initial location $s_{0}$.

Each location $s$ is associated with a set of $\mathsf{BD}$s. Let $\mathsf{BDSet}(s)$ denote the of $\mathsf{BD}$s that are associated with $s$. If $s$ is the lowest location in the partial order, then $\mathsf{BDSet}(s)=\{\mathsf{BD}(s,R_{i},R_{j})\mid R_{i}\preceq R_{j}\}$.

Computing $\mathsf{BD}(s,R_{i},R_{j})$ for a location $s$ and pair of regions $R_{i}\preceq R_{j}$. The proof proceeds by first computing $\mathsf{BD}(s,R_{i},R_{j})$ for locations $s$ which are lowest in the partial order, followed by computing $\mathsf{BD}(s^{\prime},R_{i},R_{j})$ for locations $s^{\prime}$ which are higher in the order. For any location $s$, $\mathsf{Beh}(s)$ has the form $\varphi_{1}\>\mathsf{W}\varphi_{2}$ or $\varphi_{1}\>\mathsf{U}^{\mathsf{ns}}\varphi_{2}$, or $\varphi$, where $\varphi,\varphi_{1},\varphi_{2}$ are disjunctions of conjunctions over $\Phi(S\cup\Sigma\cup X)$, where $S$ is the set of locations with or without the binding construct $x.$, and $X$ is a set of clock constraints of the form $x\sim c$. Each conjunct has the form $\psi\wedge x\in R$ where $\psi\in\Phi(\Sigma\cup S)$ and $R\in\mathcal{R}$. Let $\varphi_{1}=\bigvee(P_{i}\wedge C_{i}),\varphi_{2}=\bigvee(Q_{j}\wedge E_{j})$ where $P_{i},Q_{j}\in\Phi(\Sigma\cup S)$ and $C_{i},E_{j}\in\mathcal{R}$. Let $\mathcal{C}$ and $\mathcal{E}$ be a shorthand notation to represent any $C_{k},E_{l}$.

For $R_{i}\preceq R_{j}$, and a location $s$, $\mathsf{BD}(s,R_{i},R_{j})$ is empty if $\mathsf{Beh}(s)$ has no constraint $x\in R_{i}$ occurring in $\mathcal{C},\mathcal{E}$, and if control cannot exit $s$ in $R_{j}$. If $\mathsf{Beh}(s)$ has no $\>\mathsf{U},\mathsf{W}$ modalities, then $\mathsf{BD}(s,R_{i},R_{i})$ is computed when $\mathsf{Beh}(s)=\bigvee(Q_{j}\wedge E_{j})$ and there is some $E_{l}$ with $x\in R_{i}$. In this case, $\mathsf{BD}(s,R_{i},R_{i})[R_{i}]=Q_{l}$, and the remaining entries are $\top$ representing “dont care”. If $\mathsf{Beh}(s)$ has $\>\mathsf{U},\mathsf{W}$ modalities, then $\mathsf{BD}(s,R_{i},R_{j})$ is computed when (1) there is a constraint $x\in R_{i}$ in $\mathcal{C}$ or $\mathcal{E}$ (this allows us to start observing the behaviour in region $R_{i}$) (2) there is a constraint $x\in R_{j}$ in some $\mathcal{E}$ (this allows us to exit the control location $s$ while in region $R_{j}$). If so, the $\mathsf{BD}$ $\mathsf{Beh}(s,R_{i},R_{j})$ is a $2K+1$ tuple with (i) formula $\top$ in regions $R_{0},\dots,R_{i-1},R_{j+1},\dots,R_{K}^{+}$ (denoting dont care), (ii)If $C_{k}=E_{l}=(x\in R_{j})$ for some $C_{k},E_{l}$, then the LTL formula in region $R_{j}$ is $P_{k}\>\mathsf{U}Q_{l}$ if $s$ is not an accepting location, and is $P_{k}\>\mathsf{W}Q_{l}$ if $s$ is an accepting location, (iii)If no $C_{k}$ is equal to any $E_{l}$ for any $k,l$, and if $E_{l}=(x\in R_{j})$ for some $l$, then the formula in region $R_{j}$ is $Q_{l}$. If $C_{m}=(x\in R_{i})$ for some $m$, then the formula for region $R_{i}$ is $\Box^{\mathsf{ns}}P_{m}$. If there is some $C_{h}=(x\in R_{w})$ for $i<w<j$, then the formula in region $R_{w}$ is $\Box^{\mathsf{ns}}P_{h}\vee\epsilon$, where $\epsilon$ signifies the fact that there may be no points in regions $R_{w}$. If there are no $C_{m}$’s such that $C_{m}=(x\in R_{w})$ for $R_{i}\prec R_{w}\prec R_{j}$, then the formulae in region $R_{w}$ is $\epsilon$. We allow $\epsilon$ as a special symbol in LTL to signify that there is no behaviour in a region.

$\mathsf{BD}(s,R_{i},R_{j})$** for location $s$ lowest in po**. Let $s$ be a location that is lowest in the partial order. The locations $s_{\ell},s_{a}$ in Example 4.2 are lowest in the partial order, and $\mathsf{Beh}(s_{\ell})=b\>\mathsf{W}\bot=\Box^{\mathsf{ns}}b$, $\mathsf{Beh}(s_{a}){=}[(a\vee b)\wedge(x<1)]\>\mathsf{U}^{\mathsf{ns}}[(a\vee b)\wedge(x>1)]$. In general, if $s$ is the lowest in the partial order, then $\mathsf{Beh}(s)$ has the form $\varphi_{1}\>\mathsf{W}\varphi_{2}$ or $\varphi_{1}\>\mathsf{U}^{\mathsf{ns}}\varphi_{2}$, or $\varphi$, where $\varphi,\varphi_{1},\varphi_{2}$ are disjunctions of conjunctions over $\Phi(\Sigma\cup X)$. Each conjunct has the form $\psi\wedge x\in R$ where $\psi\in\Phi(\Sigma)$ and $R\in\mathcal{R}$. In example 4.2, the regions are $R_{0}=[0,0],R_{1}=(0,1),R_{2}=[1,1],R^{+}_{1}=(1,\infty)$. $\mathsf{Beh}(s_{\ell},R_{1},R^{+}_{1})=(\top,\Box^{\mathsf{ns}}b,\Box^{\mathsf{ns}}b\vee\epsilon,b\>\mathsf{W}\bot)$, and $\mathsf{Beh}(s_{a},R_{0},R^{+}_{1})=(\Box^{\mathsf{ns}}(a\vee b),\Box^{\mathsf{ns}}(a\vee b)\vee\epsilon,\epsilon,(a\vee b))$. If $\epsilon$ in the sole entry in a region, it represents that there is no behaviour in that region. If $\epsilon$ is a disjunct $\psi\vee\epsilon$, then it represents a possibility of no behaviour, or a behaviour $\psi$.

Using the $\mathsf{BD}$s of $s_{a}$, we can write the $\mathsf{SfrMTL}$ formula that describes the behaviour of $s_{a}$. This fomula is given by $\psi(s_{a})=\varphi_{R_{0}}(s_{a})\wedge\varphi_{R_{1}}(s_{a})\wedge\varphi_{R_{2}}(s_{a})\wedge\varphi_{R^{+}_{1}}(s_{a})$, where each $\varphi_{R_{i}}$ describes the behaviour starting from region $R_{i}$, while in location $s_{a}$. For a fixed region $R_{i}$, $\varphi_{R_{i}}(s_{a})$ is $\bigwedge_{R_{g}\prec R_{i}}\mathsf{Rat}_{R_{g}}\epsilon\wedge\mathsf{Rat}_{R_{i}}\Sigma^{+}\rightarrow\{\bigvee_{R_{i}\prec R_{j}}\varphi(s_{a},R_{i},R_{j})\}$, where $\varphi(s_{a},R_{i},R_{j})$ is described above. $\mathsf{Rat}_{R_{g}}\epsilon$ represents that there is no behaviour in $R_{g}$. Recall that $\varphi(s_{a},R_{i},R_{j})$ describes a possible behaviour of $s_{a}$ that starts at $R_{i}$ and ends in $R_{j}$. For instance, $\varphi_{R_{0}}(s_{a})$ is $\mathsf{Rat}_{R_{0}}\Sigma^{+}\rightarrow\{(\mathsf{Rat}_{R_{0}}(a+b)^{*}\wedge\mathsf{Rat}_{R_{1}}[(a+b)^{*}+\epsilon]\wedge\mathsf{Rat}_{R_{2}}\epsilon\wedge\mathsf{Rat}_{R^{+}_{1}}(a+b)^{*})\}$ while $\varphi_{R_{1}}(s_{a})$ is $\mathsf{Rat}_{R_{0}}\epsilon\wedge\mathsf{Rat}_{R_{1}}\Sigma^{+}\rightarrow\{(\mathsf{Rat}_{R_{1}}(a+b)^{*}\wedge\mathsf{Rat}_{R_{2}}\epsilon\wedge\mathsf{Rat}_{R^{+}_{1}}(a+b)^{*})\}$. Similarly, $\varphi_{R_{2}}(s_{a})$ is empty since $s_{a}$ has no behaviour in $R_{2}$. Finally, $\varphi_{R^{+}_{1}}(s_{a})$ is $\bigwedge_{R_{g}\prec R^{+}_{1}}\mathsf{Rat}_{R_{g}}\epsilon\wedge\mathsf{Rat}_{R^{+}_{1}}\Sigma^{+}\rightarrow\mathsf{Rat}_{R^{+}_{1}}(a+b)^{*}$. In a similar manner, we can write the $\mathsf{SfrMTL}$ formula $\psi_{s_{\ell}}$ that describes the behaviour of $s_{\ell}$ across regions.

$\mathsf{BD}(s,R_{i},R_{j})$** for a location $s$ which is higher up**. If $s$ is not the lowest in the partial order, then $\mathsf{Beh}(s)$ has locations $s^{\prime}\in\downarrow s$. $s^{\prime}$ occurs as $\mathsf{O}(s^{\prime})$ or $x.\mathsf{O}(s^{\prime})$ in $\mathsf{Beh}(s)$. We now elaborate the operations needed to combine $\mathsf{BD}$s.

Boolean Combinations of $\mathsf{BD}$s. Let $s_{1},s_{2}$ be two locations of the $\mathsf{po}$-1-clock ATA $\mathcal{A}$. Assume $\mathsf{Beh}(s_{1})=\varphi_{1}\>\mathsf{U}^{\mathsf{ns}}\varphi_{2}$ or $\varphi_{1}\>\mathsf{W}\varphi_{2}$ and $\mathsf{Beh}(s_{2})=\psi_{1}\>\mathsf{U}^{\mathsf{ns}}\psi_{2}$ or $\psi_{1}\>\mathsf{W}\psi_{2}$. We have already seen how to handle $x.\mathsf{O}\mathsf{Beh}(s_{1})$ or $x.\mathsf{O}\mathsf{Beh}(s_{2})$. So let us assume $s_{1},s_{2}$ appear in $\mathsf{Beh}(s)$ as $\mathsf{O}\mathsf{Beh}(s_{1})$ and $\mathsf{O}\mathsf{Beh}(s_{2})$.

Consider $\mathsf{BDSet}(s_{1})$ and $\mathsf{BDSet}(s_{2})$, and consider any pair of $\mathsf{BD}$s, say $\mathsf{BD}(s_{1},R_{i},R_{j})$ and $\mathsf{BD}(s_{2},R_{i},R_{j})$ from these respectively. The boolean operations are defined for each pair taken from $\mathsf{BDSet}(s_{1})$ and $\mathsf{BDSet}(s_{2})$.

Take $\mathsf{BD}(s_{1},R_{i},R_{j})$ and $\mathsf{BD}(s_{2},R_{i},R_{k})$ respectively from $\mathsf{BDSet}(s_{1})$ and $\mathsf{BDSet}(s_{2})$. We now define boolean operations $\wedge$ and $\vee$ on these $\mathsf{BD}$s.

The $\mathsf{BDSet}$ for $s_{1}\wedge s_{2}$: Consider $\mathsf{BD}_{1}=\mathsf{BD}(s_{1},R_{i},R_{j})$ and $\mathsf{BD}_{2}=\mathsf{BD}(s_{2},R_{i},R_{k})$, both which describe behaviours of $s_{1},s_{2}$ starting in region $R_{i}$. Assume $R_{j}\prec R_{k}$ (the case of $R_{k}\prec R_{j}$ is similar). To obtain a $\mathsf{BD}$ conjuncting these two, starting in region $R_{i}$, we do the following. Construct $\mathsf{BD}^{\prime}$ by conjuncting the entries of $\mathsf{BD}_{1},\mathsf{BD}_{2}$ component wise. This will ensure that we take the possible behaviour of $\mathsf{Beh}(s_{1})$ at region $R_{i}$ and conjunct it with the possible behaviour of $\mathsf{Beh}(s_{2})$ in the same region. $BD^{\prime}\in\mathsf{BDSet}(s_{1}\wedge s_{2})$. In a similar way, we can also compute the $\mathsf{BDSet}(s_{1}\vee s_{2})$.

Elimination of $\mathsf{O}\mathsf{Beh}(s^{\prime})$ from $\mathsf{BD}(s,R_{i},R_{j})$

Given any $\mathsf{Beh}(s)$ of the form $[\bigvee_{i}(P_{i}\wedge C_{i})]\>\mathsf{U}^{\mathsf{ns}}[\bigvee_{j}(Q_{j}\wedge E_{j})]$ or $[\bigvee_{i}(P_{i}\wedge C_{i})]\>\mathsf{W}[\bigvee_{j}(Q_{j}\wedge E_{j})]$ with $P_{i},Q_{j}\in\Phi(\Sigma\cup S)$, and $C_{i},E_{j}$ are clock constraints of the form $x\in R$. Assume that we have calculated $\mathsf{BD}(s^{\prime},R,R^{\prime})$ for all $s^{\prime}\in\downarrow s$ and all regions $R,R^{\prime}$. There might be some propositions of the form $\mathsf{O}\mathsf{Beh}(s^{\prime})$ as a conjunct in some entries of $\mathsf{BD}(s,R_{i},R_{j})$. This occurrence of $\mathsf{O}\mathsf{Beh}(s^{\prime})$ is eliminated by “stitching” the behaviour of $s^{\prime}$ with $\mathsf{BD}(s,R_{i},R_{j})$ as follows:

• •

We consider three cases here, depending on how $\mathsf{O}\mathsf{Beh}(s^{\prime})$ occurs in $\mathsf{BD}_{1}=\mathsf{BD}(s,R_{i},R_{j})$. As a first case, let $\mathsf{BD}_{1}=(X_{0},\ldots,X_{g-1},Q_{j}\wedge\mathsf{O}(\mathsf{Beh}(s^{\prime})),X_{g+1},\ldots,X_{2K})$.

To eliminate $\mathsf{O}\mathsf{Beh}(s^{\prime})$ from $\mathsf{BD}_{1}$, we first recall that $s^{\prime}\in\downarrow s$ and that $\mathsf{BD}(s^{\prime},R_{k},R_{l})$ has been computed for all regions $R_{k},R_{l}$. $\mathsf{Beh}(s)$ will not occur in any of these $\mathsf{BD}$s corresponding to $s^{\prime}$. 2. 2.

The first thing to check is which region ($R_{g}$ or later) where the next point can be enabled, based on the behaviour of $s^{\prime}$. There are $2K-g+1$ possibilities, depending on which region $\geq g$ the next point lies with respect to $Q_{j}\wedge\mathsf{O}(\mathsf{Beh}(s^{\prime}))$.

• –

Suppose the next point can be taken in $R_{g}$ itself. This means that from the next point, all the possible behaviours described by any of the $\mathsf{BD}$’s $\mathsf{BD}(s^{\prime},R_{g},R_{h})$ will apply along with $\mathsf{BD}_{1}$. We define a binary operation $\mathsf{combine}$ which combines two $\mathsf{BD}$s, $\mathsf{BD}_{1}=\mathsf{BD}(s,R_{i},R_{j})$ and $\mathsf{BD}_{2}=\mathsf{BD}(s^{\prime},R_{g},R_{h})$, producing a new $\mathsf{BD}$, $\mathsf{BD}_{3}=\mathsf{combine}(\mathsf{BD}_{1},\mathsf{BD}_{2})$. To combine the behaviours from the point where $\mathsf{Beh}(s^{\prime})$ is encountered, we substitute $\mathsf{O}\mathsf{Beh}(s^{\prime})$ with the $\mathsf{LTL}$ formula asserted at region $R_{g}$ in $\mathsf{BD}_{2}$. If $\mathsf{BD}_{2}[R_{g}]$ represents the $g$th component, then we replace $\mathsf{O}\mathsf{Beh}(s^{\prime})$ in $\mathsf{BD}_{1}[R_{g}]$ with $\mathsf{BD}_{2}[R_{g}]$. Thus, $\mathsf{BD}_{3}[R_{g}]=Q_{j}\wedge\mathsf{BD}_{2}[R_{g}]$. For all $R_{w}\prec R_{g}$, $\mathsf{BD}_{3}[R_{w}]=\mathsf{BD}_{1}[R_{w}]$. For all $R_{w}$ such that $R_{g}\prec R_{w}$, $\mathsf{BD}_{3}[R_{w}]=\mathsf{BD}_{1}[R_{w}]\wedge\mathsf{BD}_{2}[R_{w}]$.

• –

Now consider the case when the next point is taken a region $>R_{g}$. The next point can occur in region $R_{g+1}$ or higher. Let $b\in\{g+1,\ldots,2K\}$, and assume that the next point where $\mathsf{Beh}(s^{\prime})$ has a behaviour is in $R_{b}$. Then given $\mathsf{BD}_{1}=\mathsf{BD}(s,R_{i},R_{j})$ and $\mathsf{BD}_{2}=\mathsf{BD}(s^{\prime},R_{g},R_{h})$ such that $\mathsf{BD}_{2}[g+1],\dots,\mathsf{BD}_{2}[b-1]=\epsilon$, we obtain $\mathsf{BD}_{3}$ as follows. For all $R_{w}\prec R_{g}$, $\mathsf{BD}_{3}[R_{w}]=\mathsf{BD}_{1}[R_{w}]$. For $w=g$, $\mathsf{BD}_{3}[R_{g}]=Q_{j}\wedge\Box\bot$. The conjunction with $\Box\bot$ signifies that the next point in $R_{g}$ is not available for $s^{\prime}$, since $s^{\prime}$ has no behaviour in $R_{g}$. For all $b>w>g$, $\mathsf{BD}_{3}[R_{w}]=\mathsf{BD}_{1}[R_{w}]\wedge\epsilon=\epsilon$. This implies the next point from where the assertion $Q_{j}\wedge\mathsf{O}(\mathsf{Beh}(s^{\prime}))$ was made is in a region $\geq R_{b}$. For all $w\geq b$, $\mathsf{BD}_{3}[R_{w}]=\mathsf{BD}_{1}[R_{w}]\wedge\mathsf{BD}_{2}[R_{w}]$. This combines the assertions of both the behaviours from the next point onwards.

• •

As a second case, consider $\mathsf{BD}_{1}=[X_{0},\ldots,X_{g-1},\Box^{\mathsf{ns}}(P_{j}\wedge\mathsf{O}(\mathsf{Beh}(s^{\prime}))),X_{g+1},\ldots,X_{2K}]$. Elimination of $\mathsf{O}\mathsf{Beh}(s^{\prime})$ in this case is similar to case 1.

• •

As the third case, let $\mathsf{BD}_{1}{=}[X_{0},\ldots,X_{g-1},P_{i}{\wedge}\mathsf{O}(\mathsf{Beh}(s_{1})){\>\mathsf{U}^{\mathsf{ns}}}Q_{j}{\wedge}\mathsf{O}(\mathsf{Beh}(s_{2})),X_{g+1},\ldots,X_{2K}]$, and we have to eliminate both $\mathsf{O}\mathsf{Beh}(s_{1})$ and $\mathsf{O}\mathsf{Beh}(s_{2})$. Either $Q_{j}\wedge\mathsf{O}(\mathsf{Beh}(s_{2}))$ is true at the present point or, $P_{i}\wedge\mathsf{O}(\mathsf{Beh}(s_{1}))$ is true until some point in the future within the region $R_{g}$, at which point, $Q_{j}\wedge\mathsf{O}(\mathsf{Beh}(s_{2}))$ becomes true. Thus, $\mathsf{BD}_{1}$ can be replaced with two $\mathsf{BD}$s

• –

$\mathsf{BD}^{\prime}_{1}{=}[X_{0},\ldots,X_{g-1},Q_{j}{\wedge}\mathsf{O}(\mathsf{Beh}(s_{2})),X_{g+1},\ldots,X_{2K}]$, and

• –

$\mathsf{BD}^{\prime\prime}_{1}{=}[X_{0},\ldots,X_{g-1}$, $P_{i}{\wedge}\mathsf{O}(\mathsf{Beh}(s_{1})){\>\mathsf{U}}Q_{j}{\wedge}\mathsf{O}(\mathsf{Beh}(s_{2})),X_{g+1},\ldots,X_{2K}]$.

Elimination of $\mathsf{O}\mathsf{Beh}(s_{2})$ is done from $\mathsf{BD}^{\prime}_{1}$ as seen in case 1. Consider $\mathsf{BD}^{\prime\prime}_{1}$ which guarantees that the next point from which the assertion $P_{i}\wedge\mathsf{O}(\mathsf{Beh}(s_{1}))\>\mathsf{U}Q_{j}\wedge\mathsf{O}(\mathsf{Beh}(s_{2}))$ is made is within region $R_{g}$, and that $\mathsf{O}(\mathsf{Beh}(s_{1}))$ is called for the last time within $R_{g}$. $\mathsf{BD}^{\prime\prime}_{1}$ has to be combined with any $\mathsf{BD}(s_{1},R_{g},R_{h})$, which has a starting behaviour of $s_{1}$ from region $R_{g}$. $s_{2}$ can have an enabled transition from any point either within region $R_{g}$ or a succeeding region.

• –

Consider the case where $s_{2}$ has an enabled transition from within the region $R_{g}$. In this case, we have to combine $\mathsf{BD}^{\prime\prime}_{1}$ with some $\mathsf{BD}_{3}=\mathsf{BD}(s_{1},R_{g},R_{h})$ and with some $\mathsf{BD}_{4}=\mathsf{BD}(s_{2},R_{g},R_{j})$. Let $\mathsf{BD}_{3}=(Y_{0},\ldots,Y_{2K})$ and let $\mathsf{BD}_{4}=(Z_{0},\dots,Z_{2K})$. We now show to combine $\mathsf{BD}^{\prime\prime}_{1},\mathsf{BD}_{3}$ and $\mathsf{BD}_{4}$ obtaining a $\mathsf{BD}$ $(A_{0},\ldots,A_{2K})$.

For every $w<g$, $A_{w}=X_{w}$. For $w=g$, $A_{g}$ is obtaining by replacing $\mathsf{O}\mathsf{Beh}(s_{1})$ with $Y_{g}$ and $\mathsf{O}\mathsf{Beh}(s_{2})$ with $Z_{g}$ For all $w>g$, $A_{w}=X_{w}\wedge Y_{w}\wedge Z_{w}$.

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Now consider the case where $s_{2}$ has an enabled transition from a region $R_{b}$ such that $R_{g}\prec R_{b}$. In this case, $A_{w}=X_{w}$ for $w<g$. The main difference with the earlier case is that we have to assert that from the last point in $R_{g}$, the next point only occurs in the region $R_{b}$. Thus all the regions between $R_{g}$ and $R_{b}$ should be $\epsilon$ in $(A_{0},\ldots,A_{2K})$. That is, $A_{w}=\epsilon$ for $g<w<b$. For $w=g$, $A_{g}=(P_{i}\wedge Y_{g})\>\mathsf{U}(Q_{j}\wedge\Box\bot)$, where $P_{i},Q_{j}$ are obtained from $\mathsf{BD}^{\prime\prime}_{1}[R_{g}]$. Here again, conjuncting $\Box\bot$ with $Q_{j}$ signifies that the next point is not enabled for $s_{2}$. Finally, for $w\geq b$, $A_{w}=X_{w}\wedge Y_{w}\wedge Z_{w}$.

Note that elimination of $\mathsf{O}Beh(s^{\prime})$ from any $\mathsf{BD}$ in the set $\mathsf{BDSet}(s)$ results in stitching some $\mathsf{BD}$ from $\mathsf{BDSet}(s^{\prime})$ to certain elements of $\mathsf{BDSet}(s)$. At the end of the stitching, we obtain $\mathsf{BDSet}(s)$ such that in each $\mathsf{BD}$ of $\mathsf{BDSet}(s)$, $\mathsf{O}\mathsf{Beh}(s^{\prime})$ has been replaced.

Obtaining $\mathsf{SfrMTL}$ Formulae

Finally, we show that given a $\mathsf{BD}$ for $\mathsf{Beh}(s)$, we can construct an $\mathsf{SfrMTL}$ formula, $\psi_{s}$, equivalent to $x.\mathsf{O}(s)$. That is, $\rho,i\models\psi_{s}$ if and only if $\rho,i,\nu\models x.\mathsf{O}(\mathsf{Beh}(s))$, for any $\nu$. Recall that $\mathsf{Beh}(s)$ is a 1-$\mathsf{TPTL}$ formula, as computed in lemma 4.3. We give a constructive proof as follows:

Assume $\rho,i,\nu\models x.\mathsf{O}(\mathsf{Beh}(s))$. Note that according to the syntax of $\mathsf{TPTL}$, every constraint $x\in I$ checks the time elapse between the last point where $x$ was frozen. Thus satisfaction of formulae of the form $x.\phi$ at a point is independent of the clock valuation. $\rho,i,\nu\models x.\mathsf{O}(\mathsf{Beh}(s))$ iff $\rho,i,\nu[x\leftarrow\tau_{i}]\models\mathsf{O}\mathsf{Beh}(s)$. We have precomputed $\mathsf{BD}(s,R_{i},R_{j})$ for all regions $R_{i}\preceq R_{j}$; and $\mathsf{BD}(s,R_{i},R_{j})$ is guided by the 1-$\mathsf{TPTL}$ formula $\mathsf{Beh}(s)$. The entry in region $R_{i}$ of $\mathsf{BD}(s,R_{i},R_{j})$ depends on the behaviour allowed in region $R_{i}$ from location $s$; likewise, the entry in each region $R_{g}$ of $\mathsf{BD}(s,R_{i},R_{j})$ is obtained by looking up $\mathsf{Beh}(s)$. In case $\mathsf{Beh}(s)$ does not admit any behaviour in a region $R_{g}$, then the $g$th entry in $\mathsf{BD}(s,R_{i}R_{j})$ is $\epsilon$. Thus, $\rho,i,\nu\models x.\mathsf{O}(\mathsf{Beh}(s))$ iff for all $w\in{0,\ldots,2K}$, such that $\mathsf{Beh}(s)$ has an allowed behaviour in region $R_{w}$, $\rho,i+1,\tau_{i}\models(x\in R_{w})$. In addition, we also know that there is some $\mathsf{BD}(s,R_{w},R_{j})$ such that $\mathsf{BD}[R_{k}]$ is the $\mathsf{LTL}$ formula that describes the behaviour in region $R_{k}$ of location $s$.

Note that, $\rho,i+1,\tau_{i}{\models}(x\in R_{w})$ is true, iff, $\rho,i\models\bigwedge\limits_{g\in\{1,\ldots,w-1\}}[\mathsf{Rat}_{R_{g}}\epsilon]\wedge\mathsf{Rat}_{R_{w}}\Sigma^{+}$. This is true iff $\rho,i{\models}\bigvee\limits_{\mathsf{BD}=\mathsf{BD}(s,R_{w},R_{j})}\bigwedge\limits_{k\in\{1,\ldots,2K\}}\mathsf{Rat}_{R_{k}}(\mathsf{re}(\mathsf{BD}[R_{k}]))$, where $\mathsf{re}(\mathsf{BD}[R_{k}])$ is a star-free rational expression equivalent to the LTL formula $\mathsf{BD}[R_{k}]$.

Thus, $\rho,i,\nu{\models}x.\mathsf{O}(\mathsf{Beh}(s))$, iff, $\rho,i{\models}(\psi_{1}{\rightarrow}\psi_{2})$ where

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$\psi_{1}{=}{\bigwedge\limits_{w\in\{0,\ldots,2K\}\setminus E}}\bigwedge\limits_{g\in\{1,\ldots,w-1\}}\mathsf{Rat}_{R_{g}}\epsilon{\wedge}{\mathsf{Rat}_{R_{w}}\Sigma^{+}}$ and

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$\psi_{2}{=}\bigvee\limits_{\mathsf{BD}=\mathsf{BD}(s,R_{w},R_{j})}\bigwedge\limits_{k\in\{1,\ldots,2K\}}\mathsf{Rat}_{R_{k}}(\mathsf{re}(\mathsf{BD}[R_{k}]))$.

where $E$ is the set of regions where $\mathsf{Beh}(s)$ has no behaviour. The $\mathsf{SfrMTL}$ formula $\psi_{s_{0}}$ is one which begins in the initial location $s_{0}$, stitches the behaviours of locations $s_{j}$ that appear in a run from $s_{0}$ such that $L(\psi_{s_{0}})$ is non-empty iff the language accepted by the $\mathsf{po}$-1-clock ATA $\mathcal{A}$ is non-empty, and $L(\psi_{s_{0}})=L(\mathcal{A})$.

Consider the $\mathsf{po}$-1-clock ATA $\mathcal{A}=(\{a,b\},\{s_{0},s_{a},s_{\ell}\},s_{0},\{s_{0},s_{\ell}\},\delta)$ with transitions $\delta(s_{0},b)=s_{0},\delta(s_{0},a)=(s_{0}\wedge x.s_{a})\vee s_{\ell},$ $\delta(s_{a},a)=(s_{a}\wedge x<1)\vee(x>1)=\delta(s_{a},b),$ and $\delta(s_{\ell},b)=s_{\ell},\delta(s_{\ell},a)=\bot$.

Consider the subset of $L(\mathcal{A})$ consisting of timed words whose first symbol occurs at a time $>1$. We write a $\mathsf{SfrMTL}$ formula that captures this subclass.

Let us consider the formula we obtain if we consider allowed behaviours from $s_{0}$ that begin in the region $R^{+}_{1}$; this is the subset of $\mathsf{BDSet}(s_{0})$ consisting of $\mathsf{BD}(s_{0},R^{+}_{1},R^{+}_{1})=(\top,\top,\top,[(a\wedge x.\mathsf{O}\mathsf{Beh}(s_{a}))\vee b]\>\mathsf{W}(a\wedge\mathsf{O}\mathsf{Beh}(s_{\ell})))$. We look at the $\mathsf{SfrMTL}$ formula $\psi_{s_{a}}$ corresponding to $x.\mathsf{O}\mathsf{Beh}(s_{a})$, which is given by

$\mathsf{Rat}_{R_{0}}\Sigma^{+}\rightarrow\{\mathsf{Rat}_{R_{0}}(a+b)^{*}\wedge\mathsf{Rat}_{R_{1}}((a+b)^{*}+\epsilon)\wedge\mathsf{Rat}_{R_{2}}\epsilon\wedge\mathsf{Rat}_{R^{+}_{1}}(a+b)^{*}\}$ $\wedge$

$\mathsf{Rat}_{R_{0}}\epsilon\wedge\mathsf{Rat}_{R_{1}}\Sigma^{+}\rightarrow\{\mathsf{Rat}_{R_{0}}\top\wedge\mathsf{Rat}_{R_{1}}(a+b)^{*}\wedge\mathsf{Rat}_{R_{2}}\emptyset\wedge\mathsf{Rat}_{R^{+}_{1}}(a+b)^{*}\}$ $\wedge$

$\mathsf{Rat}_{R_{0}}\epsilon\wedge\mathsf{Rat}_{R_{1}}\epsilon\wedge\mathsf{Rat}_{R_{2}}\epsilon\wedge\mathsf{Rat}_{R^{+}_{1}}\Sigma^{+}\rightarrow\mathsf{Rat}_{R^{+}_{1}}(a+b)^{*}$

This formula $\psi_{a}$ is plugged in place of $x.\mathsf{O}\mathsf{Beh}(s_{a})$ in $\mathsf{BD}(s_{0},R^{+}_{1},R^{+}_{1})$. We now combine $\mathsf{BD}(s_{\ell},R^{+}_{1},R^{+}_{1})\in\mathsf{BDSet}(s_{\ell})$ with $\mathsf{BD}(s_{0},R^{+}_{1},R^{+}_{1})[R^{+}_{1}]$ to obtain the combined behaviour of locations $s_{\ell}$ from the next point along with that of $s_{0}$. We know that $\mathsf{BD}(s_{\ell},R^{+}_{1},R^{+}_{1})=(\top,\top,\top,b\>\mathsf{W}\bot)$. Thus, we obtain $\mathsf{BD}(s_{0},R^{+}_{1},R^{+}_{1})$ after combining with $\mathsf{BD}(s_{\ell},R^{+}_{1},R^{+}_{1})$ and $\psi_{s_{a}}$ as $(\top,\top,\top,[(a\wedge\psi_{s_{a}})\vee b]\>\mathsf{W}(a\wedge(b\>\mathsf{W}\bot)))$. Translating this into an $\mathsf{SfrMTL}$ formula, we obtain the formula $\varphi_{R^{+}_{1}}(s_{0})$

$\mathsf{Rat}_{R_{0}}\epsilon\wedge\mathsf{Rat}_{R_{1}}\epsilon\wedge\mathsf{Rat}_{R_{2}}\epsilon\wedge\mathsf{Rat}_{R^{+}_{1}}\Sigma^{+}\rightarrow\mathsf{Rat}_{R^{+}_{1}}\mathsf{re}([(a\wedge\psi_{s_{a}})\vee b]\>\mathsf{W}(a\wedge(b\>\mathsf{W}\bot)))$.

$\varphi_{R^{+}_{1}}(s_{0})$ is the formula which captures the subset of $L(\mathcal{A})$ which consists of timed words of the form $(a_{1},t_{1})(a_{2},t_{2})\dots(a_{n},t_{n})$ such that $t_{1}>1$. We can also write the formulae $\varphi_{R_{0}}(s_{0})$, $\varphi_{R_{1}}(s_{0})$, $\varphi_{R_{2}}(s_{0})$, which capture respectively, the subset of words of $L(\mathcal{A})$ which consists of timed words of the form $(a_{1},t_{1})(a_{2},t_{2})\dots(a_{n},t_{n})$ where $t_{1}=0$, $0<t_{1}<1$ and $t_{1}=1$ respectively. Thus, $L(\mathcal{A})$ is the union of the languages $L(\varphi_{R^{+}_{1}}(s_{0})),L(\varphi_{R_{0}}(s_{0})),L(\varphi_{R_{2}}(s_{0}))$ and $L(\varphi_{R_{1}}(s_{0}))$.