On Verifying Timed Hyperproperties

TL;DR

This paper investigates the satisfiability and model-checking problems for HyperMTL, a timed logic for hyperproperties, revealing decidability results and limitations depending on semantics and restrictions.

Contribution

It introduces HyperMTL with asynchronous and synchronous semantics, analyzing decidability of satisfiability and model-checking, and identifies conditions for decidability in timed hyperproperties.

Findings

Satisfiability for HyperMTL is decidable similar to HyperLTL.

Model checking is undecidable for most cases unless the specification is alternation-free.

Decidability can be achieved under certain conditions with quantifier alternations in synchronous semantics or bounded time domain.

Abstract

We study the satisfiability and model-checking problems for timed hyperproperties specified with HyperMTL, a timed extension of HyperLTL. Depending on whether interleaving of events in different traces is allowed, two possible semantics can be defined for timed hyperproperties: asynchronous and synchronous. While the satisfiability problem can be decided similarly to HyperLTL regardless of the choice of semantics, we show that the model-checking problem, unless the specification is alternation-free, is undecidable even when very restricted timing constraints are allowed. On the positive side, we show that model checking HyperMTL with quantifier alternations is possible under certain conditions in the synchronous semantics, or when there is a fixed bound on the length of the time domain.

Tables

Table 1. Table 1: Decidability of model checking untimed or one-clock TA s against (one-clock) HyperMTL in the synchronous semantics; NS stands for Non-Singular constraints and RoT stands for Reset-on-Testing.

	untimed	NS$+$RoT	NS	RoT
untimed	Dec. (Proposition 1)	Dec. for ${\forall }^{\ast }{\exists }^{\ast }$ (Proposition 2)	Undec. for $\exists \forall \forall$ (Proposition 3)	Undec. for $\exists \forall \forall$ (Proposition 3)
NS+RoT	Undec. for $\exists \forall \forall$ (Theorem 4.3)	Undec.	Undec.	Undec.
NS	Undec.	Undec.	Undec.	Undec.
RoT	Undec.	Undec.	Undec.	Undec.

Equations

$$\forall {\pi }_a\forall {\pi }_b\overline{\mathbf{G}}(I_a=I_b)\Rightarrow \overline{\mathbf{G}}(O_a=O_b).$$

$$\psi :=\top \mid p\mid {\psi }_1\wedge {\psi }_2\mid \neg \psi \mid {\psi }_1{\mathbf{U}}_I{\psi }_2\mid {\psi }_1{\mathbf{S}}_I{\psi }_2$$

$$\varphi=\forall\pi_{a}\,\forall\pi_{b}\,\big{(}\operatorname{\mathbf{\overline{G}}}(\top_{b}\Rightarrow\psi^{L}_{b}\wedge\psi^{=}_{U,C})\wedge\operatorname{\mathbf{\overline{G}}}(\top_{a}\wedge\bot_{b}\Rightarrow\psi^{H}_{a})\Rightarrow\operatorname{\mathbf{\overline{G}}}(\top_{b}\Rightarrow\psi^{=}_{\mathit{out}(U)})\big{)}$$

Full text

1 Introduction

Background.

One of the most popular specification formalisms for reactive systems is Linear Temporal Logic (LTL), first introduced into computer science by Pnueli [53] in the late 1970s. The success of LTL can be attributed to the fact that its satisfiability and model-checking problems are of lower complexity ($\mathrm{PSPACE}$-complete, as compared with non-elementary for the equally expressive first-order logic of order) and it enjoys simple translations into automata and excellent tool support (e.g., [37, 15]).

While LTL is adequate for describing features of individual execution traces, many security policies in practice are based on relations between two (or more) execution traces. A standard example of such properties is observational determinism [55, 59, 39]: for every pair of execution traces, if the low-security inputs agree in both execution traces, then the low-security outputs in both execution traces must agree as well. Such properties are called hyperproperties [17]: a model of the property is not a single execution trace but a set of execution traces. HyperLTL [16], obtained from LTL by adding trace quantifiers, has been proposed as a specification formalism to express hyperproperties. For example, operational determinism can be expressed as the HyperLTL formula:

[TABLE]

HyperLTL inherits almost all the benefits of LTL; in particular, tools that support HyperLTL verification can be built by leveraging existing tools for LTL.

For many applications, however, in addition to the occurences and orders of events, timing has to be accounted as well. For example, one may want to verify that in every execution trace of the system, whenever a request $\mathtt{req}$ is issued, the corresponding acknowledgement $\mathtt{ack}$ is received within the next $5$ time units. Timed automata [4] and timed logics [41, 8, 5] are introduced exactly for this purpose. In the context of security, timing anomalies caused by different high-security inputs is a realistic attack vector that can be exploited to obtain sensitive information; this kind of timing side-channel attacks also play significant roles in high-profile exploits like Meltdown [47] and Spectre [40]. In order to detect such undesired characteristics of systems, one needs to be able to reason about timed hyperproperties.

Example 1 ([46])

An AND gate with two inputs $A$, $B$ and an output $C$ and respective delays $T_{A}$, $T_{B}$, and $T_{C}$ can be modelled as the timed automaton below (suppose that $T_{A}<T_{B}$ and $T_{B}-T_{A}<T_{C}$):

Of course, once $A$ turned out to be [math] (i.e. $A^{0}$ has happened), the output $C$ must be [math] as well. But the time when $C_{0}$ happens (assuming $C=0$) also depends on the value of $A$; in other words, in addition to the value of $C$, a low-security user can, when $C=0$, also infer the value of $A$ (while she or he should not be able to). In this simple example, however, the timing side channel can be removed by adding $z:=0$ on the self-loop on the lower-right location.

Contributions.

We propose HyperMTL, obtained by adding trace quantifiers to MTL, as a specification formalism for timed hyperproperties. We consider systems modelled as timed automata, and thus system behaviours are sequences of events that happen at different instants in time; this gives two possible pointwise semantics of HyperMTL: asynchronous and synchronous. We show that, as far as satisfiability is concerned, HyperMTL is similar to HyperLTL, i.e. satisfiability is decidable for fragments not containing $\forall\exists$, regardless of which semantics is assumed. However, in contrast with HyperLTL (whose model-checking problem is decidable), model checking HyperMTL is undecidable if there is at least one quantifier alternation in the specification, even when the timing constraints used in either the system or the specification are very restricted. Still, the alternation-free fragment of HyperMTL, which is arguably sufficient to capture many timed hyperproperties of practical interest, has a decidable model-checking problem. Finally, we identify several subcases where HyperMTL model checking is decidable for larger fragments, such as when the synchronous semantics is assumed, the model is untimed, and the specification belongs to a certain subclass of one-clock timed automata, or when the time domain is bounded a priori by some $N\in\mathbb{N}_{>0}$.

Related work.

Since the pioneering work of Clarkson and Schneider [17], there have been great interest in specifying and verifying hyperproperties in the past few years. The framework based on HyperLTL [16] is possibly the most popular for this purpose, thanks to its expressiveness, flexibility, and relative ease of implementation. In addition to satisfiability [25, 24] and model checking [16, 29], tools for monitoring HyperLTL also exist [3, 26, 27].

Our formulation of HyperMTL is very closely related to HyperSTL [49] originally proposed in the context of quality assurance of cyber-physical systems. While [49] focusses on testing, we are concerned with the decidability of verification problems. On the other hand, the semantics of HyperSTL is defined over sets of continuous signals, i.e. state-based; as noted in [49], however, the price to pay for the extra generality is that implementing a model checker for HyperSTL is very difficult, especially for systems modelled in proprietary frameworks (such as Simulink®). Practical reasoning of HyperMTL, by contrast, can be carried out easily with existing highly optimised timed automata verification back ends, e.g., Uppaal [45]. Indeed, a prototype model checker based on Uppaal for the synchronous semantics of HyperMTL (with some restrictions) is reported in [34], although it does not consider the decidability of verification problems. Another relevant work [31], also based on Uppaal, checks noninterference in systems modelled as timed automata (similar to Example 2; see below). Their approach, however, is specifically tailored to noninterference and does not generalise.

It is also possible to extend hyperlogics in other quantitative dimensions orthogonal to time. HyperPCTL [2] can express probabilisitic hyperproperties, e.g., the probability distribution of the low-security outputs are independent of the high-security inputs. In [28], specialised algorithms are developed for verifying quantitative hyperproperties, e.g., there is a bound on the number of traces with the same low-security inputs but different low-level outputs. The current paper is complementary to these works.

2 Timed hyperproperties

Timed words.

A timed word (or a trace) over a finite alphabet $\Sigma$ is a finite sequence of events $(\sigma_{1},\tau_{1})\dots(\sigma_{n},\tau_{n})\in(\Sigma\times\mathbb{R}_{\geq 0})^{\ast}$ with $\tau_{1}\dots\tau_{n}$ an increasing sequence of non-negative real numbers (‘timestamps’), i.e. $\tau_{i}<\tau_{i+1}$ for all $i$, $1\leq i<n$. For $t\in\mathbb{R}_{\geq 0}$ and a timed word $\rho=(\sigma_{1},\tau_{1})\dots(\sigma_{n},\tau_{n})$, we write $t\in\rho$ iff $t=\tau_{i}$ for some $i$, $1\leq i\leq n$. We denote by $T\Sigma^{\ast}$ the set of all timed words over $\Sigma$. A timed language (or a trace property) is a subset of $T\Sigma^{\ast}$.

Timed automata.

Let $X$ be a finite set of clocks ($\mathbb{R}_{\geq 0}$-valued variables). A valuation $v$ for $X$ maps each clock $x\in X$ to a value in $\mathbb{R}_{\geq 0}$. The set $G(X)$ of clock constraints (guards) $g$ over $X$ is generated by $g:=\top\mid g\land g\mid x\bowtie c$ where ${\bowtie}\in\{{\leq},{<},{\geq},{>}\}$, $x\in X$, and $c\in\mathbb{N}_{\geq 0}$. The satisfaction of a guard $g$ by a valuation $v$ (written $v\models g$) is defined in the usual way. For $t\in\mathbb{R}_{\geq 0}$, we let $v+t$ be the valuation defined by $(v+t)(x)=v(x)+t$ for all $x\in X$. For $\lambda\subseteq X$, we let $v[\lambda\leftarrow 0]$ be the valuation defined by $(v[\lambda\leftarrow 0])(x)=0$ if $x\in\lambda$, and $(v[\lambda\leftarrow 0])(x)=v(x)$ otherwise.

A timed automaton (TA) over $\Sigma$ is a tuple $\mathcal{A}=\langle\Sigma,S,s_{0},X,\Delta,F\rangle$ where $S$ is a finite set of locations, $s_{0}\in S$ is the initial location, $X$ is a finite set of clocks, $\Delta\subseteq S\times\Sigma\times G(X)\times 2^{X}\times S$ is the transition relation, and $F$ is the set of accepting locations. We say that $\mathcal{A}$ is deterministic iff for each $s\in S$ and $\sigma\in\Sigma$ and every pair of transitions $(s,\sigma,g^{1},\lambda^{1},s^{1})\in\Delta$ and $(s,\sigma,g^{2},\lambda^{2},s^{2})\in\Delta$, $g^{1}\land g^{2}$ is not satisfiable. A state of $\mathcal{A}$ is a pair $(s,v)$ of a location $s\in S$ and a valuation $v$ for $X$. A run of $\mathcal{A}$ on a timed word $(\sigma_{1},\tau_{1})\dots(\sigma_{n},\tau_{n})\in T\Sigma^{\ast}$ is a sequence of states $(s_{0},v_{0})\dots(s_{n},v_{n})$ where (i) $v_{0}(x)=0$ for all $x\in X$ and (ii) for each $i$, $0\leq i<n$, there is a transition $(s_{i},\sigma_{i+1},g,\lambda,s_{i+1})$ such that $v_{i}+(\tau_{i+1}-\tau_{i})\models g$ (let $\tau_{0}=0$) and v_{i+1}=\big{(}v_{i}+(\tau_{i+1}-\tau_{i})\big{)}[\lambda\leftarrow 0]. A run of $\mathcal{A}$ is accepting iff it ends in a state $(s,v)$ with $s\in F$. A timed word is accepted by $\mathcal{A}$ iff $\mathcal{A}$ has an accepting run on it. We denote by $\llbracket\mathcal{A}\rrbracket$ the timed language of $\mathcal{A}$, i.e. the set of all timed words accepted by $\mathcal{A}$. Two fundamental results on TAs are that the emptiness problem is decidable ($\mathrm{PSPACE}$-complete), but the universality problem is undecidable [4].

Timed logics.

The set of MTL formulae over a finite set of atomic propositions AP are generated by

[TABLE]

where $p\in\textup{{{AP}}}$ and $I\subseteq\mathbb{R}_{\geq 0}$ is a non-singular interval with endpoints in $\mathbb{N}_{\geq 0}\cup\{\infty\}$.111In the literature, this logic (with the requirement that constraining intervals must be non-singular) is usually referred to as MITL [5], but we simply call it MTL in this paper for notational simplicity. Also note that our undecidability results carry over to the fragment with only future operators. We omit the subscript $I$ when $I=[0,\infty)$ and sometimes write pseudo-arithmetic expressions for constraining intervals, e.g., ‘$<3$’ for $[0,3)$. The other Boolean operators are defined as usual: $\bot\equiv\neg\top$ and $\psi_{1}\lor\psi_{2}\equiv\neg(\neg\psi_{1}\land\neg\psi_{2})$. We also define the dual temporal operators \psi_{1}\widetilde{\mathbin{\mathbf{U}}}_{I}\psi_{2}\equiv\neg\big{(}(\neg\psi_{1})\mathbin{\mathbf{U}}_{I}(\neg\psi_{2})\big{)} and \psi_{1}\widetilde{\mathbin{\mathbf{S}}}_{I}\psi_{2}\equiv\neg\big{(}(\neg\psi_{1})\mathbin{\mathbf{S}}_{I}(\neg\psi_{2})\big{)}. Using these operators, every MTL formula $\psi$ can be transformed into an MTL formula $\mathit{nnf}(\psi)$ in negative normal form, i.e. $\neg$ is only applied to atomic propositions. To ease the presentation, we will also use the usual shortcuts like $\operatorname{\mathbf{F}}_{I}\psi\equiv\top\mathbin{\mathbf{U}}_{I}\psi$, $\operatorname{\mathbf{G}}_{I}\psi\equiv\neg\operatorname{\mathbf{F}}_{I}\neg\psi$, $\operatorname{\mathbf{X}}\psi\equiv\bot\mathbin{\mathbf{U}}\psi$, and ‘weak-future’ variants of temporal operators, e.g., $\operatorname{\mathbf{\overline{F}}}\psi\equiv\psi\vee\operatorname{\mathbf{F}}\psi$. Given an MTL formula $\psi$ over AP, a timed word $\rho$ over $\Sigma_{\textup{{{AP}}}}=2^{\textup{{{AP}}}}$, and $t\in\mathbb{R}_{\geq 0}$, we define the MTL satisfaction relation $\models$ as follows:222The formulation of the pointwise semantics of MTL here deviates slightly from the standard one (cf. [7, 52]) to enable interleaving of events in different traces.

• •

$(\rho,t)\models\top$ iff $t\in\rho$;

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$(\rho,t)\models\bot$ iff $t\notin\rho$;

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$(\rho,t)\models p$ iff $t\in\rho$ and $p\in\sigma_{i}$;

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$(\rho,t)\models\neg p$ iff $t\in\rho$ and $p\notin\sigma_{i}$;

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$(\rho,t)\models\psi_{1}\land\psi_{2}$ iff $(\rho,t)\models\psi_{1}$ and $(\rho,t)\models\psi_{2}$;

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$(\rho,t)\models\psi_{1}\lor\psi_{2}$ iff $(\rho,t)\models\psi_{1}$ or $(\rho,t)\models\psi_{2}$;

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$(\rho,t)\models\psi_{1}\mathbin{\mathbf{U}}_{I}\psi_{2}$ iff there exists $t^{\prime}>t$ such that $t^{\prime}-t\in I$, $(\rho,t^{\prime})\models\top$, $(\rho,t^{\prime})\models\psi_{2}$, and $(\rho,t^{\prime\prime})\models\psi_{1}$ for all $t^{\prime\prime}$ such that $t^{\prime\prime}\in(t,t^{\prime})$ and $(\rho,t^{\prime\prime})\models\top$;

• •

$(\rho,t)\models\psi_{1}\widetilde{\mathbin{\mathbf{U}}}_{I}\psi_{2}$ iff for all $t^{\prime}>t$ such that $t^{\prime}-t\in I$ and $(\rho,t^{\prime})\models\top$, either $(\rho,t^{\prime})\models\psi_{2}$ or $(\rho,t^{\prime\prime})\models\psi_{1}$ for some $t^{\prime\prime}$ such that $t^{\prime\prime}\in(t,t^{\prime})$ and $(\rho,t^{\prime\prime})\models\top$;

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$(\rho,t)\models\psi_{1}\mathbin{\mathbf{S}}_{I}\psi_{2}$ iff there exists $t^{\prime}$, $0\leq t^{\prime}<t$ such that $t-t^{\prime}\in I$, $(\rho,t^{\prime})\models\top$, $(\rho,t^{\prime})\models\psi_{2}$, and $(\rho,t^{\prime\prime})\models\psi_{1}$ for all $t^{\prime\prime}$ such that $t^{\prime\prime}\in(t^{\prime},t)$ and $(\rho,t^{\prime\prime})\models\top$;

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$(\rho,t)\models\psi_{1}\widetilde{\mathbin{\mathbf{S}}}_{I}\psi_{2}$ iff for all $t^{\prime}$, $0\leq t^{\prime}<t$ such that $t-t^{\prime}\in I$ and $(\rho,t^{\prime})\models\top$, either $(\rho,t^{\prime})\models\psi_{2}$ or $(\rho,t^{\prime\prime})\models\psi_{1}$ for some $t^{\prime\prime}$ such that $t^{\prime\prime}\in(t^{\prime},t)$ and $(\rho,t^{\prime\prime})\models\top$;

• •

$(\rho,t)\models\neg\psi$ iff $(\rho,t)\models\mathit{nnf}(\neg\psi)$.

We say that $\rho$ satisfies $\psi$ ($\rho\models\psi$) iff $(\rho,0)\models\psi$, and we write $\llbracket\psi\rrbracket$ for the timed language of $\psi$, i.e. the set of all timed words satisfying $\psi$. It is well known that any MTL formula can be translated into a TA accepting the same timed language [6]; this implies that MTL satisfiability is decidable ($\mathrm{EXPSPACE}$-complete).

Adding trace quantifiers.

Let $V$ be an infinite supply of trace variables, the set of HyperMTL formulae over AP are generated by {IEEEeqnarray*}rCl φ& := ∃π  φ∣∀π  φ∣*ψ

ψ* := ⊤∣⊤π∣pπ∣ψ_1 ∧ψ_2 ∣¬ψ∣ψ_1 U_I ψ_2 ∣ψ_1 S_I ψ_2

where $\pi\in V$, $p\in\textup{{{AP}}}$, and $I\subseteq\mathbb{R}_{\geq 0}$ is a non-singular interval with endpoints in $\mathbb{N}_{\geq 0}\cup\{\infty\}$ (to ease the notation, we will usually write, e.g., $p_{a}$ for $p_{\pi_{a}}$). Without loss of generality we forbid the reuse of trace variables, i.e. each trace quantifier must use a fresh trace variable. Syntatic sugar is defined as in MTL, e.g., $\operatorname{\mathbf{F}}_{I}\psi\equiv\top\mathbin{\mathbf{U}}_{I}\psi$. A HyperMTL formula is closed if it does not have free occurrences of trace variables. Following [23], we refer to fragments of HyperMTL by their quantifier patterns, e.g., $\exists^{\ast}\forall^{\ast}$-HyperMTL. Finally, note that trace quantifiers can be added to TAs in the same manner.

In contrast with TAs and MTL formulae, which define trace properties, HyperMTL formulae define (timed) hyperproperties, i.e. sets of trace properties. Depending on whether one requires timestamps in quantified traces to match exactly (i.e. all quantified traces must synchronise), two possible semantics can be defined accordingly.

Asynchronous semantics.

A trace assignment over $\Sigma$ is a partial mapping from $V$ to $T\Sigma^{\ast}$. We write $\Pi_{\emptyset}$ for the empty trace assignment and $\Pi[\pi\mapsto\rho]$ for the trace assignment that maps $\pi$ to $\rho$ and $\pi^{\prime}$ to $\Pi(\pi^{\prime})$ for all $\pi^{\prime}\neq\pi$. Given a HyperMTL formula $\varphi$ over AP, a trace set $T$ over $\Sigma_{\textup{{{AP}}}}$, a trace assignment $\Pi$ over $\Sigma_{\textup{{{AP}}}}$, and $t\in\mathbb{R}_{\geq 0}$, we define the HyperMTL asynchronous satisfaction relation $\models$ as follows (we omit the cases where the definitions are obvious):

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$(T,t)\models_{\Pi}\top$ iff $t\in\rho$ for some $\rho\in\mathit{range}(\Pi)$;

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$(T,t)\models_{\Pi}\top_{\pi}$ iff $t\in\rho$ for $\rho=\Pi(\pi)$;

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$(T,t)\models_{\Pi}p_{\pi}$ iff $t\in\rho$ for $\rho=\Pi(\pi)$ and $p\in\sigma_{i}$ for the event $(\sigma_{i},t)$ in $\rho$;

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$(T,t)\models_{\Pi}\exists\pi\,\varphi$ iff there is a trace $\rho\in T$ such that $(T,t)\models_{\Pi[\pi\mapsto\rho]}\varphi$;

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$(T,t)\models_{\Pi}\forall\pi\,\varphi$ iff for all traces $\rho\in T$, $(T,t)\models_{\Pi[\pi\mapsto\rho]}\varphi$.

We say that $T$ satisfies a closed HyperMTL formula $\varphi$ in the asynchronous semantics ($T\models\varphi$) iff $(T,0)\models_{\Pi_{\emptyset}}\varphi$.

Example 2 (Noninterference in event-based systems [32])

A system operating on sequences of commands issued by different users can be modelled as a deterministic finite automaton $\mathcal{A}$ over $\Sigma=U\times C$ where $U$ is the set of users and $C$ is the set of commands. Additionally, let $\mathsf{Obs}$ be the set of observations and $\mathit{out}:S\times U\rightarrow\mathsf{Obs}$ be the observation function for what can be observed at each location by each user. Let there be a partition of $U$ into two disjoint sets of users $U_{H}\subseteq U$ and $U_{L}\subseteq U$. Noninterference requires that for each $w\in\Sigma^{\ast}$ where $w$ ends with a command issued by a user in $U_{L}$ and $\mathcal{A}$ reaches $s$ after reading $w$, the subsequence $w^{\prime}$ obtained by removing all the commands issued by the users in $U_{H}$ results in a location $s^{\prime}$ such that the observation $\mathit{out}(s^{\prime},u_{L})$ of each user $u_{L}\in U_{L}$ is identical to $\mathit{out}(s,u_{L})$. For our purpose, we can combine $\mathcal{A}$ and $\mathit{out}$ (in the expected way) into an automaton $\mathcal{A}^{\prime}$ over $\Sigma_{\textup{{{AP}}}}$ where $\textup{{{AP}}}=(U\times C)\uplus(U\times\mathsf{Obs})$ (atomic propositions in $U\times\mathsf{Obs}$ reflect the observations at the location that has just been entered). Checking noninterference then amounts to model checking $\mathcal{A}^{\prime}$ (whose locations are all accepting) against the following HyperMTL formula in the asynchronous semantics:

[TABLE]

(where $\psi^{L}_{b}$ asserts that the command in $\pi_{b}$ is issued by a user in $U_{L}$, $\psi^{=}_{U,C}$ says that the two synchronised commands in $\pi_{a}$ and $\pi_{b}$ agree on $U$ and $C$, etc.). Compared with the state-based formulation in [16], allowing interleaving of events leads to a much simpler and clearer specification.

Synchronous semantics.

A less general semantics can be defined for HyperMTL formulae where each trace quantifier only ranges over traces that synchronise with the traces in the current trace assignment. For example, the second quantifier in $\exists\pi_{a}\,\exists\pi_{b}\,\psi$ requires $\pi_{b}$ to satisfy $(\pi_{a},t)\models\top_{a}\Leftrightarrow(\pi_{b},t)\models\top_{b}$ for all $t\in\mathbb{R}_{\geq 0}$. The HyperMTL synchronous satisfaction relation $\models^{\mathit{sync}}$ can, in fact, be expressed in the asynchronous semantics by explicitly requiring newly quantified traces to synchronise in the quantifier-free part of the formula. More precisely, for a closed HyperMTL formula $\varphi=\mathcal{Q}\,\varphi^{\prime}$ where $\mathcal{Q}$ denotes a block of quantifiers of the same type (i.e. all existential or all universal) and $\varphi^{\prime}$ is a possibly open HyperMTL formula, and a set $V$ of trace variables, let (abusing notation slightly) \mathit{sync}(\varphi,V)=\mathcal{Q}\,\big{(}\operatorname{\mathbf{\overline{G}}}(\bigwedge_{\pi\in\mathcal{Q}\cup V}\top_{\pi})\wedge\mathit{sync}(\varphi^{\prime},\mathcal{Q}\cup V)\big{)} when $\mathcal{Q}$ are existential, \mathit{sync}(\varphi)=\mathcal{Q}\,\big{(}\operatorname{\mathbf{\overline{G}}}(\bigwedge_{\pi\in\mathcal{Q}\cup V}\top_{\pi})\Rightarrow\mathit{sync}(\varphi^{\prime},\mathcal{Q}\cup V)\big{)} when $\mathcal{Q}$ are universal, and $\mathit{sync}(\psi,V)=\psi$ when $\psi$ is quantifier-free. The following lemma holds subject to rewriting the formula into prenex normal form.

Lemma 1

For any trace set $T$ over $\Sigma_{\textup{{{AP}}}}$ and closed HyperMTL formula $\varphi$ over AP, $T\models^{\mathit{sync}}\varphi$ iff $T\models\mathit{sync}(\varphi,\emptyset)$.

While the synchronous semantics may seem quite restricted (intuitively, the chance that two random traces in a timed system have exactly the same timestamps is certainly slim!), one can argue that it already suffices for many applications if stuttering steps are allowed. We will see later that for alternation-free HyperMTL, the asynchronous semantics can ‘simulated’ in the synchronous semantics using a ‘weak inverse’ of Lemma 1.

Satisfiability and model checking.

Given a closed HyperMTL formula $\varphi$ over AP, the satisfiability problem asks whether there is a non-empty trace set $T\subseteq T\Sigma_{\textup{{{AP}}}}^{\ast}$ satisfying it, i.e. $T\models\varphi$ (or $T\models^{\mathit{sync}}\varphi$, if the synchronous semantics is assumed). Given a TA $\mathcal{A}$ over $\Sigma_{\textup{{{AP}}}}$ and a closed HyperMTL formula $\varphi$ over AP, the model-checking problem asks whether $\llbracket\mathcal{A}\rrbracket\models\varphi$ (or $\llbracket\mathcal{A}\rrbracket\models^{\mathit{sync}}\varphi$). Our focus in this paper is on the decidability of these problems, as their complexity (when they are decidable) follow straightforwardly from standard results on MTL [5] and HyperLTL [16, 23].

3 Satisfiability

To ‘emulate’ interleaving of events (of a concurrent or distributed system, say) in a synchronous, state-based setting, it is natural and necessary to introduce stuttering steps. In the context of verification, it is often a desirable trait for a temporal logic to be stutter-invariant [44, 43], so that it cannot be used to differentiate traces that are essentially ‘the same’. As a simple attempt to reconcile the asynchronous and synchronous semantics of HyperMTL, we can make use of silent events in the same spirit to enable synchronisation of interleaving traces while preserving the semantics. More precisely, let $\mathit{stutter}(\rho)$ for a trace $\rho\in T\Sigma_{\textup{{{AP}}}}^{\ast}$ be the maximal set of traces $\rho^{\prime}\in T\Sigma_{\textup{{{AP}}}_{\epsilon}}^{\ast}$ ($\textup{{{AP}}}_{\epsilon}=\textup{{{AP}}}\cup\{p^{\epsilon}\}$) such that

• •

for every event $(\sigma_{i},\tau_{i})$ in $\rho^{\prime}$, either $\sigma_{i}=\{p^{\epsilon}\}$ or $p^{\epsilon}\notin\sigma_{i}$;

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$\rho$ can be obtained from $\rho^{\prime}$ by deleting all the $\{p^{\epsilon}\}$-events.

This extends to trace sets $T\subseteq T\Sigma_{\textup{{{AP}}}}^{\ast}$ in the obvious way. For a closed alternation-free HyperMTL formula $\varphi=\mathcal{Q}\psi$ over AP, let $\mathit{stutter}(\varphi)=\mathcal{Q}\psi^{\prime\prime}$ be the HyperMTL formula over $\textup{{{AP}}}_{\epsilon}$ obtained by (recursively) replacing in $\psi$, e.g., all subformulae $\psi_{1}\mathbin{\mathbf{U}}_{I}\psi_{2}$ with $(\bigvee_{\pi\in\mathcal{Q}}\neg p_{\pi}^{\epsilon}\Rightarrow\psi_{1})\mathbin{\mathbf{U}}_{I}(\bigvee_{\pi\in\mathcal{Q}}\neg p_{\pi}^{\epsilon}\wedge\psi_{2})$, to give $\psi^{\prime}$, and finally let \psi^{\prime\prime}=\operatorname{\mathbf{\overline{G}}}(\bigvee_{\pi\in\mathcal{Q}}\neg p^{\epsilon}_{\pi})\wedge\big{(}\bigwedge_{\pi\in\mathcal{Q}}\operatorname{\mathbf{\overline{G}}}(p^{\epsilon}_{\pi}\Rightarrow\bigwedge_{p\in\textup{{{AP}}}}\neg p_{\pi})\big{)}\wedge\psi^{\prime} when $\mathcal{Q}$ are existential and \psi^{\prime\prime}=\operatorname{\mathbf{\overline{G}}}(\bigvee_{\pi\in\mathcal{Q}}\neg p^{\epsilon}_{\pi})\wedge\big{(}\bigwedge_{\pi\in\mathcal{Q}}\operatorname{\mathbf{\overline{G}}}(p^{\epsilon}_{\pi}\Rightarrow\bigwedge_{p\in\textup{{{AP}}}}\neg p_{\pi})\big{)}\Rightarrow\psi^{\prime} when $\mathcal{Q}$ are universal. Intuitively, $\psi^{\prime\prime}$ ensures that the traces involved are well-formed (i.e. satisfy the first condition above), and its own satisfaction is insensitive to the addition of silent events. The following lemma follows from a simple structural induction.

Lemma 2

For any trace set $T$ over $\Sigma_{\textup{{{AP}}}}$ and closed alternation-free HyperMTL formula $\varphi=\mathcal{Q}\psi$ over AP ($\mathcal{Q}$ is either a block of existential quantifiers or universal quantifiers and $\psi$ is quantifier-free), $T\models\varphi$ iff $\mathit{stutter}(T)\models^{\mathit{sync}}\mathit{stutter}(\varphi)$.

The following two lemmas then follow from Lemma 2 and the fact that for alternation-free HyperMTL formulae, satisfiability in the synchronous semantics can be reduced (in the same way as HyperLTL) to MTL satisfiability.

Lemma 3

The satisfiability problem for $\exists^{\ast}$-HyperMTL is decidable.

Lemma 4

The satisfiability problem for $\forall^{\ast}$-HyperMTL is decidable.

Lemma 2, however, does not extend to larger fragments of HyperMTL. For example, consider $T=\{(\{p\},1)(\{r\},3),(\{q\},2)\}$ and $\varphi=\exists\pi_{a}\,\forall\pi_{b}\,(\operatorname{\mathbf{\overline{F}}}p_{a}\wedge\neg\operatorname{\mathbf{\overline{F}}}q_{b})$. Now it is obvious that $T\centernot\models\varphi$, but since $(\{p\},1)(\{r\},3)\in\mathit{stutter}(T)$, we have $\mathit{stutter}(T)\models^{\mathit{sync}}\mathit{stutter}(\varphi)$ (provided that the definition of $\mathit{stutter}(\cdot)$ is extended to general HyperMTL formulae, as in Lemma 1). Still, it is not hard to see that the crucial observation used in $\exists^{\ast}\forall^{\ast}$-HyperLTL satisfiability (if $\exists\pi_{0}\,\dots\exists\pi_{k}\,\forall\pi_{0}^{\prime}\,\dots\forall\pi_{\ell}^{\prime}\,\psi$ is satisfiable, then it is also satisfiable by the trace set $\{\pi_{0},\dots\pi_{k}\}$) extends to HyperMTL in the asynchronous semantics; the following lemma then follows from Lemma 3.

Lemma 5

The satisfiability problem for $\exists^{\ast}\forall^{\ast}$-HyperMTL is decidable.

Finally, note that the undecidability of $\forall\exists$-HyperLTL carries over to HyperMTL: in the synchronous semantics, the reduction in [23] applies directly with some trivial modifications (as we work with finite traces); undecidability then holds for the case of asynchronous semantics as well, by Lemma 1.

Lemma 6

The satisfiability problem for $\forall\exists$-HyperMTL is undecidable.

Theorem 3.1

The satisfiability problem for HyperMTL is decidable if the formula does not contain $\forall\exists$.

4 Model checking

The alternation-free case.

Without loss of generality, we consider only the case of $\exists^{\ast}$-HyperMTL in the asynchronous semantics. By Lemma 1, checking $\llbracket\mathcal{A}\rrbracket\models\varphi$ (for a TA $\mathcal{A}$ over $\Sigma_{\textup{{{AP}}}}$ and a closed $\exists^{\ast}$-HyperMTL formula $\varphi$ over AP) is equivalent to checking $\mathit{stutter}(\llbracket\mathcal{A}\rrbracket)\models^{\mathit{sync}}\mathit{stutter}(\varphi)$. To this end, we define $\mathit{stutter}(\mathcal{A})$ as the TA over $\Sigma_{\textup{{{AP}}}_{\epsilon}}$ obtained from $\mathcal{A}$ by adding a self-loop labelled with $\{p^{\epsilon}\}$ to each location; it should be clear that $\llbracket\mathit{stutter}(\mathcal{A})\rrbracket=\mathit{stutter}(\llbracket\mathcal{A}\rrbracket)$. In this way, the problem reduces to model checking $\exists^{\ast}$-HyperMTL in the synchronous semantics which, as the model-checking problem for $\exists^{\ast}$-HyperLTL, can be reduced to MTL model checking.

Theorem 4.1

Model checking alternation-free HyperMTL is decidable.

The general case.

Recall that the model-checking problem for HyperLTL is decidable even when the specification involves arbitrary nesting of quantifiers. This is unfortunately not the case for HyperMTL: allowing only one quantifier alternation already leads to undecidability. To see this, recall that any TA can be written as a formula $\exists X\,\psi$ where $X$ is a set of (new) atomic propositions and $\psi$ is an MTL formula [35, 54]. The undecidable TA universality problem—given a TA $\mathcal{A}$ over $\Sigma$, deciding whether $\llbracket\mathcal{A}\rrbracket=T\Sigma^{\ast}$—can thus be reduced to model checking HyperMTL: one simply checks whether there exists an $X$-labelling for every timed word over $\Sigma$ so that $\psi$ is satisfied. Here we show that model checking HyperMTL is, essentially, a harder problem; in the case of asynchronous semantics, it remains undecidable even when both the model and the specification are deterministic and only one of them uses a single clock (i.e. the other is untimed); by contrast, TA universality (over finite timed words) is decidable when the TA is deterministic [4] or uses only one clock [51].

We adapt the undecidability proof of the reactive synthesis problem for MTL in [14], which itself is by reduction from the halting problem for deterministic channel machines (DCMs), known to be undecidable [13]. Note that the reactive synthesis problem is decidable when the specification is deterministic [20] (as a matter of fact, the quantifier-free part $\psi$ in the encoding mentioned above is already in $\textup{{{LTL}}}_{\operatorname{\vartriangleleft}}$); in this sense, quantification over traces is more powerful than quantification over strategies (for all possible strategies of the environment, there is a strategy of the controller that is winning). For our purpose, we introduce the $\operatorname{\vartriangleleft}_{I}$ operator, in which we allow $I$ to be singular (note that this is merely syntactic sugar and does not increase the expressiveness of MTL [35, 54]):

• •

$(\rho,t)\models\operatorname{\vartriangleleft}_{I}\varphi$ iff there exists $t^{\prime}$, $0\leq t^{\prime}<t$ such that $t-t^{\prime}\in I$, $(\rho,t^{\prime})\models\top$, $(\rho,t^{\prime})\models\varphi$, and $(\rho,t^{\prime\prime})\centernot\models\varphi$ for all $t^{\prime\prime}$ such that $t^{\prime\prime}\in(t^{\prime},t)$ and $(\rho,t^{\prime\prime})\models\top$.

Let $\textup{{{LTL}}}_{\operatorname{\vartriangleleft}}$ be the fragment of MTL where all timed subformulae must be of the form $\operatorname{\vartriangleleft}_{I}\varphi$, and all $\varphi$’s in such subformulae must be ‘pure past’ formulae; these requirements ensure that $\textup{{{LTL}}}_{\operatorname{\vartriangleleft}}$, in which we will write the quantifier-free part of the specification, translates into deterministic TAs [19]. To ease the understanding, we will first do the proof for the case of asynchronous semantics and then adapt it to the case of synchronous semantics.

Theorem 4.2

Model checking $\exists^{\ast}\forall^{\ast}$-HyperMTL and $\forall^{\ast}\exists^{\ast}$-HyperMTL are undecidable in the asynchronous semantics.

Proof

A DCM $\mathcal{S}=\langle S,s_{0},s_{\mathit{halt}},M,\Delta\rangle$ can be seen as a finite automaton equipped with an unbounded fifo channel: $S$ is a finite set of locations, $s_{0}$ is the initial location, $s_{\mathit{halt}}$ is the halting location (such that $s_{\mathit{halt}}\neq s_{0}$), $M$ is a finite set of messages, and $\Delta\subseteq S\times\{m!,m?\mid m\in M\}\times S$ is the transition relation satisfying the following determinism hypothesis: (i) $(s,a,s^{\prime})\in\Delta$ and $(s,a,s^{\prime\prime})\in\Delta$ implies $s^{\prime}=s^{\prime\prime}$; (ii) if $(s,m!,s^{\prime})\in\Delta$ then it is the only outgoing transition from $s$. Without loss of generality, we further assume that there is no incoming transition to $s_{0}$, no outgoing transition from $s_{\mathit{halt}}$, and $(s_{0},a,s^{\prime})\in\Delta$ implies that $a\in\{m!\mid m\in M\}$ and $s^{\prime}\neq s_{\mathit{halt}}$. The semantics of $\mathcal{S}$ can be described with a graph $G(\mathcal{S})$ with vertices $\{(s,x)\mid s\in S\text{, }x\in M^{\ast}\}$ and edges defined as follows: (i) $(s,x)\rightarrow(s^{\prime},xm)$ if $(s,m!,s^{\prime})\in\Delta$; (ii) $(s,mx)\rightarrow(s^{\prime},x)$ if $(s,m?,s^{\prime})\in\Delta$. In other words, $m!$ ‘writes’ a copy of $m$ to the channel and $m?$ ‘reads’ a copy of $m$ off the channel. We say that $\mathcal{S}$ halts if there is a path in $G(\mathcal{S})$ from $(s_{0},\epsilon)$ to $(s_{\mathit{halt}},x)$ (a halting computation of $\mathcal{S}$) for some $x\in M^{\ast}$.

The idea, as in many similar proofs (e.g., [52]), is to encode a halting computation of $\mathcal{S}$ as a trace where each $m?$ is preceded by a corresponding $m!$ exactly $1$ time unit earlier, and each $m!$ is followed by an $m?$ exactly $1$ time unit later if $s_{\mathit{halt}}$ has not been reached yet. To this end, let the model $\mathcal{A}$ be an (untimed) finite automaton over $\Sigma=2^{\textup{{{AP}}}}$ where $\textup{{{AP}}}=\{m^{!},m^{?}\mid m\in M\}\cup\{p^{\mathit{begin}},p^{\mathit{end}},p^{\mathit{read}},p^{1},q^{1}\}$ and whose set of locations is $S\cup\{s_{1}\}$, where $s_{1}$ is a new non-accepting location. The transitions of $\mathcal{A}$ follow $\mathcal{S}$: for each $m\in M$, $s\xrightarrow{\{m^{?}\}}s^{\prime}$ is a transition of $\mathcal{A}$ iff $(s,m?,s^{\prime})\in\Delta$, and similarly for $m!$—except for those going out of $s_{0}$ or going into $s_{\mathit{halt}}$, on which we further require $p^{\mathit{begin}}$ or $p^{\mathit{end}}$ to hold, respectively. Let $s_{0}$ be the initial location and $s_{\mathit{halt}}$ be the only accepting location, and finally add transitions $s_{0}\xrightarrow{\{p^{\mathit{read}}\}}s_{\mathit{halt}}$ and $s_{0}\xrightarrow{\{p^{1}\}}s_{1}\xrightarrow{\{q^{1}\}}s_{\mathit{halt}}$. It is clear that $\mathcal{A}$ is deterministic and it admits only three types of traces:

From $s_{0}$ through some other locations of $\mathcal{S}$ and finally $s_{\mathit{halt}}$, i.e. those respecting the transition relation, but not necessarily the semantics, of $\mathcal{S}$. 2. 2.

From $s_{0}$ to $s_{\mathit{halt}}$ in a single transition (on which $p^{\mathit{read}}$ holds). 3. 3.

From $s_{0}$ to $s_{1}$ and then $s_{\mathit{halt}}$.

It remains to write a specification $\varphi$ such that $\llbracket\mathcal{A}\rrbracket\models\varphi$ exactly when $\mathcal{A}$ accepts a trace of type ($1$) that also respects the semantics of $\mathcal{S}$. This is where the traces of types ($2$) and ($3$) come into play: for example, if a trace of type ($1$) issues a read $m?$ without a corresponding write $m!$, then a trace of type ($3$) can be used to ‘pinpoint’ the error. More precisely, let $\varphi=\exists\pi_{a}\,\forall\pi_{b}\,(\psi_{1}\wedge\psi_{2}\wedge\psi_{3}\wedge\psi_{4})$ where

• •

$\psi_{1}=\operatorname{\mathbf{\overline{F}}}p^{\mathit{end}}_{a}$ ensures that $\pi_{a}$ is of type ($1$);

• •

$\psi_{2}=\operatorname{\mathbf{\overline{F}}}(p^{\mathit{read}}_{b}\wedge\psi_{\mathit{R}})\Rightarrow\operatorname{\mathbf{\overline{F}}}(p^{\mathit{read}}_{b}\wedge\operatorname{\vartriangleleft}_{\geq 1}p^{\mathit{begin}}_{a})$, where $\psi_{\mathit{R}}=\bigvee\{m^{?}_{a}\mid m\in M\}$, is a simple sanity check which ensures that in $\pi_{a}$, each $m^{?}$ must happen at time $\geq t+1$ if $p^{\mathit{begin}}$ happens at $t$;

• •

\psi_{3}=\bigwedge_{m\in M}\Big{(}\operatorname{\mathbf{\overline{F}}}(q^{1}_{b}\wedge m_{a}^{?})\Rightarrow\big{(}\operatorname{\mathbf{\overline{F}}}(p^{\mathit{begin}}_{a}\wedge\operatorname{\mathbf{\overline{F}}}p^{1}_{b})\wedge\operatorname{\mathbf{\overline{F}}}(q^{1}_{b}\wedge\operatorname{\vartriangleleft}_{=1}p^{1}_{b})\Rightarrow\operatorname{\mathbf{\overline{F}}}(p^{1}_{b}\wedge m_{a}^{!})\big{)}\Big{)} ensures that each $m^{?}$, if it happens at $t$, is preceded by a corresponding $m^{!}$ at $t-1$ in $\pi_{a}$;

• •

\psi_{4}=\bigwedge_{m\in M}\Big{(}\operatorname{\mathbf{\overline{F}}}(p^{1}_{b}\wedge m_{a}^{!})\Rightarrow\operatorname{\mathbf{\overline{F}}}(p^{\mathit{end}}_{a}\wedge\operatorname{\vartriangleleft}_{<1}p^{1}_{b})\vee\big{(}\operatorname{\mathbf{\overline{F}}}(q^{1}_{b}\wedge\operatorname{\vartriangleleft}_{=1}p^{1}_{b})\Rightarrow\operatorname{\mathbf{\overline{F}}}(q^{1}_{b}\wedge m_{a}^{?})\big{)}\Big{)} ensures that each $m^{!}$ at $t$ is followed by a corresponding $m^{?}$ at $t+1$ (unless $p^{\mathit{end}}$ happens first) in $\pi_{a}$.

Now observe that the only timed subformulae are $\operatorname{\vartriangleleft}_{\geq 1}p^{\mathit{begin}}_{a}$, $\operatorname{\vartriangleleft}_{=1}p^{1}_{b}$, and $\operatorname{\vartriangleleft}_{<1}p^{1}_{b}$. As $p^{1}$ and $p^{\mathit{read}}$ cannot happen in the same trace ($\pi_{b}$), it is not hard to see that the reduction remains correct if we replace these by $\operatorname{\vartriangleleft}_{\geq 1}(p^{\mathit{begin}}_{a}\vee p^{1}_{b})$, $\operatorname{\vartriangleleft}_{=1}(p^{\mathit{begin}}_{a}\vee p^{1}_{b})$, and $\operatorname{\vartriangleleft}_{<1}(p^{\mathit{begin}}_{a}\vee p^{1}_{b})$ (respectively) to obtain $\psi_{2}^{\prime}$, $\psi_{3}^{\prime}$, and $\psi_{4}^{\prime}$. It follows that $\psi_{1}\wedge\psi_{2}^{\prime}\wedge\psi_{3}^{\prime}\wedge\psi_{4}^{\prime}$ can be translated into a one-clock deterministic TA. Finally, it is possible to move all the timing constraint into the model and use an untimed LTL formula as the specification: in the model, ensure that $p^{1}$ and $q^{1}$ are separated by exactly $1$ time unit, and add $s_{0}\xrightarrow{\{p^{2}\}}s_{1}\xrightarrow{\{q^{2}\}}s_{\mathit{halt}}$ such that $p^{2}$ and $q^{2}$ are separated by $<1$ time unit; in the specification, use $p^{2}$, $q^{2}$ to rule out those $\pi_{a}$’s with some $m^{?}$ at $<1$ time unit from $p^{\mathit{begin}}$.

Now we consider the synchronous semantics. The corresponding result is weaker in this case, as we will see in the next section that in several subcases the problem becomes decidable. Still, the reduction above can be made to work if the model has one clock and an extra trace quantifier is allowed.

Theorem 4.3

Model checking $\exists^{\ast}\forall^{\ast}$-HyperMTL and $\forall^{\ast}\exists^{\ast}$-HyperMTL are undecidable in the synchronous semantics.

Proof

We use a modified model $\mathcal{A}^{\prime}$ whose set of locations is $S\cup\{s_{1},s_{2},s_{3},s_{4}\}$; the transitions are similar to $\mathcal{A}$ in the proof of Theorem 4.2, but we now use a clock $x$ in the path $s_{0}\xrightarrow[x:=0]{\{p^{1}\}}s_{1}\xrightarrow[x\geq 1,x:=0]{\{q^{1}\}}s_{\mathit{halt}}$, the paths $s_{0}\xrightarrow[x:=0]{\{p^{2}\}}s_{2}\xrightarrow[x\leq 1,x:=0]{\{q^{2}\}}s_{\mathit{halt}}$, $s_{0}\xrightarrow[x:=0]{\{p^{3}\}}s_{3}\xrightarrow[x>1,x:=0]{\{q^{3}\}}s_{\mathit{halt}}$, and $s_{0}\xrightarrow[x:=0]{\{p^{4}\}}s_{4}\xrightarrow[x<1,x:=0]{\{q^{4}\}}s_{\mathit{halt}}$ are added, and $s_{0}\xrightarrow{\{p^{\mathit{read}}\}}s_{\mathit{halt}}$ is removed. Moreover, a self-loop labelled with $\{p^{\epsilon}\}$ is added to each of $s_{0}$, $s_{1}$, $s_{2}$, $s_{3}$, $s_{4}$, and $s_{\mathit{halt}}$. The specification is $\varphi^{\prime}=\exists\pi_{a}\,\forall\pi_{b}\,\forall\pi_{c}\bigwedge_{1\leq i\leq 9}\psi_{i}^{\prime}$ where $\bigwedge_{1\leq i\leq 9}\psi_{i}^{\prime}$ is the following untimed LTL formula:

• •

$\psi_{1}^{\prime}=\operatorname{\mathbf{\overline{F}}}p^{\mathit{end}}_{a}$;

• •

$\psi_{2}^{\prime}=\operatorname{\mathbf{\overline{F}}}(q^{4}_{b}\wedge\psi_{\mathit{R}})\Rightarrow\neg\operatorname{\mathbf{\overline{F}}}(p^{4}_{b}\wedge p^{\mathit{begin}}_{a})$ where $\psi_{\mathit{R}}=\bigvee\{m^{?}_{a}\mid m\in M\}$;

• •

\psi_{3}^{\prime}=\bigwedge_{m\in M}\big{(}\operatorname{\mathbf{\overline{F}}}(q^{1}_{b}\wedge q^{2}_{c}\wedge m_{a}^{?})\wedge\operatorname{\mathbf{\overline{F}}}(p^{1}_{b}\wedge p^{2}_{c})\Rightarrow\operatorname{\mathbf{\overline{F}}}(p^{1}_{b}\wedge p^{2}_{c}\wedge m_{a}^{!})\big{)};

• •

$\psi_{4}^{\prime}=\operatorname{\mathbf{\overline{F}}}(q^{3}_{b}\wedge\psi_{\mathit{R}})\Rightarrow\neg\operatorname{\mathbf{\overline{F}}}(p^{3}_{b}\wedge\operatorname{\mathbf{X}}q^{3}_{b})$;

• •

$\psi_{5}^{\prime}=\operatorname{\mathbf{\overline{F}}}(q^{3}_{b}\wedge q^{4}_{c}\wedge\psi_{\mathit{R}})\Rightarrow\neg\operatorname{\mathbf{\overline{F}}}(p^{3}_{b}\wedge\operatorname{\mathbf{X}}p^{4}_{c})$;

• •

$\psi_{6}^{\prime}=\operatorname{\mathbf{\overline{F}}}(p^{4}_{b}\wedge\psi_{\mathit{W}})\Rightarrow\neg\operatorname{\mathbf{\overline{F}}}(q^{4}_{b}\wedge\neg\operatorname{\mathbf{X}}\top)$ where $\psi_{\mathit{W}}=\bigvee\{m^{!}_{a}\mid m\in M\}$;

• •

\psi_{7}^{\prime}=\bigwedge_{m\in M}\Big{(}\operatorname{\mathbf{\overline{F}}}(p^{1}_{b}\wedge p^{2}_{c}\wedge m_{a}^{!})\wedge\operatorname{\mathbf{\overline{F}}}(q^{1}_{b}\wedge q^{2}_{c})\Rightarrow\operatorname{\mathbf{\overline{F}}}\big{(}q^{1}_{b}\wedge q^{2}_{c}\wedge(m_{a}^{?}\vee p_{a}^{\epsilon})\big{)}\Big{)};

• •

$\psi_{8}^{\prime}=\operatorname{\mathbf{\overline{F}}}(p^{3}_{b}\wedge\psi_{\mathit{W}})\Rightarrow\neg\operatorname{\mathbf{\overline{F}}}(p^{3}_{b}\wedge\operatorname{\mathbf{X}}q^{3}_{b})$;

• •

$\psi_{9}^{\prime}=\operatorname{\mathbf{\overline{F}}}(p^{3}_{b}\wedge p^{4}_{c}\wedge\psi_{\mathit{W}})\Rightarrow\neg\operatorname{\mathbf{\overline{F}}}(q^{3}_{b}\wedge\operatorname{\mathbf{X}}q^{4}_{c})$.

In this modified reduction, $\psi_{1}^{\prime}$, $\psi_{2}^{\prime}$ play similar roles as $\psi_{1}$, $\psi_{2}$ in the proof of Theorem 4.2. $\psi_{3}^{\prime}$ ensures that if each $m^{?}$ at $t$ is preceded by an event at $t-1$, then $m^{!}$ must hold there. $\psi_{4}^{\prime}$ and $\psi_{5}^{\prime}$ ensures that each $m^{?}$ at $t$ is actually preceded by an event at $t-1$. The roles of $\psi_{6}^{\prime}$, $\psi_{7}^{\prime}$, $\psi_{8}^{\prime}$, and $\psi_{8}^{\prime}$ are analogous (note the use of silent events at the end of $\pi_{a}$).

5 Decidable subcases

While the negative results in the previous section may be disappointing, we stress again that model checking alternation-free HyperMTL is no harder than MTL model checking, and it can in fact be carried out with algorithms and tools for the latter. In any case, we now identify several subcases where model checking is decidable beyond the alternation-free fragment.

Untimed model + untimed specification.

The first case we consider is when both the model and the specification are untimed, and the asynchronous semantics is assumed (this case is simply HyperLTL model checking in the synchronous semantics). Our algorithm follows the lines of [16] and is essentially based on self-composition (cf. [10], and many others; see the references in [16]) of the model; the difficulty here, however, is to handle interleaving of events. Let the model $\mathcal{A}$ be a finite automaton over $\Sigma_{\textup{{{AP}}}}$ and the specification be a (untimed) closed HyperMTL formula over AP. Without loss of generality, we assume the specification to be $\varphi=\exists\pi_{1}\,\forall\pi_{2}\,\dots\exists\pi_{k-1}\,\forall\pi_{k}\,\psi$, which can be rewritten into $\exists\pi_{1}\,\neg\exists\pi_{2}\,\neg\dots\neg\exists\pi_{k-1}\,\neg\exists\pi_{k}\,\neg\psi$. We start by translating $\mathit{stutter}(\neg\psi)$ (in which we replace all occurrences of $\top_{i}$ with $\neg p^{\epsilon}_{i}$, i.e. regarded here simply as an MTL formula over $(\textup{{{AP}}}_{\epsilon})^{k}=\{p_{i}\mid p\in\textup{{{AP}}}_{\epsilon},1\leq i\leq k\}$) into the equivalent finite automaton over $\Sigma_{(\textup{{{AP}}}_{\epsilon})^{k}}$, and take its product with (i) the automaton for \operatorname{\mathbf{\overline{G}}}(\bigvee_{1\leq i\leq k}\neg p^{\epsilon}_{i})\wedge\big{(}\bigwedge_{1\leq i\leq k}\operatorname{\mathbf{\overline{G}}}(p^{\epsilon}_{i}\Rightarrow\bigwedge_{p\in\textup{{{AP}}}}\neg p_{i})\big{)} and (ii) the automaton obtained from $\mathit{stutter}(\mathcal{A})$ by extending the alphabet to $\Sigma_{(\textup{{{AP}}}_{\epsilon})^{k}}$ and renaming all the occurrences of $p$ to $p_{k}$, to obtain $\mathcal{B}$. Now let $\mathcal{C}$ be the projection of $\mathcal{B}$ onto $(\textup{{{AP}}}_{\epsilon})^{k-1}=\{p_{i}\mid p\in\textup{{{AP}}}_{\epsilon},1\leq i\leq k-1\}$ (this step corresponds to $\exists$ in $\neg\exists\pi_{k}$). By construction, $\mathcal{B}$ accepts only traces that are well-formed in dimensions $1$ to $k-1$, and so does $\mathcal{C}$; but $\mathcal{C}$ may accept traces containing $\{p^{\epsilon}_{i}\mid 1\leq i\leq k-1\}$-events. We replace these events by $\epsilon$ (the ‘real’ silent event, which can be removed with the standard textbook constructions, e.g., [38]) to obtain $\mathcal{C}^{\prime}$. Finally, we complement $\mathcal{C}^{\prime}$ to obtain $\mathcal{C}^{\prime\prime}$ (this step corresponds to $\neg$ in $\neg\exists\pi_{k}$). We can then start over by taking the product of $\mathcal{C}^{\prime\prime}$, the automaton for \operatorname{\mathbf{\overline{G}}}(\bigvee_{1\leq i\leq k-1}\neg p^{\epsilon}_{i})\wedge\big{(}\bigwedge_{1\leq i\leq k-1}\operatorname{\mathbf{\overline{G}}}(p^{\epsilon}_{i}\Rightarrow\bigwedge_{p\in\textup{{{AP}}}}\neg p_{i})\big{)}, and the automaton obtained from $\mathit{stutter}(\mathcal{A})$ by extending the alphabet to $\Sigma_{(\textup{{{AP}}}_{\epsilon})^{k-1}}$ and renaming all the occurrences of $p$ to $p_{k-1}$; the resulting automaton is the new $\mathcal{B}$. We continue this process until the outermost quantifier $\exists\pi_{1}$ is reached, when we test the emptiness of $\mathcal{B}$ (at this point, it is an automaton over $\Sigma_{\textup{{{AP}}}_{\epsilon}}$).

Proposition 1

Model checking HyperMTL is decidable when the model and the specification are both untimed.

One clock + one alternation.

The algorithm outlined in the previous case crucially depends on the fact that both $\mathcal{A}$ and $\varphi$ are untimed, hence their product (in the sense detailed in the previous case) can be complemented. When the synchronous semantics is assumed and there are is only one quantifier alternation in $\varphi$, it might be the case that we do not actually need complementation. For example, if $\mathcal{A}$ is untimed and $\varphi=\forall\pi_{a}\,\exists\pi_{b}\,\exists\pi_{c}\,\psi$ where $\psi$ translates into a one-clock TA, the corresponding model-checking problem clearly reduces to universality for one-clock TAs, which is decidable but non-primitive recursive [1].333This case is undecidable in the asynchronous semantics by Theorem 4.2; as explained above, the algorithm may introduce $\epsilon$-transitions in the asynchronous semantics, while universality for one-clock TAs with $\epsilon$-transitions is undecidable [1]. This observation applies to other cases as well, such as when $\mathcal{A}$ is a one-clock TA and $\varphi=\exists\pi_{a}\,\forall\pi_{b}\,\psi$ where $\psi$ is untimed; here model checking reduces to language inclusion between two one-clock TAs.

Untimed model + MIA specification.

The main obstacle in applying the algorithm above to larger fragments of HyperMTL, as should be clear now, is that universal quantifiers amount to complementations, which are not possible in general in the case of TAs. Moreover, we note that the usual strategy of restricting to deterministic models and specifications does not help, as the projection step in the algorithm necessarily introduces non-determinism. To make the algorithm work for larger fragments, we essentially need a class of automata that is both closed under projection and complementable. Fortunately, there is a subclass of one-clock TAs that satisfies these conditions. We consider two additional restrictions on one-clock TAs:

• •

Non-Singular (NS): a one-clock TA is NS if all the guards are non-singular.

• •

Reset-on-Testing (RoT): a one-clock TA is RoT if whenever the guard of a transition is not $\top$, the single clock $x$ must be reset on that transition.

One-clock TAs satisfying both NS and RoT are called metric interval automata (MIAs), which are determinisable [22]. Since the projection operation cannot invalidate NS and RoT, the algorithm above can be applied when the synchronous semantics is assumed, $\mathcal{A}$ is untimed, $\psi$ or $\neg\psi$ translates to a MIA, and only one complementation is involved; in this case it runs in elementary time.

Proposition 2

Model checking $\forall^{\ast}\exists^{\ast}$-HyperMTL ($\exists^{\ast}\forall^{\ast}$-HyperMTL) is decidable in the synchronous semantics when the model is untimed and $\psi$ ($\neg\psi$) translates into a MIA in the specification $\varphi=\forall\pi_{1}\,\dots\exists\pi_{k}\,\psi$ ($\varphi=\exists\pi_{1}\,\dots\forall\pi_{k}\,\psi$).

On the other hand, we can adapt the proof of Theorem 4.3 to show that model checking an untimed model against an $\exists^{\ast}\forall^{\ast}$-HyperMTL specification $\varphi$ in the synchronous semantics, when the quantifier-free part $\psi$ (instead of $\neg\psi$) translates into a MIA, remains undecidable.

Proposition 3

Model checking $\exists^{\ast}\forall^{\ast}$-HyperMTL is undecidable in the synchronous semantics when the model is untimed and $\psi$ in the specification $\varphi=\exists\pi_{1}\,\dots\forall\pi_{k}\,\psi$ translates into a MIA.

Proof (sketch)

Similar to the proof of Theorem 4.3: we add a new initial location $s_{0}^{\prime}$ and the transitions $s_{0}^{\prime}\xrightarrow{\{p^{\epsilon}\}}s_{0}$, self-loops labelled with $\{p^{\epsilon}\}$ to $s_{0}$ and $s_{\mathit{halt}}$, and $s_{0}\xrightarrow{\{p^{3}\}}s_{3}\xrightarrow{\{q^{3}\}}s_{\mathit{halt}}$, $s_{0}\xrightarrow{\{p^{4}\}}s_{4}\xrightarrow{\{q^{4}\}}s_{\mathit{halt}}$, $s_{0}\xrightarrow{\{q^{5}\}}s_{\mathit{halt}}$. In the quantifier-free part $\psi^{\prime\prime}$ of the specification $\varphi^{\prime\prime}=\exists\pi_{a}\,\forall\pi_{b}\,\forall\pi_{c}\,\psi^{\prime\prime}$ we use, e.g., \bigwedge_{m\in M}\Big{(}\big{(}\operatorname{\mathbf{\overline{F}}}(q^{3}_{b}\wedge q^{4}_{c}\wedge m_{a}^{?})\wedge\operatorname{\mathbf{\overline{F}}}(q^{3}_{b}\wedge\operatorname{\vartriangleleft}_{>1}p^{3}_{b})\wedge\operatorname{\mathbf{\overline{F}}}(q^{4}_{c}\wedge\operatorname{\vartriangleleft}_{<1}p^{4}_{c})\big{)}\vee\big{(}\operatorname{\mathbf{\overline{F}}}(q^{3}_{b}\wedge q^{5}_{c}\wedge m_{a}^{?})\wedge\operatorname{\mathbf{\overline{F}}}(q^{3}_{b}\wedge\operatorname{\vartriangleleft}_{>1}p^{3}_{b})\big{)}\Rightarrow\operatorname{\mathbf{\overline{F}}}\big{(}p^{3}_{b}\wedge\operatorname{\mathbf{F}}(m_{a}^{!}\wedge\operatorname{\mathbf{F}}p^{4}_{c})\big{)}\vee\big{(}p^{3}_{b}\wedge\operatorname{\mathbf{F}}(m_{a}^{!}\wedge\operatorname{\mathbf{F}}q^{5}_{c})\big{)}\Big{)}. Finally, note that $\psi^{\prime\prime}$ can be translated into a MIA.

The decidability results in the synchronous semantics are summarised in Table 1.

Bounded time domains.

We end this section by showing that when there is an a priori bound $N$ (where $N$ is a positive integer) on the length of the time domain, the model-checking problem for full HyperMTL becomes decidable; in fact, in the case of synchronous semantics it reduces to the satisfiability problem for QPTL [56]. From a practical point of view, this implies that time-bounded HyperMTL verification (at least for the $\exists^{\ast}\forall^{\ast}$-fragment, say) can be carried out with highly efficient, off-the-shelf tools that works with LTL and (untimed) automata, such as SPOT [21], GOAL [57], and Owl [42].

We assume the asynchronous semantics. For a given $N$, we consider all traces in which all timestamps are less than $N$. Denote by $\llbracket\mathcal{A}\rrbracket_{[0,N)}$ the set of all such traces in $\llbracket\mathcal{A}\rrbracket$; the model-checking problem then becomes deciding whether $\llbracket\mathcal{A}\rrbracket_{[0,N)}\models\varphi$. As before, we assume $\varphi$ to be $\exists\pi_{1}\,\neg\exists\pi_{2}\,\neg\dots\neg\exists\pi_{k-1}\,\neg\exists\pi_{k}\,\neg\psi$. Following [50, 36], we can use the stacking construction to obtain, from the conjunction $\psi^{\prime}$ of $\mathit{stutter}(\neg\psi)$ and \operatorname{\mathbf{\overline{G}}}(\bigvee_{\pi\in\mathcal{Q}}\neg p^{\epsilon}_{\pi})\wedge\big{(}\bigwedge_{\pi\in\mathcal{Q}}\operatorname{\mathbf{\overline{G}}}(p^{\epsilon}_{\pi}\Rightarrow\bigwedge_{p\in\textup{{{AP}}}}\neg p_{\pi})\big{)}, an equi-satisfiable untimed (QPTL) formula $\overline{\varphi}=\exists W\,\overline{\psi^{\prime}}$ over the stacked alphabet $\overline{(\textup{{{AP}}}_{\epsilon})^{k}}\cup\overline{Q}$ (where $\overline{(\textup{{{AP}}}_{\epsilon})^{k}}=\{p_{i,j}\mid p\in\textup{{{AP}}}_{\epsilon}\text{, }1\leq i\leq k\text{, }0\leq j<N\}$ and $\overline{Q}=\{q_{j}\mid 0\leq j<N\}$). We apply the following modifications to $\overline{\varphi}$ to obtain $\overline{\varphi^{\prime}}$:

• •

introduce atomic propositions $\{p^{\epsilon}_{i}\mid 1\leq i\leq k\}$ and add the formula \bigwedge_{1\leq i\leq k}\operatorname{\mathbf{\overline{G}}}\Big{(}\big{(}\bigwedge_{0\leq j<N}(q_{j}\Rightarrow p^{\epsilon}_{i,j})\big{)}\Leftrightarrow p^{\epsilon}_{i}\Big{)} as a conjunct;

• •

introduce atomic propositions $\{q_{i,j}\mid 1\leq i\leq k\text{, }0\leq j<N\}$ and add the formula \bigwedge_{1\leq i\leq k}\operatorname{\mathbf{\overline{G}}}\big{(}\bigwedge_{0\leq j<N}(\neg p^{\epsilon}_{i}\wedge q_{j}\Leftrightarrow q_{i,j})\big{)} as a conjunct;

• •

project away $\{p^{\epsilon}_{i,j}\mid 1\leq i\leq k\text{, }0\leq j<N\}$ and $\overline{Q}$.

• •

replace all occurrences of $p^{\epsilon}_{i}$ by $\bot_{i}$.

Now, as we mentioned earlier, we can write $\mathcal{A}$ as an (MSO[$<,+1$] [50]) formula $\varphi_{\mathcal{A}}=\exists X_{\mathcal{A}}\,\psi_{\mathcal{A}}$ where $X_{\mathcal{A}}$ is a set of atomic propositions such that $\textup{{{AP}}}\cap X_{\mathcal{A}}=\emptyset$ and $\psi_{\mathcal{A}}$ is an MTL formula over $\textup{{{AP}}}\cup X_{\mathcal{A}}$. Let $\overline{\varphi_{\mathcal{A}}}$ be its stacked counterpart $\exists\overline{X_{\mathcal{A}}}\,\exists Y\,\overline{\psi_{\mathcal{A}}}$; we translate $\overline{\varphi_{\mathcal{A}}}$ back into an untimed automaton $\overline{\mathcal{A}}$ over the stacked alphabet $\overline{\textup{{{AP}}}}\cup\overline{Q}$. The problem thus reduces to untimed model checking of $\overline{\mathcal{A}}$ against $\exists\pi_{1}\,\forall\pi_{2}\,\dots\exists\pi_{k-1}\,\forall\pi_{k}\,\overline{\varphi^{\prime}}$ in the asynchronous semantics, which is decidable by Proposition 1 ($\overline{\varphi^{\prime}}$ has outermost existential propositional quantifiers, but clearly the equivalent automaton can be used directly in the algorithm).

Finally, note that the proof is simpler for the case of synchronous semantics: we can simply work with a (non-stuttering) MSO[$<,+1$] formula in all the intermediate steps without translating it into an automaton, and then check the satisfiability of the final formula by stacking it into a QPTL formula.

Proposition 4

Model checking HyperMTL is decidable when the time domain is $[0,N)$, where $N$ is a given positive integer.

6 Conclusion

We studied the satisfiability and model-checking problems for HyperMTL over sets of timed words. While satisfiability can be solved similarly as for HyperLTL, model checking turned out to be undecidable when the specification involves at least one quantifier alternation; this holds even for otherwise fairly restricted models and specifications. On the other hand, we showed that model checking beyond the alternation-free fragment is possible if (i) interleaving of events in different traces is disallowed, or (ii) the time domain is $N$-bounded for a fixed positive integer $N$. We leave as future work to investigate whether a suitable notion of ‘timing fuzziness’ (e.g., [9, 33, 18]) can be incorporated, either to recover decidability of model checking or better align with practical applications, e.g., monitoring of cyber-physical systems [12, 11]. Another possible direction is to consider the case where the number of events [48] (or more generally, the number of events in any interval of fixed length [58, 30]) in any trace is bounded, which may be sufficient for modelling many real-world systems.