Complete Abstractions for Checking Language Inclusion

TL;DR

This paper introduces a novel abstract interpretation framework for deciding language inclusion problems, leveraging overapproximations via quasiorders, applicable to regular, context-free, and other language classes, with connections to existing algorithms.

Contribution

It develops a general method using quasiorder-based overapproximations for deciding language inclusion, unifying and extending existing algorithms, and introduces a new fixpoint-based inclusion checking algorithm.

Findings

Decidability of language inclusion under certain abstraction conditions.

Systematic design of inclusion algorithms for various language classes.

Connection of new methods to existing antichain algorithms.

Abstract

We study the language inclusion problem $L_1 \subseteq L_2$ where $L_1$ is regular or context-free. Our approach relies on abstract interpretation and checks whether an overapproximating abstraction of $L_1$, obtained by overapproximating the Kleene iterates of its least fixpoint characterization, is included in $L_2$. We show that a language inclusion problem is decidable whenever this overapproximating abstraction satisfies a completeness condition (i.e., its loss of precision causes no false alarm) and prevents infinite ascending chains (i.e., it guarantees termination of least fixpoint computations). This overapproximating abstraction of languages can be defined using quasiorder relations on words, where the abstraction gives the language of all the words "greater than or equal to" a given input word for that quasiorder. We put forward a range of such quasiorders that allow us to systematically design decision procedures for different language inclusion problems such as regular languages into regular languages or into trace sets of one-counter nets, and context-free languages into regular languages. In the case of inclusion between regular languages, some of the induced inclusion checking procedures correspond to well-known state-of-the-art algorithms like the so-called antichain algorithms. Finally, we provide an equivalent language inclusion checking algorithm based on a greatest fixpoint computation that relies on quotients of languages and, to the best of our knowledge, was not previously known.

Equations

$$X\sqsubseteq Y\>\stackrel{{\scriptstyle{\mbox{\tiny$\triangle$}}}}{{\Leftrightarrow}}\>\forall x\in X,\exists y\in Y,\;y\leqslant x\enspace.$$

$$c_1{\le }_C{c}_2\iff \rho (c_1){\le }_C\rho (c_2)\iff \rho (c_1){\le }_C{c}_2$$

$$\rho (\vee X)=\rho (\vee \rho (X))\text{and}\wedge \rho (X)=\rho (\wedge \rho (X)).$$

$$\gamma (\mathrm{lfp}(\alpha f\gamma )){\le }_C\gamma (\mathrm{lfp}(\alpha f\gamma ))$$

$$\gamma \alpha f(\gamma (\mathrm{lfp}(\alpha f\gamma ))){\le }_C\gamma (\mathrm{lfp}(\alpha f\gamma ))$$

$$\mathrm{lfp}(\gamma \alpha f){\le }_C\gamma (\mathrm{lfp}(\alpha f\gamma ))$$

$$\mathrm{lfp}(\gamma \alpha f){\le }_C\mathrm{lfp}(\gamma \alpha f)$$

$$\gamma \alpha f(\mathrm{lfp}(\gamma \alpha f)){\le }_C\mathrm{lfp}(\gamma \alpha f)$$

$$\alpha \gamma \alpha f(\mathrm{lfp}(\gamma \alpha f)){\le }_A\alpha (\mathrm{lfp}(\gamma \alpha f))$$

$$\alpha f(\mathrm{lfp}(\gamma \alpha f)){\le }_A\alpha (\mathrm{lfp}(\gamma \alpha f))$$

$$\alpha f\gamma (\alpha (\mathrm{lfp}(\gamma \alpha f))){\le }_A\alpha (\mathrm{lfp}(\gamma \alpha f))$$

$$\mathrm{lfp}(\alpha f\gamma ){\le }_A\alpha (\mathrm{lfp}(\gamma \alpha f))$$

$$\gamma (\mathrm{lfp}(\alpha f\gamma )){\le }_C\gamma \alpha (\mathrm{lfp}(\gamma \alpha f))$$

$$\gamma (\mathrm{lfp}(\alpha f\gamma )){\le }_C\mathrm{lfp}(\gamma \alpha f)$$

$$\rho f=\rho f\rho \Rightarrow \rho ({\mathrm{lfp}}_x(f))={\mathrm{lfp}}_x(\rho f)={\mathrm{lfp}}_x(\rho f\rho )$$

$$\operatorname{{\textsc{Kleene}}}(\operatorname{{Conv}},f,a)\triangleq\left\{\begin{array}[]{l}x:=a;\\ \textbf{while~{}}\neg\operatorname{{Conv}}(f(x),x)\textbf{~{}do~{}}x:=f(x);\\ \textbf{return~{}}x;\end{array}\right.$$

$${\mathrm{Incl}}_{\rho }\triangleq \{(x,y)\in C\times C\mid \rho (x){\le }_C\rho (y)\}.$$

$$\forall n\in \mathbb{N},\rho \text{comp}{f}^n=(\rho \text{comp}f{)}^n\text{comp}\rho .$$

$$\rho \text{comp}{f}^{n+1}$$

$$\rho \text{comp}{f}^n\text{comp}f$$

$$(\rho \text{comp}f{)}^n\text{comp}\rho \text{comp}f$$

$$(\rho \text{comp}f{)}^n\text{comp}\rho \text{comp}f\text{comp}\rho$$

$$(\rho \text{comp}f{)}^{n+1}\text{comp}\rho$$

$$W_{S,T}^{\mathcal{A}}\triangleq \{u\in {\Sigma }^{\ast }\mid \exists q\in S,\exists q^{\prime }\in T,q\stackrel{u}{⇝}{q}^{\prime }\}.$$

$$\mathcal{L}(\mathcal{A})={\bigcup }_{q\in I}{W}_{q,F}^{\mathcal{A}}={\bigcup }_{q\in F}{W}_{I,q}^{\mathcal{A}}$$

$${\psi }_F^T(p(x))\triangleq \{\begin{array}{ll}T& \text{if p(x) holds}\\ F& \text{otherwise}\end{array}.$$

$$\mathrm{Eqn}(\mathcal{A})\triangleq \{X_q={\psi }_{\varnothing }^{\{ϵ\}}(q{\in }^{\text{scaleto}?3.5pt}F)\cup {\bigcup }_{a\in \Sigma ,q^{\prime }\in \delta (q,a)}a{X}_{q^{\prime }}\mid q\in Q\}.$$

$$\mathrm{Eqn}(\mathcal{A})=\{\begin{array}{l}{X}_1=\{ϵ\}\cup a{X}_1\cup b{X}_2\\ X_2=\varnothing \cup a{X}_1\cup b{X}_2\end{array}.$$

$$\text{vv}{\mathbf{ϵ}}^F$$

$$⟨W_{q,F}^{\mathcal{A}}{⟩}_{q\in Q}=\mathrm{lfp}(\lambda \text{vv}\mathbf{X}.\text{vv}{\mathbf{ϵ}}^F\cup {\mathrm{Pre}}_{\mathcal{A}}(\text{vv}\mathbf{X})).$$

$$◊$$

$$\text{vv}{{\mathbf{L}}_2}^I\triangleq ⟨{\psi }_{{\Sigma }^{\ast }}^{L_2}(q{\in }^{\text{scaleto}?3.5pt}I){⟩}_{q\in Q}$$

$$\mathcal{L}(\mathcal{A})\subseteq L_2\iff \mathrm{lfp}(\lambda \text{vv}\mathbf{X}.\text{vv}{\mathbf{ϵ}}^F\cup {\mathrm{Pre}}_{\mathcal{A}}(\text{vv}\mathbf{X}))\subseteq \text{vv}{{\mathbf{L}}_2}^I.$$

$$\rho ({\mathrm{Pre}}_{\mathcal{A}}(⟨X_q{⟩}_{q\in Q}))$$

$$\rho ({\bigcup }_{a\in \Sigma ,q^{\prime }\in \delta (q,a)}a{X}_{q^{\prime }})$$

$$\rho ({\bigcup }_{a\in \Sigma ,q^{\prime }\in \delta (q,a)}\rho (a{X}_{q^{\prime }}))$$

$$\rho ({\bigcup }_{a\in \Sigma ,q^{\prime }\in \delta (q,a)}\rho (a\rho (X_{q^{\prime }})))$$

$$\rho ({\bigcup }_{a\in \Sigma ,q^{\prime }\in \delta (q,a)}a\rho (X_{q^{\prime }}))$$

$$\rho ({\mathrm{Pre}}_{\mathcal{A}}(\rho (⟨X_q{⟩}_{q\in Q})))$$

$$\rho (\text{vv}{\mathbf{ϵ}}^F\cup {\mathrm{Pre}}_{\mathcal{A}}(\rho (\text{vv}\mathbf{X})))$$

Full text

Abstract interpretation, completeness, language inclusion, regular language, context-free language, one-counter net, automaton, grammar.

††copyright: acmcopyright††journal: TOCL††ccs: Theory of computation Regular languages††ccs: Theory of computation Grammars and context-free languages††ccs: Theory of computation Abstraction††ccs: Theory of computation Program reasoning††ccs: Software and its engineering Formal language definitions

1. Introduction

Language inclusion is a fundamental and classical problem (Hopcroft and Ullman, 1979, Chapter 11) which consists in deciding, given two languages $L_{1}$ and $L_{2}$, whether $L_{1}\subseteq L_{2}$ holds. Language inclusion problems are found in diverse fields ranging from compiler construction (Bauer and Eickel, 1976; Waite and Goos, 1984) to model checking (Baier and Katoen, 2008; Clarke et al., 2018). We consider languages of finite words over a finite alphabet $\Sigma$. For regular and context-free languages, the inclusion problem is well known to be PSPACE-complete (see (Hunt et al., 1976)).

The basic idea of our approach for solving a language inclusion problem $L_{1}\subseteq L_{2}$ is to leverage Cousot and Cousot’s abstract interpretation (Cousot and Cousot, 1977, 1979) for checking the inclusion of an overapproximation (i.e., a superset) of $L_{1}$ into $L_{2}$. This idea draws inspiration from the work of Hofmann and Chen (2014), who used abstract interpretation to decide language inclusion between languages of infinite words.

Let us assume that $L_{1}$ is specified as least fixpoint of an equation system $X=F_{L_{1}}(X)$ on sets of words in $\wp(\Sigma^{*})$, that is, $L_{1}=\operatorname{lfp}(F_{L_{1}})$ is viewed as limit of the possibly infinite sequence of Kleene iterates $\{F^{n}_{L_{1}}(\varnothing)\}_{n\in\mathbb{N}}$ of the transformer $F_{L_{1}}$. An approximation of $L_{1}$ is obtained by applying an overapproximation for sets of words as modeled by a closure operator $\rho:\wp(\Sigma^{*})\rightarrow\wp(\Sigma^{*})$. In abstract interpretation one such closure $\rho$ logically defines an abstract domain, which is here used for overapproximating a language by adding words to it, possibly none in case of no approximation. The language abstraction $\rho$ is then used for defining an abstract check of convergence for the Kleene iterates of $F_{L_{1}}$ whose limit is $L_{1}$, i.e., the convergence of the sequence $\{F^{n}_{L_{1}}(\varnothing)\}_{n\in\mathbb{N}}$ is checked on the abstraction $\rho$ by the condition $\rho(F^{n+1}_{L_{1}}(\varnothing))\subseteq\rho(F^{n}_{L_{1}}(\varnothing))$. If the abstraction $\rho$ does not contain infinite ascending chains then we obtain finite convergence w.r.t. this abstract check for some $F_{L_{1}}^{N}(\varnothing)$.

Therefore, this abstract interpretation-based approach finitely computes an abstraction $L_{1}^{\rho}=\rho(F_{L_{1}}^{N}(\varnothing))$ such that the abstract language inclusion check $L_{1}^{\rho}\subseteq L_{2}$ is sound because $L_{1}\subseteq L_{1}^{\rho}$ always holds. We then give conditions on $\rho$ which ensure a complete abstract inclusion check, namely, the answer to $L_{1}^{\rho}\subseteq L_{2}$ is always exact (no “false alarm” in abstract interpretation terminology):

1. (i)

$L_{2}$ is exactly represented by the abstraction $\rho$, i.e., $\rho(L_{2})=L_{2}$; 2. (ii)

$\rho$ is a complete abstraction for symbol concatenation $\lambda X\in\wp(\Sigma^{*}).\,aX$, for all $a\in\Sigma$, according to the standard notion of completeness in abstract interpretation (Cousot and Cousot, 1977); this entails that $\rho(L_{1})=L_{1}^{\rho}$ holds, so that $L_{1}^{\rho}\not\subseteq L_{2}$ implies $L_{1}\not\subseteq L_{2}$.

This approach leads us to design a general algorithmic framework for language inclusion problems which is parameterized by an underlying language abstraction.

We then focus on language abstractions $\rho$ which are induced by a quasiorder relation on words $\mathord{\leqslant}\subseteq\Sigma^{*}\times\Sigma^{*}$. Here, a language $L$ is overapproximated by adding all the words which are “greater than or equal to” some word of $L$ for $\mathord{\leqslant}$. This allows us to instantiate the above conditions (i) and (ii) for achieving a complete abstract inclusion check in terms of the quasiorder relation $\mathord{\leqslant}$. Termination, which corresponds to having finitely many Kleene iterates, is guaranteed by requiring that the relation $\mathord{\leqslant}$ is a well-quasiorder.

We define well-quasiorders satisfying the conditions (i) and (ii) which are directly derived from the standard Nerode equivalence relations on words. These quasiorders have been first investigated by Ehrenfeucht et al. (1983) and have been later generalized and extended by de Luca and Varricchio (1994; 2011). In particular, drawing from a result by de Luca and Varricchio (1994), we show that the language abstractions induced by the Nerode quasiorders are the most general ones (intuitively, optimal) which fit in our algorithmic framework for checking language inclusion. While these quasiorder abstractions do not depend on some finite representation of languages (e.g., some class of automata), we provide quasiorders which instead exploit an underlying language representation given by a finite automaton. In particular, by selecting suitable well-quasiorders for the class of language inclusion problems at hand we are able to systematically derive decision procedures of the inclusion problem $L_{1}\subseteq L_{2}$ for the following cases:

1. (1)

both $L_{1}$ and $L_{2}$ are regular; 2. (2)

$L_{1}$ is regular and $L_{2}$ is the trace language of a one-counter net; 3. (3)

$L_{1}$ is context-free and $L_{2}$ is regular.

These decision procedures, here systematically designed by instantiating our framework, are then related to existing language inclusion checking algorithms. We study in detail the case where both languages $L_{1}$ and $L_{2}$ are regular and represented by finite state automata. When our decision procedure for $L_{1}\subseteq L_{2}$ is derived from a well-quasiorder on $\Sigma^{*}$ by exploiting an automaton-based representation of $L_{2}$, it turns out that we obtain the well-known “antichain algorithm” by De Wulf et al. (2006). Also, by including a simulation relation in the definition of the well-quasiorder we derive a decision procedure that partially matches the language inclusion algorithm by Abdulla et al. (2010), and in turn also that by Bonchi and Pous (2013). It is also worth pointing out that for the case in which $L_{1}$ is regular and $L_{2}$ is the set of traces of a one-counter net, our systematic instantiation provides an alternative proof for the decidability of the corresponding language inclusion problem (Jančar et al., 1999).

Finally, we leverage a standard duality result between abstract least and greatest fixpoint checking (Cousot, 2000) and put forward a greatest fixpoint approach (instead of the above least fixpoint-based procedures) for the case where both $L_{1}$ and $L_{2}$ are regular languages. Here, we exploit the properties of the overapproximating abstraction induced by the quasiorder relation in order to show that the Kleene iterates converging to the greatest fixpoint are finitely many. Interestingly, the Kleene iterates of the greatest fixpoint are finitely many whether you apply the overapproximating abstraction or not, and this is shown by relying on a second type of completeness in abstract interpretation called forward completeness (Giacobazzi and Quintarelli, 2001).

Structure of the Article

In Section 2 we recall the needed basic notions and background on order theory, abstract interpretation and formal languages. Section 3 defines a general method for checking the convergence of Kleene iterates on an abstract domain, which provides the basis for designing in Section 4 an abstract interpretation-based framework for checking language inclusion, in particular by relying on abstractions that are complete for concatenation of languages. This general framework is instantiated in Section 5 to the class of abstractions induced by well-quasiorders on words, thus yielding effective inclusion checking algorithms for regular languages and traces of one-counter nets. Section 6 shows that one specific instance of our algorithmic framework turns out to be equivalent to the well-known antichain algorithm for language inclusion by De Wulf et al. (2006). The instantiation of the framework for checking the inclusion of context-free languages into regular languages is described in Section 7. Section 8 shows how to derive a new language inclusion algorithm which relies on the computation of a greatest fixpoint rather than a least fixpoint. Finally, Section 9 outlines some directions for future work.

This article is an extended and revised version of the conference paper (Ganty et al., 2019), that includes full proofs, additional detailed examples, a simplification of some technical notions, and a new application for checking the inclusion of context-free languages into regular languages.

2. Background

2.1. Order Theory

If $X$ is any set then $\wp(X)$ denotes its powerset. If $X$ is a subset of some universe set $U$ then $X^{c}$ denotes the complement of $X$ with respect to $U$ when $U$ is implicitly given by the context. If $f:X\rightarrow Y$ is a function between sets and $S\in\wp(X)$ then $f(S)\triangleq\{f(x)\in Y\mid x\in S\}$ denotes its image on a subset $S$. A composition of two functions $f$ and $g$ is denoted both by $fg$ and $f\comp g$.

$\langle{D,\mathord{\leqslant}}\rangle$ is a quasiordered set (qoset) when $\mathord{\leqslant}$ is a quasiorder (qo) relation on $D$, i.e. a reflexive and transitive binary relation $\mathord{\leqslant}\subseteq D\times D$. In a qoset $\langle{D,\mathord{\leqslant}}\rangle$ we will also use the following induced equivalence relation $\sim_{D}$: for all $d,d^{\prime}\in D$, d\sim_{D}d^{\prime}\mbox{\raisebox{0.0pt}[4.30554pt][4.30554pt]{\stackrel{{\scriptstyle{\mbox{\tiny$\triangle$}}}}{{\Leftrightarrow}}}}d\leqslant d^{\prime}\>\wedge\>d^{\prime}\leqslant d. A qoset satisfies the ascending (resp. descending) chain condition (ACC, resp. DCC) if there is no countably infinite sequence of distinct elements $\{x_{i}\}_{i\in\mathbb{N}}$ such that, for all $i\in\mathbb{N}$, $x_{i}\leqslant x_{i{+}1}$ (resp. $x_{i{+}1}\leqslant x_{i}$). A qoset is called ACC (DCC) when it satisfies the ACC (DCC).

A qoset $\langle{D,\mathord{\leqslant}}\rangle$ is a partially ordered set (poset) when $\mathord{\leqslant}$ is antisymmetric. A subset $X\subseteq D$ of a poset is directed if $X$ is nonempty and every pair of elements in $X$ has an upper bound in $X$. A poset $\langle{D,\mathord{\leqslant}}\rangle$ is a directed-complete partial order (CPO) if it has the least upper bound (lub) of all its directed subsets. A poset is a join-semilattice if it has the lub of all its nonempty finite subsets (therefore binary lubs are enough). A poset is a complete lattice if it has the lub of all its arbitrary (possibly empty) subsets; in this case, let us recall that it also has the greatest lower bound (glb) of all its arbitrary subsets.

An antichain in a qoset $\langle{D,\mathord{\leqslant}}\rangle$ is a subset $X\subseteq D$ such that any two distinct elements in $X$ are incomparable for $\leqslant$. We denote the set of antichains of a qoset $\langle{D,\mathord{\leqslant}}\rangle$ by $\operatorname{{AC}}_{\langle{D,\mathord{\leqslant}}\rangle}\triangleq\{X\subseteq D\mid X\text{ is an antichain}\}$. A qoset $\langle{D,\mathord{\leqslant}}\rangle$ is a well-quasiordered set (wqoset), and $\mathord{\leqslant}$ is called well-quasiorder (wqo) on $D$, when for every countably infinite sequence of elements $\{x_{i}\}_{i\in\mathbb{N}}$ there exist $i,j\in\mathbb{N}$ such that $i<j$ and $x_{i}\leqslant x_{j}$. Equivalently, $\langle{D,\mathord{\leqslant}}\rangle$ is a wqoset iff $D$ is DCC and $D$ has no infinite antichain. For every qoset $\langle{D,\mathord{\leqslant}}\rangle$, let us define the following binary relation $\sqsubseteq_{\leqslant}$ on the powerset: given $X,Y\in\wp(D)$,

[TABLE]

A minor of a subset $X\subseteq D$, denoted by $\lfloor{X}\rfloor$, is a subset of the minimal elements of $X$ w.r.t. $\leqslant$, i.e. $\lfloor{X}\rfloor\subseteq\operatorname{{min}}_{\leqslant}(X)\triangleq\{x\in X\mid\forall y\in X,y\leqslant x\Rightarrow y=x\}$, such that $X\sqsubseteq\lfloor{X}\rfloor$ holds. Therefore, a minor $\lfloor{X}\rfloor$ of $X\subseteq D$ is always an antichain in $D$. Let us recall that every subset $X$ of a wqoset $\langle{D,\leqslant}\rangle$ has at least one minor set, all minor sets of $X$ are finite, $\lfloor{\{x\}}\rfloor=\{x\}$, $\lfloor{\varnothing}\rfloor=\varnothing$, and if $\langle{D,\mathord{\leqslant}}\rangle$ is additionally a poset then there exists exactly one minor set of $X$. It turns out that $\langle{\operatorname{{AC}}_{\langle{D,\mathord{\leqslant}}\rangle},\sqsubseteq}\rangle$ is a qoset, which is ACC if $\langle{D,\leqslant}\rangle$ is a wqoset and is a poset if $\langle{D,\leqslant}\rangle$ is a poset.

For the sake of clarity, we overload the notation and use the same symbol for a function/relation and its componentwise (i.e. pointwise) extension on product domains, e.g., if $f:X\rightarrow Y$ then $f$ also denotes the standard product function $f:X^{n}\rightarrow Y^{n}$ which is componentwise defined by $\lambda\langle{x_{1},\ldots,x_{n}}\rangle\in X^{n}.\langle{f(x_{1}),\ldots,f(x_{n})}\rangle$. A vector $\vv{\bm{x}}$ in some product domain $D^{|S|}$ indexed by a finite set $S$ is also denoted by $\langle{x_{i}}\rangle_{i\in S}$ and, for some $i\in S$, $\vv{\bm{x}}_{\!\!i}$ denotes its component $x_{i}$.

Let $\langle{X,\leqslant}\rangle$ be a qoset and $f:X\rightarrow X$ be a function. $f$ is monotonic when $x\leqslant y$ implies $f(x)\leqslant f(y)$. For all $n\in\mathbb{N}$, the $n$-th power $f^{n}:X\rightarrow X$ of $f$ is inductively defined by: $f^{0}\triangleq\lambda x.x$; $f^{n+1}\triangleq f\comp f^{n}$ (or, equivalently, $f^{n+1}\triangleq f^{n}\comp f$). The denumerable sequence of Kleene iterates of $f$ starting from an initial value $a\in X$ is given by $\langle{f^{n}(a)}\rangle_{n\in\mathbb{N}}$. If $\langle{X,\leqslant}\rangle$ is a poset and $a\in X$ then $\operatorname{lfp}_{a}(f)$ (resp. $\operatorname{gfp}_{a}(f)$) denotes the least (resp. greatest) fixpoint of $f$ which is greater (resp. less) than or equal to $a$, when this exists; in particular, $\operatorname{lfp}(f)$ (resp. $\operatorname{gfp}(f)$) denotes the least (resp. greatest) fixpoint of $f$, when this exists. If $\langle{X,\leqslant}\rangle$ is an ACC (resp. DCC) CPO, $a\leqslant f(a)$ (resp. $f(a)\leqslant a$) holds and $f$ is monotonic then the Kleene iterates $\langle{f^{n}(a)}\rangle_{n\in\mathbb{N}}$ finitely converge to $\operatorname{lfp}_{a}(f)$ (resp. $\operatorname{gfp}_{a}(f)$), i.e., there exists $k\in\mathbb{N}$ such that for all $n\geq k$, $f^{n}(a)=f^{k}(a)=\operatorname{lfp}_{a}(f)$ (resp. $\operatorname{gfp}_{a}(f)$). In particular, if $\bot$ (resp. $\top$) is the least (greatest) element of $\langle{X,\leqslant}\rangle$ then $\langle{f^{n}(\bot)}\rangle_{n\in\mathbb{N}}$ (resp. $\langle{f^{n}(\top)}\rangle_{n\in\mathbb{N}}$) finitely converges to $\operatorname{lfp}(f)$ (resp. $\operatorname{gfp}(f)$).

2.2. Abstract Interpretation

Let us recall some basic notions on closure operators and Galois Connections commonly used in abstract interpretation (see, e.g., (Cousot and Cousot, 1979; Miné, 2017; Rival and Yi, 2020)). Closure operators and Galois Connections are equivalent notions and, therefore, they are both used for defining the notion of approximation in abstract interpretation, where closure operators allow us to define and reason on abstract domains independently of a specific representation for abstract values which is required by Galois Connections.

Let $\langle{C,\mathord{\leq_{C}},\vee,\wedge}\rangle$ be a complete lattice, where $\vee$ and $\wedge$ denote, resp., lub and glb. An upper closure operator, or simply closure, on $\langle{C,\mathord{\leq_{C}}}\rangle$ is a function $\rho:C\to C$ which is:

(i) monotonic,

(ii) idempotent: $\rho(\rho(x))=\rho(x)$ for all $x\in C$, and

(iii) extensive: $x\leq_{C}\rho(x)$ for all $x\in C$.

The set of all upper closed operators on $C$ is denoted by $\operatorname{uco}(C)$. We often write $c\in\rho(C)$, or simply $c\in\rho$, to denote that there exists $c^{\prime}\in C$ such that $c=\rho(c^{\prime})$, and recall that this happens iff $\rho(c)=c$. If $\rho\in\operatorname{uco}(C)$ then for all $c_{1}\in C$, $c_{2}\in\rho$ and $X\subseteq C$, it turns out that:

[TABLE]

In abstract interpretation, a closure operator $\rho\in\operatorname{uco}(C)$ on a concrete domain $C$ plays the role of abstraction function for objects of $C$. Given two closures $\rho,\rho^{\prime}\in\operatorname{uco}(C)$, $\rho$ is a coarser abstraction than $\rho^{\prime}$ (or, equivalently, $\rho^{\prime}$ is a more precise abstraction than $\rho$) iff the image of $\rho$ is a subset of the image of $\rho^{\prime}$, i.e. $\rho(C)\subseteq\rho^{\prime}(C)$, and this happens iff for any $x\in C$, $\rho^{\prime}(x)\leq_{C}\rho(x)$.

Let us recall that a Galois Connection (GC) or adjunction between two posets $\langle{C,\leq_{C}}\rangle$, called concrete domain, and $\langle{A,\leq_{A}}\rangle$, called abstract domain, consists of two functions $\alpha\colon C\rightarrow A$ and $\gamma\colon A\rightarrow C$ such that $\alpha(c)\leq_{A}a\>\Leftrightarrow\>c\leq_{C}\gamma(a)$ always holds. A Galois Connection is denoted by $\langle{C,\leq_{C}}\rangle\galois{\alpha}{\gamma}\langle{A,\leq_{A}}\rangle$. The function $\alpha$ is called the left-adjoint of $\gamma$, and, dually, $\gamma$ is called the right-adjoint of $\alpha$. This terminology is justified by the fact that if some function $\alpha:C\rightarrow A$ admits a right-adjoint $\gamma:A\rightarrow C$ then this is unique, and this dually holds for left-adjoints. It turns out that in a GC between complete lattices, $\gamma$ is always co-additive (i.e., it preserves arbitrary glb’s) while $\alpha$ is always additive (i.e., it preserves arbitrary lub’s). Moreover, an additive function $\alpha:C\rightarrow A$ uniquely determines its right-adjoint by $\gamma\triangleq\lambda a\ldotp\vee_{C}\{c\in C\mid\alpha(c)\leq_{A}a\}$ and, dually, a co-additive function $\gamma:A\rightarrow C$ uniquely determines its left-adjoint by $\alpha\triangleq\lambda c\ldotp\wedge_{A}\{a\in A\mid c\leq_{C}\gamma(a)\}$.

The following remark is folklore in abstract interpretation and a proof is here provided for the sake of completeness.

Lemma 2.1.

Let $\langle{C,\leq_{C}}\rangle\galois{\alpha}{\gamma}\langle{A,\leq_{A}}\rangle$ be a GC between complete lattices and $f\colon C\rightarrow C$ be a monotonic function. Then, $\gamma(\operatorname{lfp}(\alpha f\gamma))=\operatorname{lfp}(\gamma\alpha f)$.

Proof.

Let us first show that $\operatorname{lfp}(\gamma\alpha f)\leq_{C}\gamma(\operatorname{lfp}(\alpha f\gamma))$:

[TABLE]

Then, let us prove that $\gamma(\operatorname{lfp}(\alpha f\gamma))\leq_{C}\operatorname{lfp}(\gamma\alpha f)$:

[TABLE]

2.3. Languages

Let $\Sigma$ be an alphabet, i.e., a finite nonempty set of symbols. A word (or string) on $\Sigma$ is a finite (possibly empty) sequence of symbols in $\Sigma$, where $\epsilon$ denotes the empty sequence. $\Sigma^{*}$ denotes the set of finite words on $\Sigma$. A language on $\Sigma$ is a subset $L\subseteq\Sigma^{*}$. Concatenation of words and languages is denoted by simple juxtaposition, that is, the concatenation of words $u,v\in\Sigma^{*}$ is denoted by $uv\in\Sigma^{*}$, while the concatenation of languages $L,L^{\prime}\subseteq\Sigma^{*}$ is denoted by $LL^{\prime}\triangleq\{uv\mid u\in L,\,v\in L^{\prime}\}$. By considering a word as a singleton language, we also concatenate words with languages, for example $uL$ and $uLv$.

A finite automaton (FA) is a tuple $\mathcal{A}=\langle{Q,\delta,I,F,\Sigma}\rangle$ where: $\Sigma$ is an alphabet, $Q$ is a finite set of states, $I\subseteq Q$ is a subset of initial states, $F\subseteq Q$ is a subset of final states, and $\delta\colon Q\times\Sigma\rightarrow\wp(Q)$ is a transition relation. The notation $q\stackrel{{\scriptstyle a}}{{\rightarrow}}q^{\prime}$ is also used to denote that $q^{\prime}\in\delta(q,a)$. If $u\in\Sigma^{*}$ and $q,q^{\prime}\in Q$ then $q\stackrel{{\scriptstyle u}}{{\leadsto}}q^{\prime}$ means that the state $q^{\prime}$ is reachable from $q$ by following the string $u$. More formally, by induction on the length of $u\in\Sigma^{*}$: (i) if $u=\epsilon$ then $q\stackrel{{\scriptstyle\epsilon}}{{\leadsto}}q^{\prime}$ iff $q=q^{\prime}$; (ii) if $u=av$ with $a\in\Sigma,v\in\Sigma^{*}$ then $q\stackrel{{\scriptstyle av}}{{\leadsto}}q^{\prime}$ iff $\exists q^{\prime\prime}\in\delta(q,a),\;q^{\prime\prime}\stackrel{{\scriptstyle v}}{{\leadsto}}q^{\prime}$. The language generated by a FA $\mathcal{A}$ is ${\mathcal{L}(\mathcal{A})}\triangleq\{u\in\Sigma^{*}\mid\exists q_{i}\in I,\exists q_{f}\in F,\;q_{i}\stackrel{{\scriptstyle u}}{{\leadsto}}q_{f}\}$. An example of FA is depicted in Fig. 1.

3. Kleene Iterates with Abstract Inclusion Check

Abstract interpretation can be applied to solve a generic inclusion checking problem by leveraging backward complete abstractions (Cousot and Cousot, 1977, 1979; Giacobazzi et al., 2000; Ranzato, 2013). A closure $\rho\in\operatorname{uco}(C)$ is called backward complete for a concrete monotonic function ${f:C\rightarrow C}$ when $\rho f=\rho f\rho$ holds. Since $\rho f(c)\leq_{C}\rho f\rho(c)$ always holds for all $c\in C$ (because $\rho$ is extensive and monotonic and $f$ is monotonic), the intuition is that backward completeness models an ideal situation where no loss of precision is accumulated in the computations of $\rho f$ when its concrete input objects $c$ are approximated by $\rho(c)$. It is well known (Cousot and Cousot, 1979) that backward completeness implies completeness of least fixpoints, namely for all $x\in C$ such that $x\leq_{C}f(x)$,

[TABLE]

provided that these least fixpoints exist (this is the case, e.g., when $C$ is a CPO).

Given an initial value $a\in C$, let us define the following iterative procedure:

[TABLE]

which computes the Kleene iterates of $f$ starting from $a$ and stops when a convergence relation $\operatorname{{Conv}}\subseteq C\times C$ for two consecutive Kleene iterates $f^{n+1}(a)$ and $f^{n}(a)$ holds. When $\operatorname{{Conv}}=\operatorname{{Incl}}\triangleq\{(x,y)\mid x\leq_{C}y\}$ is the convergence relation and $a\leq_{C}f(a)$ holds, the procedure $\operatorname{{\textsc{Kleene}}}(\operatorname{{Incl}},f,a)$ returns $\operatorname{lfp}_{a}(f)$ if the Kleene iterates finitely converge. Hence, termination of $\operatorname{{\textsc{Kleene}}}(\operatorname{{Eq}},f,a)$ is guaranteed when $C$ is an ACC CPO.

Given a closure $\rho\in\operatorname{uco}(C)$, let us consider the following abstract convergence relation induced by $\rho$:

[TABLE]

Hence, $\operatorname{{\textsc{Kleene}}}(\operatorname{{Incl}}_{\rho},f,a)$ terminates if eventually $\rho(f(x))\leq_{C}\rho(x)$ holds. Notice that $\mathord{\operatorname{{Incl}}}\subseteq\mathord{\operatorname{{Incl}}_{\rho}}$ always holds by monotonicity of $\rho$ and $\mathord{\operatorname{{Incl}}}=\mathord{\operatorname{{Incl}}_{\rho}}$ iff $\rho=\mathrm{id}$.

Theorem 3.1.

Let $\rho\in\operatorname{uco}(C)$ be such that $\rho$ is backward complete for $f$ and $\rho(C)$ does not contain infinite ascending chains. Let $a\in C$ such that $a\leq_{C}f(a)$ holds. Then, the procedure $\operatorname{{\textsc{Kleene}}}(\operatorname{{Incl}}_{\rho},f,a)$ terminates and $\rho(\operatorname{{\textsc{Kleene}}}(\operatorname{{Incl}}_{\rho},f,a))=\rho(\operatorname{lfp}_{a}(f))=\operatorname{lfp}_{a}(\rho f)$.

Proof.

Let us first prove by induction the following property:

[TABLE]

For $n=0$, we have that $\rho\comp f^{0}=\rho=(\rho\comp f)^{0}\comp\rho$. For $n+1$,

[TABLE]

Then, let us observe that $\operatorname{lfp}_{a}(\rho f)=\operatorname{lfp}_{\rho(a)}(\rho f)$: this is a consequence of the fact that $\rho(f(x))=x\wedge a\leq_{C}x$ iff $\rho(f(x))=x\wedge\rho(a)\leq_{C}x$, because $\rho(f(x))=x\wedge a\leq_{C}x$ implies $\rho(f(x))=x\wedge\rho(a)\leq_{C}\rho(x)=\rho(\rho(f(x)))=\rho(f(x))=x$.

Since $a\leq_{C}f(a)$, we have that $\langle{f^{n}(a)}\rangle_{n\in\mathbb{N}}$ is an ascending chain, so that, by monotonicity of $\rho$, $\langle{\rho(f^{n}(a))}\rangle_{n\in\mathbb{N}}$ is an ascending chain in $\rho(C)$. Since $\rho(C)$ does not contain infinite ascending chains, there exists $N=\min(\{n\in\mathbb{N}\mid\rho(f^{n+1}(a))\leq_{C}\rho(f^{n}(a))\})$. This means that $\operatorname{{\textsc{Kleene}}}(\operatorname{{Incl}}_{\rho},f,a)$ terminates after $N+1$ iterations and outputs $f^{N}(a)$. We prove by induction on $N\in\mathbb{N}$ that $N=\min(\{n\in\mathbb{N}\mid(\rho\comp f)^{n+1}(\rho(a))\leq_{C}(\rho\comp f)^{n}(\rho(a))\})$.

• $(N=0):$

We have that $\rho(f^{1}(a))\leq_{C}\rho(f^{0}(a))$, namely, $\rho(f(a))\leq_{C}\rho(a)$. Then, by backward completeness, $\rho(f(\rho(a)))\leq_{C}\rho(a)$, namely, $(\rho\comp f)^{1}(\rho(a))\leq_{C}(\rho\comp f)^{0}(\rho(a))$.

• $(N+1):$

We have that $\rho(f^{N+2}(a))\leq_{C}\rho(f^{N+1}(a))$, so that by (5), $(\rho\comp f)^{N+2}(\rho(a))\leq_{C}(\rho\comp f)^{N+1}(\rho(a))$. Moreover, $N+1$ is the minimum natural number such that $(\rho\comp f)^{n+1}(\rho(a))\leq_{C}(\rho\comp f)^{n}(\rho(a))$ holds, because if $(\rho\comp f)^{k+1}(\rho(a))\leq_{C}(\rho\comp f)^{k}(\rho(a))$ for some $k\leq N$, then, by (5), we would have that $\rho(f^{k+1}(a))\leq_{C}\rho(f^{k}(a))$, thus contradicting the minimality of $N+1$ for $\rho(f^{n+1}(a))\leq_{C}\rho(f^{n}(a))$.

Since $a\leq_{C}f(a)$ implies, by backward completeness, $\rho(a)\leq_{C}\rho(f(a))=(\rho\comp f)(\rho(a)))$, and $N=\min(\{n\in\mathbb{N}\mid(\rho\comp f)^{n+1}(\rho(a))\leq_{C}(\rho\comp f)^{n}(\rho(a))\})$, it turns out that $(\rho\comp f)^{N}(\rho(a))={\operatorname{lfp}_{\rho(a)}(\rho f)}=\operatorname{lfp}_{a}(\rho f)$. Thus, by (5), we obtain $\operatorname{lfp}_{a}(\rho f)=(\rho\comp f)^{N}(\rho(a))=\rho(f^{N}(a))=\rho(\operatorname{{\textsc{Kleene}}}(\operatorname{{Incl}}_{\rho},f,a))$. Finally, by (4), $\rho(\operatorname{{\textsc{Kleene}}}(\operatorname{{Incl}}_{\rho},f,a))=\operatorname{lfp}_{a}(\rho f)=\rho(\operatorname{lfp}_{a}(f))$. ∎

We will apply the order-theoretic algorithmic scheme provided by $\operatorname{{\textsc{Kleene}}}$ under the hypotheses of Theorem 3.1 to a number of different language inclusion problems $L_{1}\subseteq L_{2}$, where $L_{1}$ can be expressed as least fixpoint of a monotonic function on $\wp(\Sigma^{*})$. This will allow us to systematically design several language inclusion algorithms which rely on different backward complete abstractions of the complete lattice $\langle{\wp(\Sigma^{*}),\subseteq}\rangle$.

4. An Algorithmic Framework for Language Inclusion

4.1. Languages as Fixed Points

Let $\mathcal{A}=\langle{Q,\delta,I,F,\Sigma}\rangle$ be a FA. Given $S,T\subseteq Q$, define the set of words leading from some state in $S$ to some state in $T$ as follows:

[TABLE]

When $S=\{q\}$ or $T=\{q^{\prime}\}$ we slightly abuse the notation and write $W^{\mathcal{A}}_{q,T}$, $W^{\mathcal{A}}_{S,q^{\prime}}$, or $W^{\mathcal{A}}_{q,q^{\prime}}$. Also, we omit the automaton $\mathcal{A}$ in superscripts when this is clear from the context. The language accepted by $\mathcal{A}$ is therefore ${\mathcal{L}(\mathcal{A})}\triangleq W^{\mathcal{A}}_{I,F}$. Observe that

[TABLE]

where, as usual, $\textstyle{\bigcup\varnothing}=\varnothing$.

Let us recall how to define the language accepted by an automaton as a solution of a set of equations (Schützenberger, 1963). Given a generic Boolean predicate $p(x)$ for a variable $x$ ranging in some set (typically a membership predicate $x\in^{\scaleto{?}{3.5pt}}Z$) and two generic sets $T$ and $F$, we define the following parametric choice function:

[TABLE]

The FA $\mathcal{A}$ induces the following set of equations, where the $X_{q}$’s are variables of type $X_{q}\in\wp(\Sigma^{*})$ and are indexed by states $q\in Q$ of $\mathcal{A}$:

[TABLE]

Thus, the functions $\lambda\langle{X_{q^{\prime}}}\rangle_{q^{\prime}\in Q}.\>{\psi^{\{\epsilon\}}_{\varnothing}(q\in^{\scaleto{?}{3.5pt}}F)}\cup{\textstyle\bigcup_{a\in\Sigma,\,q^{\prime}\in\delta(q,a)}}aX_{q^{\prime}}$ in the right-hand side of the equations in $\operatorname{{Eqn}}(\mathcal{A})$ have type $\wp(\Sigma^{*})^{|Q|}\rightarrow\wp(\Sigma^{*})$. Since $\langle{\wp(\Sigma^{*})^{|Q|},\subseteq}\rangle$ is a (product) complete lattice (as $\langle{\wp(\Sigma^{*}),\subseteq}\rangle$ is a complete lattice) and all the right-hand side functions in $\operatorname{{Eqn}}(\mathcal{A})$ are clearly monotonic, the least solution $\langle{Y_{q}}\rangle_{q\in Q}\in\wp(\Sigma^{*})^{|Q|}$ of $\operatorname{{Eqn}}(\mathcal{A})$ does exist and it is easy to check that for every $q\in Q$, $Y_{q}=W^{\mathcal{A}}_{q,F}$ holds.

It is worth noticing that, by relying on right concatenations rather than left ones $aX_{q^{\prime}}$ used in $\operatorname{{Eqn}}(\mathcal{A})$, one could also define a set of symmetric equations whose least solution coincides with $\langle{W_{I,q}^{\mathcal{A}}}\rangle_{q\in Q}$ instead of $\langle{W_{q,F}^{\mathcal{A}}}\rangle_{q\in Q}$.

Example 4.1.

Let us consider the automaton $\mathcal{A}$ in Figure 1. The set of equations induced by $\mathcal{A}$ are as follows:

[TABLE]

It is notationally convenient to formulate the equations in $\operatorname{{Eqn}}(\mathcal{A})$ by exploiting an “initial” vector $\vv{\bm{\epsilon}}^{{\!{\scriptstyle{F}}}}\in\wp(\Sigma^{*})^{|Q|}$ and a predecessor function $\operatorname{{Pre}}_{\mathcal{A}}\colon\wp(\Sigma^{*})^{|Q|}{\rightarrow}\wp(\Sigma^{*})^{|Q|}$ defined as follows:

[TABLE]

The intuition for the function $\operatorname{{Pre}}_{\mathcal{A}}$ is that given the language $W_{q^{\prime},F}^{\mathcal{A}}$ and a transition $q^{\prime}\in\delta(q,a)$, we have that $aW^{\mathcal{A}}_{q^{\prime},F}\subseteq W^{\mathcal{A}}_{q,F}$ holds, i.e., given a subset $X_{q}^{\prime}$ of the language generated by $\mathcal{A}$ from some state $q^{\prime}$, the function $\operatorname{{Pre}}_{\mathcal{A}}$ computes a subset $X_{q}$ of the language generated by $\mathcal{A}$ for its predecessor state $q$. Notice that if all the components of $\vv{\bm{X}}$ are finite sets of words then $\operatorname{{Pre}}_{\mathcal{A}}(\vv{\bm{X}})$ is still a vector of finite sets. Since $\epsilon\in W_{q,F}^{\mathcal{A}}$ for all $q\in F$, the least fixpoint computation can start from the vector $\vv{\bm{\epsilon}}^{{\!{\scriptstyle{F}}}}$ and iteratively apply $\operatorname{{Pre}}_{\mathcal{A}}$. Therefore, it turns out that

[TABLE]

Together with Equation (6), it follows that ${\mathcal{L}(\mathcal{A})}$ is given by the union of the component languages of the vector $\operatorname{lfp}(\lambda\vv{\bm{X}}\ldotp\vv{\bm{\epsilon}}^{{\!{\scriptstyle{F}}}}\cup\operatorname{{Pre}}_{\mathcal{A}}(\vv{\bm{X}}))$ that are indexed by the initial states in $I$.

Example 4.2 (Continuation of Example 4.1).

The fixpoint characterization of $\langle{W_{q,F}^{\mathcal{A}}}\rangle_{q\in Q}$ is:

[TABLE]

4.2. Language Inclusion using Fixed Points

Consider a language inclusion problem $L_{1}\subseteq L_{2}$, where $L_{1}={\mathcal{L}(\mathcal{A})}$ for some FA $\mathcal{A}=\langle{Q,\delta,I,F,\Sigma}\rangle$. The language $L_{2}$ can be formalized as a vector in $\wp(\Sigma^{*})^{|Q|}$ as follows:

[TABLE]

whose components indexed by initial states in $I$ are $L_{2}$ and those indexed by noninitial states are $\Sigma^{*}$. Then, as a consequence of (6), (7) and (8), we have that

[TABLE]

Theorem 4.3.

If $\rho\in\operatorname{uco}(\wp(\Sigma^{*}))$ is backward complete for $\lambda X\in\wp(\Sigma^{*})\ldotp aX$ for all $a\in\Sigma$, then, for all FAs $\mathcal{A}=\langle{Q,\delta,I,F,\Sigma}\rangle$ on the alphabet $\Sigma$, $\rho$ is backward complete for $\operatorname{{Pre}}_{\mathcal{A}}$ and $\lambda\vv{\bm{X}}\ldotp\vv{\bm{\epsilon}}^{{\!{\scriptstyle{F}}}}\cup\operatorname{{Pre}}_{\mathcal{A}}(\vv{\bm{X}})$.

Proof.

First, it turns out that:

[TABLE]

As a consequence, $\rho$ is backward complete for $\lambda\vv{\bm{X}}\ldotp\vv{\bm{\epsilon}}^{{\!{\scriptstyle{F}}}}\cup{\operatorname{{Pre}}}_{\mathcal{A}}(\vv{\bm{X}})$:

[TABLE]

∎

Then, by resorting to least fixpoint transfer of completeness (4), we also obtain the following consequence.

Corollary 4.4.

If $\rho\in\operatorname{uco}(\wp(\Sigma^{*}))$ is backward complete for $\lambda X\in\wp(\Sigma^{*})\ldotp aX$ for all $a\in\Sigma$ then $\rho(\operatorname{lfp}(\lambda\vv{\bm{X}}\ldotp\vv{\bm{\epsilon}}^{{\!{\scriptstyle{F}}}}\cup\operatorname{{Pre}}_{\mathcal{A}}(\vv{\bm{X}})))=\operatorname{lfp}(\lambda\vv{\bm{X}}\ldotp\rho(\vv{\bm{\epsilon}}^{{\!{\scriptstyle{F}}}}\cup\operatorname{{Pre}}_{\mathcal{A}}(\vv{\bm{X}})))$.

Note that if $\rho$ is backward complete for $\lambda X.aX$ for all $a\in\Sigma$ and $L_{2}\in\rho$ then, by Theorem 3.1 and Corollary 4.4, the equivalence (9) becomes

[TABLE]

4.2.1. Right Concatenation

Let us consider the symmetric case of right concatenation $\lambda X.Xa$. Recall that $W_{I,q}=\{w\in\Sigma^{*}\mid\exists q_{i}\in I,q_{i}\stackrel{{\scriptstyle w}}{{\leadsto}}q\}$ and that $W_{I,q}={\psi^{\{\epsilon\}}_{\varnothing}(q\in^{\scaleto{?}{3.5pt}}I)}\cup{\textstyle\bigcup_{a\in\Sigma,a\in W_{q^{\prime},q}}}W_{I,q^{\prime}}a$ holds. Correspondingly, we define a set of fixpoint equations on $\wp(\Sigma^{*})$ which is based on right concatenation and is symmetric to the equations defined in (7):

[TABLE]

In this case, if $\vv{\bm{Y}}=\langle{Y_{q}}\rangle_{q\in Q}$ is the least fixpoint solution of $\operatorname{{Eqn}^{r}}(\mathcal{A})$ then $Y_{q}=W^{\mathcal{A}}_{I,q}$ for every $q\in Q$. Also, by defining $\vv{\bm{\epsilon}}^{{\!{\scriptstyle{I}}}}\in\wp(\Sigma^{*})^{|Q|}$ and $\operatorname{{Post}}_{\mathcal{A}}\colon\wp(\Sigma^{*})^{|Q|}{\rightarrow}\wp(\Sigma^{*})^{|Q|}$ as follows:

[TABLE]

we have that

[TABLE]

Thus, by (6), it turns out that ${\mathcal{L}(\mathcal{A})}={\textstyle\bigcup_{q_{f}\in F}}W_{I,q_{f}}$ holds, that is, ${\mathcal{L}(\mathcal{A})}$ is the union of the component languages of the vector $\operatorname{lfp}(\lambda\vv{\bm{X}}\ldotp\vv{\bm{\epsilon}}^{{\!{\scriptstyle{I}}}}\cup\operatorname{{Post}}_{\mathcal{A}}(\vv{\bm{X}}))$ indexed by the final states in $F$.

Example 4.5.

Consider again the FA $\mathcal{A}$ in Figure 1. The set of right equations for $\mathcal{A}$ is as follows:

[TABLE]

so that

[TABLE]

In a language inclusion problem ${\mathcal{L}(\mathcal{A})}\subseteq L_{2}$, we consider the vector $\vv{\bm{L_{2}}}^{{\!{\scriptstyle{F}}}}\triangleq\langle{{\psi^{L_{2}}_{\Sigma^{*}}(q\in^{\scaleto{?}{3.5pt}}F)}}\rangle_{q\in Q}\in\wp(\Sigma^{*})^{|Q|}$, so that, by (11), it turns out that:

[TABLE]

We therefore have the following symmetric version of Theorem 4.3 for right concatenation.

Theorem 4.6.

If $\rho\in\operatorname{uco}(\wp(\Sigma^{*}))$ is backward complete for $\lambda X\ldotp Xa$ for all $a\in\Sigma$ then, for all FAs $\mathcal{A}$ on the alphabet $\Sigma$, $\rho$ is backward complete for $\lambda\vv{\bm{X}}\ldotp\vv{\bm{\epsilon}}^{{\!{\scriptstyle{I}}}}\cup\operatorname{{Post}}_{\mathcal{A}}(\vv{\bm{X}})$.

4.3. A Language Inclusion Algorithm with Abstract Inclusion Check

Let us now apply the general Theorem 3.1 to design an algorithm that solves a language inclusion problem ${\mathcal{L}(\mathcal{A})}\subseteq L_{2}$ by exploiting a language abstraction $\rho$ that satisfies a list of requirements of backward completeness and computability.

Theorem 4.7.

Let $\mathcal{A}=\langle{Q,\delta,I,F,\Sigma}\rangle$ be a FA, $L_{2}\in\wp(\Sigma^{*})$ and $\rho\in\operatorname{uco}(\wp(\Sigma^{*}))$. Assume that the following properties hold:

1. (i)

The closure $\rho$ is backward complete for $\lambda X\in\wp(\Sigma^{*})\ldotp aX$, for all $a\in\Sigma$, and satisfies $\rho(L_{2})=L_{2}$. 2. (ii)

$\rho(\wp(\Sigma^{*}))$* does not contain infinite ascending chains.* 3. (iii)

If $X,Y\in\wp(\Sigma^{*})$ are finite sets of words then the inclusion $\rho(X)\subseteq^{\scaleto{?}{3.5pt}}\rho(Y)$ is decidable. 4. (iv)

If $Y\in\wp(\Sigma^{*})$ is a finite set of words then the inclusion $\rho(Y)\subseteq^{\scaleto{?}{3.5pt}}L_{2}$ is decidable.

Then,

\begin{array}[]{l}\langle{Y_{q}}\rangle_{q\in Q}:=\operatorname{{\textsc{Kleene}}}(\operatorname{{Incl}}_{\rho},\lambda\vv{\bm{X}}.\vv{\bm{\epsilon}}^{{\!{\scriptstyle{F}}}}\!\cup\operatorname{{Pre}}_{\mathcal{A}}(\vv{\bm{X}}),\vv{\bm{\varnothing}})\emph{;}\\ \emph{{return}}\>\operatorname{{Incl}}_{\rho}(\langle{Y_{q}}\rangle_{q\in Q},\vv{\bm{L_{2}}}^{{\!{\scriptstyle{I}}}})\emph{;}\end{array}**

is a decision algorithm for ${\mathcal{L}(\mathcal{A})}\subseteq L_{2}$.

Proof.

Conditions (i), (ii) and (iii) guarantee that the hypotheses of Theorem 3.1 are satisfied. Thus, $\operatorname{{\textsc{Kleene}}}(\operatorname{{Incl}}_{\rho},\lambda\vv{\bm{X}}\ldotp\vv{\bm{\epsilon}}^{{\!{\scriptstyle{F}}}}\!\cup\operatorname{{Pre}}_{\mathcal{A}}(\vv{\bm{X}}),\vv{\bm{\varnothing}})$ is an algorithm that terminates with output $\langle{Y_{q}}\rangle_{q\in Q}$ and

[TABLE]

Moreover, by (9), ${\mathcal{L}(\mathcal{A})}\subseteq L_{2}\Leftrightarrow\rho({\mathcal{L}(\mathcal{A})})\subseteq\rho(L_{2})=L_{2}\Leftrightarrow\rho(\operatorname{lfp}(\lambda\vv{\bm{X}}\ldotp\vv{\bm{\epsilon}}^{{\!{\scriptstyle{F}}}}\cup\operatorname{{Pre}}_{\mathcal{A}}(\vv{\bm{X}})))\subseteq\vv{\bm{L_{2}}}^{{\!{\scriptstyle{I}}}}\Leftrightarrow\rho(\langle{Y_{q}}\rangle_{q\in Q})\subseteq\rho(\vv{\bm{L_{2}}}^{{\!{\scriptstyle{I}}}})\Leftrightarrow\operatorname{{Incl}}_{\rho}(\langle{Y_{q}}\rangle_{q\in Q},\vv{\bm{L_{2}}}^{{\!{\scriptstyle{I}}}})$. Finally, by condition (iv), $\operatorname{{Incl}}_{\rho}(\langle{Y_{q}}\rangle_{q\in Q},\vv{\bm{L_{2}}}^{{\!{\scriptstyle{I}}}})$ is decidable. ∎

It is worth noticing that Theorem 4.7 can also be stated in a symmetric version for $\lambda\vv{\bm{X}}\ldotp\vv{\bm{\epsilon}}^{{\!{\scriptstyle{I}}}}\cup\operatorname{{Post}}_{\mathcal{A}}(\vv{\bm{X}})$ similarly to Theorem 4.6.

5. Instantiating the Framework with Quasiorders

We instantiate the general algorithmic framework of Section 4 to the class of closure operators induced by quasiorder relations on words.

5.1. Word-based Abstractions

Let $\mathord{\leqslant}\subseteq\Sigma^{*}\times\Sigma^{*}$ be a quasiorder relation on words in $\Sigma^{*}$. The corresponding closure operator $\rho_{\leqslant}\in\operatorname{uco}(\wp(\Sigma^{*}))$ is defined as follows:

[TABLE]

Thus, $\rho_{\leqslant}(X)$ is the $\leqslant$-upward closure of $X$ and it is easy to check that $\rho_{\leqslant}$ is indeed a closure on the complete lattice $\langle{\wp(\Sigma^{*}),\subseteq}\rangle$.

Following (de Luca and Varricchio, 1994), a quasiorder $\leqslant$ on $\Sigma^{*}$ is left-monotonic (resp. right-monotonic) if

[TABLE]

Also, $\leqslant$ is called monotonic if it is both left- and right-monotonic. It turns out that $\leqslant$ is left-monotonic (resp. right-monotonic) iff

[TABLE]

In fact, if $x_{1}\leqslant x_{2}$ then (13) implies that for all $y\in\Sigma^{*}$, $yx_{1}\leqslant yx_{2}$: by induction on the length $|y|\in\mathbb{N}$, we have that: (i) if $y=\epsilon$ then $yx_{1}\leqslant yx_{2}$; (ii) if $y=av$ with $a\in\Sigma,v\in\Sigma^{*}$ then, by inductive hypothesis, $vx_{1}\leqslant vx_{2}$, so that by (13), $yx_{1}=avx_{1}\leqslant avx_{2}=yx_{2}$.

Definition 5.1 ($L$-Consistent Quasiorder).

Let $L\in$ $\wp(\Sigma^{*})$. A quasiorder $\mathord{\leqslant_{L}}\subseteq\Sigma^{*}\times\Sigma^{*}$ is called left (resp. right) $L$-consistent when:

1. (a)

$\mathord{\leqslant}_{L}\cap(L\times\neg L)=\varnothing$; 2. (b)

$\mathord{\leqslant}_{L}$ is left-monotonic (resp. right-monotonic).

Also, $\mathord{\leqslant}_{L}$ is called $L$-consistent when it is both left and right $L$-consistent.

It turns out that a quasiorder is $L$-consistent iff it induces a closure which includes $L$ in its image and it is backward complete for concatenation.

Lemma 5.2.

Let $L\in\wp(\Sigma^{*})$ and $\mathord{\leqslant_{L}}$ be a quasiorder on $\Sigma^{*}$. Then, $\mathord{\leqslant_{L}}$ is a left (resp. right) $L$-consistent quasiorder on $\Sigma^{*}$ if and only if

1. (a)

$\rho_{\leqslant_{L}}(L)=L$, and 2. (b)

$\rho_{\leqslant_{L}}$* is backward complete for $\lambda X\ldotp aX$ (resp. $\lambda X\ldotp Xa$) for all $a\in\Sigma$.*

Proof.

We consider the left case, the right case is symmetric.

1. (a)

The inclusion $L\subseteq\rho_{\leqslant_{L}}(L)$ always holds because $\rho_{\leqslant_{L}}$ is an upper closure. Then, it turns out that $\rho_{\leqslant_{L}}(L)=L$ iff $\rho_{\leqslant_{L}}(L)\subseteq L$ iff $\forall v\in\Sigma^{*}$, $(\exists u\in L,\,u\leqslant_{L}v)\>\Rightarrow\>v\in L$ iff $\mathord{\leqslant}_{L}\cap(L\times\neg L)=\varnothing$. Thus, $\rho_{\leqslant_{L}}(L)=L$ iff condition (a) of Definition 5.1 holds. 2. (b)

We first prove that if $\mathord{\leqslant}_{L}$ is left-monotonic then for all $X\in\wp(\Sigma^{*})$, $\rho_{\leqslant_{L}}(aX)=\rho_{\leqslant_{L}}(a\rho_{\leqslant_{L}}(X))$ for all $a\in\Sigma$. Monotonicity of concatenation together with monotonicity and extensivity of $\rho_{\leqslant_{L}}$ imply that $\rho_{\leqslant_{L}}(aX)\subseteq\rho_{\leqslant_{L}}(a\rho_{\leqslant_{L}}(X))$ holds. For the reverse inclusion, we have that:

[TABLE]

Conversely, assume that for all $X\in\wp(\Sigma^{*})$ and $a\in\Sigma$, $\rho_{\leqslant_{L}}(aX)=\rho_{\leqslant_{L}}(a\rho_{\leqslant_{L}}(X))$. Consider $x_{1},x_{2}\in\Sigma^{*}$ and $a\in\Sigma$. If $x_{1}\leqslant_{L}x_{2}$ then $\{x_{2}\}\subseteq\rho_{\leqslant_{L}}(\{x_{1}\})$, and in turn $a\{x_{2}\}\subseteq a\rho_{\leqslant_{L}}(\{x_{1}\})$. Then, by applying the monotonic function $\rho_{\leqslant_{L}}$, $\rho_{\leqslant_{L}}(a\{x_{2}\})\subseteq\rho_{\leqslant_{L}}(a\rho_{\leqslant_{L}}(\{x_{1}\}))$, so that, by backward completeness, $\rho_{\leqslant_{L}}(a\{x_{2}\})\subseteq\rho_{\leqslant_{L}}(a\{x_{1}\})$. Hence, $a\{x_{2}\}\subseteq\rho_{\leqslant_{L}}(a\{x_{1}\})$, namely, $ax_{1}\leqslant_{L}ax_{2}$. By (13), this shows that $\leqslant_{L}$ is left-monotonic. ∎

We can apply Theorem 4.7 to the closure $\rho_{\leqslant^{l}_{L_{2}}}$ induced by a left $L_{2}$-consistent well-quasiorder, since it satisfies all the required hypotheses, thus obtaining the following Algorithm $\mathtt{FAIncW}$ which solves the language inclusion problem ${\mathcal{L}(\mathcal{A})}\subseteq L_{2}$ for any automaton $\mathcal{A}$. This algorithm is called “word-based” because the output vector $\langle{Y_{q}}\rangle_{q\in Q}$ computed by $\operatorname{{\textsc{Kleene}}}$ consists of finite sets of words. Here, the convergence relation $\operatorname{{Incl}}_{\rho_{\leqslant^{l}_{L_{2}}}}$ of $\operatorname{{\textsc{Kleene}}}$ coincides with the relation $\mathord{\sqsubseteq_{\leqslant^{l}_{L_{2}}}}$ because $\operatorname{{Incl}}_{\rho_{\leqslant^{l}_{L_{2}}}}(X,Y)$ iff $\rho_{\leqslant^{l}_{L_{2}}}(X)\subseteq\rho_{\leqslant^{l}_{L_{2}}}(Y)$ iff $X\sqsubseteq_{\leqslant^{l}_{L_{2}}}Y$.

Theorem 5.3.

*Let $\mathcal{A}=\langle{Q,\delta,I,F,\Sigma}\rangle$ be a FA and $L_{2}\in\wp(\Sigma^{*})$ be a language such that:

(i) membership in $L_{2}$ is decidable;

(ii) there exists a decidable left $L_{2}$-consistent wqo on $\Sigma^{*}$. Then, Algorithm $\mathtt{FAIncW}$ decides the inclusion problem ${\mathcal{L}(\mathcal{A})}\subseteq L_{2}$.*

Proof.

Let $\leqslant_{L_{2}}^{l}$ be the decidable left $L_{2}$-consistent wqo on $\Sigma^{*}$. Let us check that the hypotheses (i)-(ii)-(iii) of Theorem 4.7 are satisfied.

1. (i)

It follows from hypothesis (ii) and Lemma 5.2 that $\leqslant_{L_{2}}^{l}$ is backward complete for left concatenation and satisfies $\rho_{\leqslant_{L_{2}}^{l}}(L_{2})=L_{2}$. 2. (ii)

Since $\leqslant_{L_{2}}^{l}$ is a well-quasiorder, it follows that $\{\rho_{\leqslant_{L_{2}}^{l}}(S)\mid S\in\wp(\Sigma^{*})\}$ does not contain infinite ascending chains. 3. (iii)

For finite sets for finite sets $X$ and $Y$, the abstract inclusion $\operatorname{{Incl}}_{\rho_{\leqslant^{l}_{L_{2}}}}(X,Y)$ $\Leftrightarrow$ $X\sqsubseteq_{\leqslant^{l}_{L_{2}}}Y$ is decidable since $\leqslant_{L_{2}}^{l}$ is a decidable wqo.

Moreover, it turns out that the check $\operatorname{{Incl}}_{\rho_{\leqslant^{l}_{L_{2}}}}(\langle{Y_{q}}\rangle_{q\in Q},\vv{\bm{L_{2}}}^{{\!{\scriptstyle{I}}}})$ of Theorem 4.7 is decidable and is performed by lines 2-5 of Algorithm $\mathtt{FAIncW}$. In fact, since, by Theorem 4.7, $\operatorname{{\textsc{Kleene}}}(\sqsubseteq_{\leqslant^{l}_{L_{2}}},\lambda\vv{\bm{X}}\ldotp\vv{\bm{\epsilon}}^{{\!{\scriptstyle{F}}}}\cup\operatorname{{Pre}}_{\mathcal{A}}(\vv{\bm{X}}),\vv{\bm{\varnothing}})$ terminates after a finite number of steps with output $\langle{Y_{q}}\rangle_{q\in Q}$, each set of words $Y_{q}$ of the output turns out to be finite. Also, since $\vv{\bm{L_{2}}}^{{\!{\scriptstyle{I}}}}=\langle{{\psi^{L_{2}}_{\Sigma^{*}}(q\in^{\scaleto{?}{3.5pt}}I)}}\rangle_{q\in Q})$, the abstract inclusion trivially holds for all components $Y_{q}$ with $q\notin I$. Therefore, it suffices to check whether $Y_{q}\sqsubseteq_{\leqslant^{l}_{L_{2}}}L_{2}$ holds for all $q\in I$. Since $Y_{q}\sqsubseteq_{\leqslant^{l}_{L_{2}}}L_{2}$ iff $\rho_{\leqslant^{l}_{L_{2}}}(Y_{q})\subseteq\rho_{\leqslant^{l}_{L_{2}}}(L_{2})=L_{2}$ iff $Y_{q}\subseteq L_{2}$ and since $Y_{q}$ is a finite set, $Y_{q}\sqsubseteq_{\leqslant^{l}_{L_{2}}}L_{2}$ can be decided by performing the finitely many membership check $u\in^{\scaleto{?}{3.5pt}}L_{2}$ at lines 2-5, where by hypothesis (ii), any membership check is decidable. Thus, hypothesis (iv) of Theorem 4.7 is satisfied.

Summing up, we have shown that Algorithm $\mathtt{FAIncW}$ decides the inclusion ${\mathcal{L}(\mathcal{A})}\subseteq L_{2}$. ∎

Remark 5.4.

It is worth noticing that in each iteration of $\operatorname{{\textsc{Kleene}}}(\sqsubseteq_{\leqslant^{l}_{L_{2}}},\lambda\vv{\bm{X}}\ldotp\vv{\bm{\epsilon}}^{{\!{\scriptstyle{F}}}}\cup\operatorname{{Pre}}_{\mathcal{A}}(\vv{\bm{X}}),\vv{\bm{\varnothing}})$ in Algorithm $\mathtt{FAIncW}$, in the current vector $\langle{Y_{q}}\rangle_{q\in Q}$ one could safely remove from a component $Y_{q}$ any word $w\in Y_{q}$ such that there exists a word $u\in Y_{q}$ such that $u\leqslant^{l}_{L_{2}}w$ and $u\neq w$. This enables replacing each finite set $Y_{q}$ occurring in Kleene iterates with a minor subset $\lfloor{Y_{q}}\rfloor$ w.r.t. $\leqslant^{l}_{L_{2}}$. This replacement is correct, i.e. Theorem 5.3 still holds for the corresponding modified language inclusion algorithm, because an inclusion check $X\sqsubseteq_{\leqslant^{l}_{L_{2}}}Y$ holds iff the check $\lfloor{X}\rfloor\sqsubseteq_{\leqslant^{l}_{L_{2}}}\lfloor{Y}\rfloor$ for the corresponding minor subsets holds. $\Diamond$

5.1.1. Right Concatenation

Following Section 4.2.1, a symmetric version, called $\mathtt{FAIncWr}$, of the algorithm $\mathtt{FAIncW}$ (and of Theorem 5.3) for right $L_{2}$-consistent wqos can be given as follows.

Theorem 5.5.

*Let $\mathcal{A}=\langle{Q,\delta,I,F,\Sigma}\rangle$ be a FA and $L_{2}\in\wp(\Sigma^{*})$ be a language such that:

(i) membership in $L_{2}$ is decidable;

(ii) there exists a decidable right $L_{2}$-consistent wqo on $\Sigma^{*}$. Then, Algorithm $\mathtt{FAIncWr}$ decides the inclusion problem ${\mathcal{L}(\mathcal{A})}\subseteq L_{2}$.*

In the following, we will consider different quasiorders on $\Sigma^{*}$ and we will show that they fulfill the requirements of Theorem 5.3, therefore yielding algorithms for solving language inclusion problems.

5.2. Nerode Quasiorders

The notions of left and right quotient of a language $L\in\wp(\Sigma^{*})$ w.r.t. a word $w\in\Sigma^{*}$ are standard:

[TABLE]

Correspondingly, let us define the following quasiorder relations on $\Sigma^{*}$:

[TABLE]

De Luca and Varricchio (1994, Section 2) call them, resp., the left ($\leqq_{L}^{l}$) and right ($\leqq_{L}^{r}$) Nerode quasiorders relative to $L$. The following result shows that Nerode quasiorders are the weakest (i.e., greatest w.r.t. set inclusion of binary relations) $L_{2}$-consistent quasiorders for which the algorithm $\mathtt{FAIncW}$ can be instantiated to decide a language inclusion ${\mathcal{L}(\mathcal{A})}\subseteq L_{2}$.

Lemma 5.6.

Let $L\in\wp(\Sigma^{*})$.

1. (a)

$\mathord{\leqq_{L}^{l}}$* and $\mathord{\leqq_{L}^{r}}$ are, resp., left and right $L$-consistent quasiorders. If $L$ is regular then, additionally, $\mathord{\leqq_{L}^{l}}$ and $\mathord{\leqq_{L}^{r}}$ are decidable wqos.* 2. (b)

If $\mathord{\leqslant}$ is a left (resp. right) $L$-consistent quasiorder on $\Sigma^{*}$ then $\rho_{\leqq_{L}^{l}}(\wp(\Sigma^{*}))\subseteq\rho_{\leqslant}(\wp(\Sigma^{*}))$ (resp. $\rho_{\leqq_{L}^{r}}(\wp(\Sigma^{*}))\subseteq\rho_{\leqslant}(\wp(\Sigma^{*}))$).

Proof.

Let us consider point (a). De Luca and Varricchio (1994, Section 2) observe that $\mathord{\leqq_{L}^{l}}$ and $\mathord{\leqq_{L}^{r}}$ are, resp., left and right monotonic. Moreover, De Luca and Varricchio (1994, Theorem 2.4) show that if $L$ is regular then both $\mathord{\leqq_{L}^{l}}$ and $\mathord{\leqq_{L}^{r}}$ are wqos. Let us also observe that given $u\in L$ and $v\notin L$ we have that $\epsilon\in Lu^{-1}$ and $\epsilon\in u^{-1}L$ while $\epsilon\notin Lv^{-1}$ and $\epsilon\notin v^{-1}L$. Hence, $\mathord{\leqq_{L}^{l}}$ ($\mathord{\leqq_{L}^{r}}$) is a left (right) $L$-consistent quasiorder. Finally, if $L$ is regular then both relations are clearly decidable.

Let us now consider point (b) for the left case (the right case is symmetric). By the characterization of left consistent quasiorders given by Lemma 5.2, De Luca and Varricchio (1994, Section 2, point 4) observe that $\mathord{\leqq_{L}^{l}}$ is maximum in the set of all left $L$-consistent quasiorders, i.e. every left $L$-consistent quasiorder $\leqslant$ is such that $x\leqslant y\Rightarrow x\leqq_{L}^{l}y$. As a consequence, $\rho_{\leqslant}(X)\subseteq\rho_{\leqq_{L}^{l}}(X)$ holds for all $X\in\wp(\Sigma^{*})$, namely, $\rho_{\leqq_{L}^{l}}(\wp(\Sigma^{*}))\subseteq\rho_{\leqslant}(\wp(\Sigma^{*}))$. ∎

This allows us to derive a first instantiation of Theorem 5.3. Because membership is decidable for regular languages $L_{2}$, Lemma 5.6 (a) for $\leqq^{l}_{L_{2}}$ implies that the hypotheses (i) and (ii) of Theorem 5.3 are satisfied, so that the algorithm $\mathtt{FAIncW}$ instantiated to $\leqq^{l}_{L_{2}}$ decides the inclusion ${\mathcal{L}(\mathcal{A})}\subseteq L_{2}$ when $L_{2}$ is regular. Furthermore, under these hypotheses, Lemma 5.6 (b) shows that $\leqq_{L_{2}}^{l}$ is the weakest left $L_{2}$-consistent quasiorder relation on $\Sigma^{*}$ for which the algorithm $\mathtt{FAIncW}$ can be instantiated for deciding an inclusion ${\mathcal{L}(\mathcal{A})}\subseteq L_{2}$.

Example 5.7.

We illustrate the use of the left Nerode quasiorder in Algorithm $\mathtt{FAIncW}$ for solving the language inclusion ${\mathcal{L}(\mathcal{A}_{1})}\subseteq{\mathcal{L}(\mathcal{A}_{2})}$, where $\mathcal{A}_{1}$ and $\mathcal{A}_{2}$ are the FAs shown in Figure 2. The equations for $\mathcal{A}_{1}$ are as follows:

[TABLE]

We have the following quotients (among others) for $L={\mathcal{L}(\mathcal{A}_{2})}=a^{*}(a(a+b)^{*}a+a^{+}c+ab+bb)$:

[TABLE]

Hence, among others, the following relations hold: $c\leqq_{L}^{l}a$, $c\leqq_{L}^{l}b$ and $c\leqq_{L}^{l}w$ for all $w\in(a(a+b)^{*}a+ac+ab+bb)$. Then, let us show the computation of the Kleene iterates performed by the Algorithm $\mathtt{FAIncW}$.

[TABLE]

It turns out that $\langle{\{aa,ab,ac,a,b,c\},\{\epsilon\}}\rangle\sqsubseteq_{\leqq_{L}^{l}}\langle{\{a,b,c\},\{\epsilon\}}\rangle$ because $c\leqq_{L}^{l}aa$, $c\leqq_{L}^{l}ab$ and $c\leqq_{L}^{l}ac$ hold, so that $\operatorname{{\textsc{Kleene}}}$ stops with $\vv{\bm{Y}}^{(3)}$ and outputs $\vv{\bm{Y}}=\langle{\{a,b,c\},\{\epsilon\}}\rangle$. Since $c\in\vv{\bm{Y}}_{1}$ and $c\notin{\mathcal{L}(\mathcal{A}_{2})}$, the Algorithm $\mathtt{FAIncW}$ correctly concludes that ${\mathcal{L}(\mathcal{A}_{1})}\subseteq{\mathcal{L}(\mathcal{A}_{2})}$ does not hold. $\Diamond$

5.2.1. On the Complexity of Nerode quasiorders

For the inclusion problem between languages generated by finite automata, deciding the (left or right) Nerode quasiorder relation between words can be easily shown to be as hard as the language inclusion problem itself, which is PSPACE-complete. In fact, given the automata $\mathcal{A}_{1}=(Q_{1},\delta_{1},I_{1},F_{1},\Sigma)$ and $\mathcal{A}_{2}=(Q_{2},\delta_{2},I_{2},F_{2},\Sigma)$, one can define the union automaton $\mathcal{A}_{3}\triangleq(Q_{1}\cup Q_{2}\cup\{q^{\iota}\},\delta_{3},\{q^{\iota}\},F_{1}\cup F_{2})$ where $\delta_{3}$ maps $(q^{\iota},a)$ to $I_{1}$, $(q^{\iota},b)$ to $I_{2}$ and behaves like $\delta_{1}$ or $\delta_{2}$ elsewhere. Then, it turns out that $a\leqq^{r}_{{\mathcal{L}(\mathcal{A}_{3})}}b\Leftrightarrow a^{-1}{\mathcal{L}(\mathcal{A}_{3})}\subseteq b^{-1}{\mathcal{L}(\mathcal{A}_{3})}\Leftrightarrow{\mathcal{L}(\mathcal{A}_{1})}\subseteq{\mathcal{L}(\mathcal{A}_{2})}$.

Also, for the inclusion problem of a language generated by an automaton within the trace set of a one-counter net (cf. Section 5.4), the right Nerode quasiorder is a right language-consistent well-quasiorder but it turns out to be undecidable (cf. Lemma 5.16).

5.3. State-based Quasiorders

Consider an inclusion problem ${\mathcal{L}(\mathcal{A}_{1})}\subseteq{\mathcal{L}(\mathcal{A}_{2})}$ where $\mathcal{A}_{1}$ and $\mathcal{A}_{2}$ are FAs. In the following, we study a class of well-quasiorders based on $\mathcal{A}_{2}$, that we call state-based quasiorders. These quasiorders are strictly stronger (i.e., lower w.r.t. set inclusion of binary relations) than the Nerode quasiorders defined in Section 5.2 and sidestep the untractability or undecidability of Nerode quasiorders (cf. Section 5.2.1) yet allowing to define an algorithm solving the language inclusion ${\mathcal{L}(\mathcal{A}_{1})}\subseteq{\mathcal{L}(\mathcal{A}_{2})}$.

5.3.1. Inclusion in Regular Languages.

We define the quasiorders $\leq^{l}_{\mathcal{A}}$ and $\leq^{r}_{\mathcal{A}}$ on $\Sigma^{*}$ induced by a FA $\mathcal{A}=\langle{Q,\delta,I,F,\Sigma}\rangle$ as follows: for all $u,v\in\Sigma^{*}$,

[TABLE]

where, for all $X\in\wp(Q)$, $\operatorname{pre}_{u}^{\mathcal{A}}(X)\triangleq\{q\in Q\mid u\in W^{\mathcal{A}}_{q,X}\}$ and $\operatorname{post}_{u}^{\mathcal{A}}(X)\triangleq\{q^{\prime}\in Q\mid u\in W^{\mathcal{A}}_{X,q^{\prime}}\}$ denote the standard predecessor/successor state transformers in $\mathcal{A}$. The superscripts in $\leq^{l}_{\mathcal{A}}$ and $\leq^{r}_{\mathcal{A}}$ stand, resp., for left/right because the following result holds.

Lemma 5.8.

The relations $\mathord{\leq^{l}_{\mathcal{A}}}$ and $\mathord{\leq^{r}_{\mathcal{A}}}$ are, resp., decidable left and right ${\mathcal{L}(\mathcal{A})}$-consistent wqos.

Proof.

Since, for every $u\in\Sigma^{*}$, $\operatorname{pre}^{\mathcal{A}}_{u}(F)$ is a finite and computable set, it turns out that $\mathord{\leq^{l}_{\mathcal{A}}}$ is a decidable wqo. Let us check that $\mathord{\leq^{l}_{\mathcal{A}}}$ is left ${\mathcal{L}(A)}$-consistent according to Definition 5.1 (a)-(b).

(a) By picking $u\in{\mathcal{L}(\mathcal{A})}$ and $v\notin{\mathcal{L}(\mathcal{A})}$ we have that $\operatorname{pre}^{\mathcal{A}}_{u}(F)$ contains some initial state while $\operatorname{pre}^{\mathcal{A}}_{v}(F)$ does not, hence $u\nleq^{l}_{\mathcal{A}}v$.

(b) Let us check that $\leq^{l}_{\mathcal{A}}$ is left monotonic. Observe that, for all $x\in\Sigma^{*}$, $\operatorname{pre}^{\mathcal{A}}_{x}$ is a monotonic function and that

[TABLE]

Therefore, for all $x_{1},x_{2}\in\Sigma^{*}$ and $a\in\Sigma$,

[TABLE]

The proof that $\leq_{\mathcal{A}}^{r}$ is a decidable right ${\mathcal{L}(\mathcal{A})}$-consistent quasiorder is symmetric. ∎

As a consequence, Theorem 5.3 applies to the wqo $\mathord{\leq^{l}_{\mathcal{A}_{2}}}$ (and $\mathord{\leq^{r}_{\mathcal{A}_{2}}}$), so that one can instantiate the algorithm $\mathtt{FAIncW}$ to $\mathord{\leq^{l}_{\mathcal{A}_{2}}}$ for deciding an inclusion ${\mathcal{L}(\mathcal{A}_{1})}\subseteq{\mathcal{L}(\mathcal{A}_{2})}$.

Turning back to the left Nerode wqo $\leqq_{{\mathcal{L}(\mathcal{A}_{2})}}^{l}$, it turns out that the following equivalences hold:

[TABLE]

Since $\operatorname{pre}^{\mathcal{A}_{2}}_{u}(F)\subseteq\operatorname{pre}^{\mathcal{A}_{2}}_{v}(F)$ entails $W_{I,\operatorname{pre}^{\mathcal{A}_{2}}_{u}(F)}\subseteq W_{I,\operatorname{pre}^{\mathcal{A}_{2}}_{v}(F)}$, it follows that $u\leq_{\mathcal{A}_{2}}^{l}v\Rightarrow u\leqq_{{\mathcal{L}(\mathcal{A}_{2})}}^{l}v$ and, in turn, $\rho_{\leqq_{{\mathcal{L}(\mathcal{A}_{2})}}^{l}}(\wp(\Sigma^{*}))\subseteq\rho_{\leq^{l}_{\mathcal{A}_{2}}}(\wp(\Sigma^{*}))$.

Example 5.9.

We illustrate the left state-based quasiorder by using it to solve the language inclusion ${\mathcal{L}(\mathcal{A}_{1})}\subseteq{\mathcal{L}(\mathcal{A}_{2})}$ of Example 5.7. We have, among others, the following set of predecessors of $F_{\mathcal{A}_{2}}$:

[TABLE]

Recall from Example 5.7 that, for the Nerode quasiorder, we have $c\leqq_{{\mathcal{L}(\mathcal{A}_{2})}}^{l}b$, $c\leqq_{{\mathcal{L}(\mathcal{A}_{2})}}^{l}a$ while none of these relations hold for $\leq^{l}_{\mathcal{A}_{2}}$.

Let us next show the Kleene iterates computed by Algorithm $\mathtt{FAIncW}$ when using the quasiorder $\leq^{l}_{\mathcal{A}_{2}}$.

[TABLE]

It turns out that $\langle{\{aaa,aab,aac,aa,ab,ac,a,b,c\},\{\epsilon\}}\rangle\sqsubseteq_{\leq^{l}_{\mathcal{A}_{2}}}\langle{\{aa,ab,ac,a,b,c\},\{\epsilon\}}\rangle$ so that $\operatorname{{\textsc{Kleene}}}$ outputs the vector $\vv{\bm{Y}}=\langle{\{aa,ab,ac,a,b,c\},\{\epsilon\}}\rangle$. Since $c\in\vv{\bm{Y}}_{0}$ and $c\notin{\mathcal{L}(\mathcal{A}_{2})}$, Algorithm $\mathtt{FAIncW}$ concludes that the language inclusion ${\mathcal{L}(\mathcal{A}_{1})}\subseteq{\mathcal{L}(\mathcal{A}_{2})}$ does not hold. $\Diamond$

5.3.2. Simulation-based Quasiorders.

Recall that a simulation on a FA $\mathcal{A}=\langle{Q,\delta,I,F,\Sigma}\rangle$ is a binary relation $\mathord{\preceq}\subseteq Q\times Q$ such that for all $p,q\in Q$ such that $p\preceq q$ the following two conditions hold:

1. (i)

if $p\in F$ then $q\in F$; 2. (ii)

for every transition $p\xrightarrow{a}p^{\prime}$, there exists a transition $q\xrightarrow{a}q^{\prime}$ such that $p^{\prime}\preceq q^{\prime}$.

It is well known that simulation relations are closed under arbitrary unions, where the greatest (w.r.t. inclusion) simulation relation $\mathord{\preceq_{A}}\triangleq\cup\{\mathord{\preceq}\subseteq Q\times Q\mid\mathord{\preceq}$ is a simulation on $\mathcal{A}\}$ is a quasiorder, called simulation quasiorder of $\mathcal{A}$. It is also well known that simulation implies language inclusion, i.e., if $\preceq$ is a simulation on $\mathcal{A}$ then

[TABLE]

A relation $\mathord{\leq}\subseteq Q\times Q$ on states can be lifted in the standard universal/existential way to a relation $\leq^{\forall\exists}\subseteq\wp(Q)\times\wp(Q)$ on sets of states as follows:

[TABLE]

In particular, if $\leq$ is a quasiorder then $\leq^{\forall\exists}$ is a quasiorder as well. Also, if $\preceq$ is a simulation relation then its lifting $\preceq^{\forall\exists}$ is such that $X\preceq^{\forall\exists}Y\Rightarrow W^{\mathcal{A}}_{X,F}\subseteq W^{\mathcal{A}}_{Y,F}$ holds. This suggests us to define a right simulation-based quasiorder $\preceq^{r}_{\mathcal{A}}$ on $\Sigma^{*}$ induced by a simulation $\preceq$ on $\mathcal{A}$ as follows: for all $u,v\in\Sigma^{*}$,

[TABLE]

Lemma 5.10.

Given a simulation relation $\mathord{\preceq}$ on $\mathcal{A}$, the right simulation-based quasiorder $\mathord{\preceq^{r}_{\mathcal{A}}}$ is a decidable right ${\mathcal{L}(\mathcal{A})}$-consistent wqo.

Proof.

Let $u\in{\mathcal{L}(\mathcal{A})}$ and $v\notin{\mathcal{L}(\mathcal{A})}$, so that $F\cap\operatorname{post}^{\mathcal{A}}_{u}(I)\neq\varnothing$ and $(F\cap\operatorname{post}^{\mathcal{A}}_{v}(I))=\varnothing$ hold. Hence, there exists $q\in\operatorname{post}^{\mathcal{A}}_{u}(F)\cap F$ such that $q\preceq^{r}_{\mathcal{A}}q^{\prime}$ for no $q^{\prime}\in\operatorname{post}^{\mathcal{A}}_{v}(F)$ since, by simulation, this would imply $q^{\prime}\in\operatorname{post}^{\mathcal{A}}_{v}(F)\cap F$, which would contradict $F\cap\operatorname{post}^{\mathcal{A}}_{v}(I)=\varnothing$. Therefore, $u\npreceq^{r}_{\mathcal{A}}v$ holds.

Next we show that $\preceq^{r}_{\mathcal{A}}$ is right monotonic. By (13), we check that for all $u,v\in\Sigma^{*}$ and $a\in\Sigma$, $u\preceq^{r}_{\mathcal{A}}v\Rightarrow ua\preceq^{r}_{\mathcal{A}}va$:

[TABLE]

Thus, $\preceq_{\mathcal{A}}^{r}$ is a right ${\mathcal{L}(\mathcal{A})}$-consistent quasiorder.

Finally, since $\wp(Q)$ is finite, it follows that $\preceq_{\mathcal{A}}^{r}$ is a well-quasiorder and, since $\operatorname{post}^{\mathcal{A}}_{u}(I)$ is finite and computable for every $u\in\Sigma^{*}$, it follows that $\preceq_{\mathcal{A}}^{r}$ is decidable. ∎

Thus, once again, Theorem 5.5 applies to $\mathord{\preceq^{r}_{\mathcal{A}_{2}}}$ and this allows us to instantiate the algorithm $\mathtt{FAIncWr}$ to the quasiorder $\mathord{\preceq^{r}_{\mathcal{A}_{2}}}$ for deciding an inclusion ${\mathcal{L}(\mathcal{A}_{1})}\subseteq{\mathcal{L}(\mathcal{A}_{2})}$.

Note that it is possible to define a left simulation $\preceq^{\forall\exists}_{R}$ on an automaton $\mathcal{A}$ by applying $\preceq^{\forall\exists}$ on the reverse automaton $\mathcal{A}^{R}$ of $\mathcal{A}$ where arrows are flipped and initial/final states are swapped. This left simulation induces a left simulation-based quasiorder on $\Sigma^{*}$ as follows: for all $u,v\in\Sigma^{*}$,

[TABLE]

It is straightforward to check that Theorem 5.3 applies to $\mathord{\preceq^{l}_{\mathcal{A}_{2}}}$ and, therefore, we can instantiate the Algorithm $\mathtt{FAIncW}$ for deciding ${\mathcal{L}(\mathcal{A}_{1})}\subseteq{\mathcal{L}(\mathcal{A}_{2})}$.

Example 5.11.

Let us illustrate the use of the left simulation-based quasiorder to solve the language inclusion ${\mathcal{L}(\mathcal{A}_{1})}\subseteq{\mathcal{L}(\mathcal{A}_{2})}$ of Example 5.7. For the set $F_{\mathcal{A}_{2}}$ of final states $\mathcal{A}_{2}$ we have the same set of predecessors computed in Example 5.9 and, among others, the following left simulations between these sets w.r.t. the simulation quasiorder $\preceq_{\mathcal{A}_{2}^{R}}$ of the reverse of $\mathcal{A}_{2}$ (recall that $\preceq^{\forall\exists}$ is defined w.r.t. simulations of $\mathcal{A}_{2}^{R}$):

[TABLE]

because $q_{2}\preceq_{\mathcal{A}_{2}^{R}}q_{1}$, $q_{2}\preceq_{\mathcal{A}_{2}^{R}}q_{3}$ and $q_{2}\preceq_{\mathcal{A}_{2}^{R}}q_{4}$ hold.

Let us show the computation of the Kleene iterates performed by Algorithm $\mathtt{FAIncW}$ when using the quasiorder $\sqsubseteq_{\mathord{\preceq_{\mathcal{A}_{2}}^{l}}}$ as abstract inclusion check:

[TABLE]

It turns out that $\langle{\{aa,ab,ac,a,b,c\},\{\epsilon\}}\rangle\sqsubseteq_{\preceq^{l}_{\mathcal{A}_{2}}}\langle{\{a,b,c\},\{\epsilon\}}\rangle$, so that $\operatorname{{\textsc{Kleene}}}$ outputs the vector $\vv{\bm{Y}}=\langle{\{a,b,c\},\{\epsilon\}}\rangle$. Thus, once again, since $c\in\vv{\bm{Y}}_{0}$ and $c\notin{\mathcal{L}(\mathcal{A}_{2})}$, Algorithm $\mathtt{FAIncW}$ concludes that ${\mathcal{L}(\mathcal{A}_{1})}\subseteq{\mathcal{L}(\mathcal{A}_{2})}$ does not hold. $\Diamond$

Let us observe that $u\preceq^{r}_{\mathcal{A}_{2}}v$ implies $W_{\operatorname{post}^{\mathcal{A}_{2}}_{u}(I),F}\subseteq W_{\operatorname{post}^{\mathcal{A}_{2}}_{v}(I),F}$, which is equivalent to the right Nerode quasiorder $u\leqq^{r}_{{\mathcal{L}(\mathcal{A}_{2})}}v$ for ${\mathcal{L}(\mathcal{A}_{2})}$ defined in (14), so that $u\preceq^{r}_{\mathcal{A}_{2}}v\Rightarrow u\leqq^{r}_{{\mathcal{L}(\mathcal{A}_{2})}}v$ holds. Furthermore, for the state-based quasiorder defined in (15), we have that $u\leq^{r}_{\mathcal{A}_{2}}v\Rightarrow u\preceq^{r}_{\mathcal{A}_{2}}v$ trivially holds. Summing up, the following containments relate (the right versions of) state-based, simulation-based and Nerode quasiorders:

[TABLE]

All these quasiorders are decidable ${\mathcal{L}(\mathcal{A}_{2})}$-consistent wqos so that the algorithm $\mathtt{FAIncW}$ can be instantiated to each of them for deciding an inclusion ${\mathcal{L}(\mathcal{A}_{1})}\subseteq{\mathcal{L}(\mathcal{A}_{2})}$. Examples 5.7, 5.9 and 5.11 show how $\mathtt{FAIncW}$ behaves for each of the three quasiorders considered in this section. Despite their simplicity, the examples show the differences in the behavior of the algorithm when considering the different quasiorders. In particular, we observe that the iterations of $\operatorname{{\textsc{Kleene}}}$ for $\mathord{\leqq^{r}_{{\mathcal{L}(\mathcal{A}_{2})}}}$ coincides with those for $\mathord{\preceq^{r}_{\mathcal{A}_{2}}}$ and, as expected, these Kleene iterates converge faster than those for $\mathord{\leq^{r}_{\mathcal{A}_{2}}}$. Recall that $\mathord{\leqq^{r}_{{\mathcal{L}(\mathcal{A}_{2})}}}$ is the coarsest well-quasiorder for which Algorithm $\mathtt{FAIncW}$ works, hence its corresponding Kleene iterates exhibit optimal behavior in terms of number of iterations to converge. The drawback of using the Nerode quasiorder $\mathord{\leqq^{r}_{{\mathcal{L}(\mathcal{A}_{2})}}}$ is that it requires checking language inclusion in order to decide whether two words are related, and this is a PSPACE-complete problem. Therefore, the coincidence of the Kleene iterates for $\mathord{\leqq^{r}_{{\mathcal{L}(\mathcal{A}_{2})}}}$ and $\mathord{\preceq^{r}_{\mathcal{A}_{2}}}$ is of special interest since it highlights that Algorithm $\mathtt{FAIncW}$ might exhibit optimal behavior while using a “simpler” (i.e., finer) well-quasiorder such as $\mathord{\preceq^{r}_{\mathcal{A}_{2}}}$, which is a polynomial approximation of $\mathord{\leqq^{r}_{{\mathcal{L}(\mathcal{A}_{2})}}}$.

5.4. Inclusion in Traces of One-Counter Nets.

We show that our framework can be instantiated to systematically derive an algorithm for deciding an inclusion ${\mathcal{L}(\mathcal{A})}\subseteq L_{2}$ where $L_{2}$ is the trace set of a one-counter net (OCN). This is accomplished by defining a decidable $L_{2}$-consistent quasiorder so that Theorem 5.3 can be applied.

Intuitively, an OCN is a FA endowed with a nonnegative integer counter which can be incremented, decremented or left unchanged by a transition. Formally, a one-counter net (Hofman and Totzke, 2018) is a tuple $\mathcal{O}=\langle{Q,\Sigma,\delta}\rangle$ where $Q$ is a finite set of states, $\Sigma$ is an alphabet and $\delta\subseteq Q\times\Sigma\times\{-1,0,1\}\times Q$ is a set of transitions. A configuration of $\mathcal{O}$ is a pair $qn$ consisting of a state $q\in Q$ and a value $n\in\mathbb{N}$ for the counter. Given two configurations $qn,q^{\prime}n^{\prime}\in Q\times\mathbb{N}$ we write $qn\xrightarrow{a}q^{\prime}n^{\prime}$ and call it a $a$-step (or simply step) if there exists a transition $(q,a,d,q^{\prime})\in\delta$ such that $n^{\prime}=n+d$. Given $qn\in Q\times\mathbb{N}$, the trace set $T(qn)\subseteq\Sigma^{*}$ of an OCN is defined as follows:

[TABLE]

Observe that $Z_{\epsilon}^{qn}=\{qn\}$ and $Z_{u}^{qn}$ is a finite set for every word $u\in\Sigma^{*}$.

Let us consider the poset $\langle{\mathbb{N}_{\bot}\triangleq\mathbb{N}\cup\{\bot\},\leq_{\mathbb{N}_{\bot}}}\rangle$ where $\bot\leq_{\mathbb{N}_{\bot}}n$ holds for all $n\in\mathbb{N}_{\bot}$, while for all $n,n^{\prime}\in\mathbb{N}$, $n\leq_{\mathbb{N}_{\bot}}n^{\prime}$ is the standard ordering relation between numbers. For a finite set of states $S\subseteq Q\times\mathbb{N}$, define the so-called macro state $M_{S}\colon Q\rightarrow\mathbb{N}_{\bot}$ as follows:

[TABLE]

where $\max\varnothing\triangleq\bot$. Let us define the following quasiorder $\mathord{\leq_{{qn}}^{r}}\subseteq\Sigma^{*}\times\Sigma^{*}$:

[TABLE]

Example 5.12.

Figure 3 depicts an OCN over the singleton alphabet $\Sigma=\{a\}$. For $\mathcal{O}$ we have the following sets:

[TABLE]

Hence, we have that:

[TABLE]

Therefore, the words $\epsilon,a$ and $aa$ are pairwise incomparable for $\mathord{\leq_{{q_{1}0}}^{r}}$, while we have that $aa\leq_{q_{1}0}^{r}aaa$ and $\epsilon\leq_{q_{1}0}^{r}aaa$. $\Diamond$

Lemma 5.13.

Let $\mathcal{O}$ be an OCN. For any configuration $qn$ of $\mathcal{O}$, $\mathord{\leq_{{qn}}^{r}}$ is a right $T(qn)$-consistent decidable wqo.

Proof.

It follows from Dickson’s Lemma (Sakarovitch, 2009, Section II.7.1.2) that $\mathord{\leq_{{qn}}^{r}}$ is a wqo. Since $Z_{u}^{qn}$ and $Z_{v}^{qn}$ are finite sets of configurations, the macro state functions $M_{Z_{u}^{qn}}$ and $M_{Z_{v}^{qn}}$ are computable, hence the relation $\mathord{\leq_{{qn}}^{r}}$ is decidable. If $u\in T(qn)$ and $v\notin T(qn)$ then $u\not\leq_{{qn}}^{r}v$, otherwise we would have that $M_{Z_{u}^{qn}}(q^{\prime})\neq\bot$ for some $q^{\prime}\in Q$, hence $M_{Z_{v}^{qn}}(q^{\prime})\neq\bot$, and this would be a contradiction because $Z_{v}^{qn}=\varnothing$, so that $M_{Z_{v}^{qn}}(q^{\prime})=\bot$.

Finally, let us show that $u\leq_{{qn}}^{r}v$ implies $ua\leq_{{qn}}^{r}va$ for all $a\in\Sigma$, since, by (13), this is equivalent to the fact that $\leq_{{qn}}^{r}$ is right monotonic. We proceed by contradiction. Assume that $u\leq_{{qn}}^{r}v$ and $\exists q^{\prime}\in Q$, $M_{Z^{qn}_{ua}}(q^{\prime})\not\leq_{\mathbb{N}_{\bot}}M_{Z^{qn}_{va}}(q^{\prime})$. Then, $m_{1}\triangleq\max\{n\mid pn\in Z^{qn}_{ua}\}\not\leq_{\mathbb{N}_{\bot}}m_{2}\triangleq\max\{n\mid pn\in Z^{qn}_{va}\}$, which implies, since $m_{1}\neq\bot$, that $m_{1},m_{2}\in\mathbb{N}$ and $m_{1}>m_{2}$. Thus, for all $(q,a,d,q^{\prime})\in\delta$ we have $q^{\prime}(m_{1}-d)\in Z_{u}^{qn}$ and $q^{\prime}(m_{2}-d)\in Z_{v}^{qn}$. Since $m_{1}-d>m_{2}-d$ we have that $\max\{n\mid pn\in Z_{u}^{qn}\}>\max\{n\mid pn\in Z_{v}^{qn}\}$, which contradicts $u\leq_{{qn}}^{r}v$. ∎

By Theorem 5.3, Lemma 5.13 and the decidability of membership $u\in^{\scaleto{?}{3.5pt}}T(qn)$, the following known decidability result for inclusion of regular languages into traces of OCNs (Jančar et al., 1999, Theorem 3.2) is systematically derived as a consequence of our algorithmic framework.

Corollary 5.14.

Let $\mathcal{A}$ be a FA and $\mathcal{O}$ be an OCN. For any configuration $qn$ of $\mathcal{O}$, the language inclusion problem ${\mathcal{L}(\mathcal{A})}\subseteq T(qn)$ is decidable.

Example 5.15.

Consider the OCN of Figure 3 and the problem of deciding whether $\Sigma^{*}=a^{*}$ is included into $T(q_{0}0)$, i.e., whether the trace set of $\mathcal{O}$ is universal. By considering the equation $X=Xa\cup\{\epsilon\}$ which defines $\Sigma^{*}$, it turns out that the Kleene iterates computed by Algorithm $\mathtt{FAIncW}$ when using the abstract inclusion check given by $\sqsubseteq_{\leq^{r}_{q_{0}0}}$ are as follows:

[TABLE]

We have that $Y^{(4)}\sqsubseteq_{\leq^{r}_{q_{0}0}}Y^{(3)}$ because $aa\leq^{r}_{q_{0}0}aaa$ holds, as shown in Example 5.12, so that the output of $\operatorname{{\textsc{Kleene}}}$ is $Y^{(3)}=\{aa,a,\epsilon\}$. Since $\{aa,a,\epsilon\}$ is a set of traces of $\mathcal{O}$ (i.e. $\{aa,a,\epsilon\}\subseteq T(q_{0}0)$) we conclude that $\mathcal{O}$ is universal. $\Diamond$

Moreover, by exploiting Lemma 5.13 and (Hofman et al., 2013, Theorem 20), the following result settles a conjecture made by de Luca and Varricchio (1994, Section 6) on the right Nerode quasiorder for traces of OCNs.

Lemma 5.16.

The right Nerode quasiorder $\mathord{\leqq^{r}_{T(qn)}}$ is an undecidable well-quasiorder.

Proof.

As already recalled, de Luca and Varricchio (1994, Section 2, point 4) show that $\mathord{\leqq^{r}_{T(qn)}}$ is maximum in the set of all right $T(qn)$-consistent quasiorders, so that $u\leq^{r}_{{qn}}v$ $\Rightarrow$ $u\leqq^{r}_{T(qn)}v$, for all $u,v\in\Sigma^{*}$. By Lemma 5.13, $\leq^{r}_{{qn}}$ is a wqo, so that $\mathord{\leqq^{r}_{T(qn)}}$ is a wqo as well. Undecidability of $\mathord{\leqq^{r}_{T(qn)}}$ follows from the undecidability of the trace inclusion problem for nondeterministic OCNs (Hofman et al., 2013, Theorem 20) by an argument similar to the automata case. ∎

It is worth remarking that, by Lemma 5.6 (a), the left and right Nerode quasiorders $\mathord{\leqq^{l}_{T(qn)}}$ and $\mathord{\leqq^{r}_{T(qn)}}$ are $T(qn)$-consistent. However, the left Nerode quasiorder does not need to be a wqo, otherwise $T(qn)$ would be regular.

We conclude this section by conjecturing that our framework could be instantiated for extending Corollary 5.14 to traces of Petri Nets, a result which is already known to be true (Jančar et al., 1999).

6. A Novel Perspective on the Antichain Algorithm

In this section we will show how to solve the language inclusion problem by computing Kleene iterates in an abstract domain of $\wp(\Sigma^{*})$ as defined by a Galois connection. This is of practical interest since it allows us to decide a language inclusion problem ${\mathcal{L}(\mathcal{A})}\subseteq L_{2}$ by manipulating an automaton representation for $L_{2}$.

6.1. A Language Inclusion Algorithm Using Galois Connections

The next result provides a formulation of Theorem 4.7 by using a Galois Connection $\langle{\wp(\Sigma^{*}),\subseteq}\rangle\galois{\alpha}{\gamma}\langle{D,\leq_{D}}\rangle$ rather than a closure operator $\rho\in\operatorname{uco}(\wp(\Sigma^{*})$ and shows how to design an algorithm that solves a language inclusion ${\mathcal{L}(\mathcal{A})}\subseteq L_{2}$ by computing Kleene iterates on the abstract domain $D$.

Theorem 6.1.

Let $\mathcal{A}=\langle{Q,\delta,I,F,\Sigma}\rangle$ be a FA and $L_{2}\in\wp(\Sigma^{*})$. Let $\langle{D,\leq_{D}}\rangle$ be a poset and $\langle{\wp(\Sigma^{*}),\subseteq}\rangle\galois{\alpha}{\gamma}\langle{D,\leq_{D}}\rangle$ be a GC. Assume that the following properties hold:

1. (i)

$L_{2}\in\gamma(D)$* and for all $a\in\Sigma$ and $X\in\wp(\Sigma^{*})$, $\gamma\alpha(aX)=\gamma\alpha(a\gamma\alpha(X))$.* 2. (ii)

$\langle{D,\leq_{D},\sqcup,\bot_{D}}\rangle$* is an effective domain, meaning that: $\langle{D,\leq_{D},\sqcup,\bot_{D}}\rangle$ is an ACC join-semilattice with bottom $\bot_{D}$, every element of $D$ has a finite representation, the binary relation $\leq_{D}$ is decidable and the binary lub $\sqcup$ is computable.* 3. (iii)

There is an algorithm, say $\operatorname{{Pre}}^{\sharp}$, which computes $\alpha\comp\operatorname{{Pre}}_{\mathcal{A}}\comp\gamma$. 4. (iv)

There is an algorithm, say $\epsilon^{\sharp}$, which computes $\alpha(\vv{\bm{\epsilon}}^{{\!{\scriptstyle{F}}}})$. 5. (v)

There is an algorithm, say $\operatorname{{Incl^{\sharp}}}$, which decides $\vv{\bm{X}}^{\sharp}\leq_{D}\alpha(\vv{\bm{L_{2}}}^{{\!{\scriptstyle{I}}}})$, for all $\vv{\bm{X}}^{\sharp}\in\alpha(\wp(\Sigma^{*}))^{|Q|}$.

Then,

$\langle{Y^{\sharp}_{q}}\rangle_{q\in Q}:=\operatorname{{\textsc{Kleene}}}(\leq_{D},\lambda\vv{\bm{X}}^{\sharp}\ldotp\epsilon^{\sharp}\sqcup\operatorname{{Pre}}^{\sharp}(\vv{\bm{X}}^{\sharp}),\vv{\bm{\bot_{D}}})$;**

return* $\operatorname{{Incl^{\sharp}}}(\langle{Y^{\sharp}_{q}}\rangle_{q\in Q})$*;**

is a decision algorithm for ${\mathcal{L}(\mathcal{A})}\subseteq L_{2}$.

Proof.

Let $\rho\triangleq\gamma\alpha\in\operatorname{uco}(\wp(\Sigma^{*}))$, so that hypothesis (i) can be stated as $\rho(L_{2})=L_{2}$ and $\rho(aX)=\rho(a\rho(X))$, and this allows us to apply Corollary 4.4. It turns out that:

[TABLE]

Thus, by hypotheses (ii), (iii) and (iv), it turns out that $\operatorname{{\textsc{Kleene}}}(\leq_{D},\lambda\vv{\bm{X}}^{\sharp}\ldotp\epsilon^{\sharp}\sqcup\operatorname{{Pre}}^{\sharp}(\vv{\bm{X}}^{\sharp}),\vv{\bm{\bot_{D}}})$ is an algorithm computing the least fixpoint $\operatorname{lfp}(\lambda\vv{\bm{X}}^{\sharp}\ldotp\alpha(\vv{\bm{\epsilon}}^{{\!{\scriptstyle{F}}}})\sqcup\alpha(\operatorname{{Pre}}_{\mathcal{A}}(\gamma(\vv{\bm{X}}^{\sharp}))))$. In particular, (ii), (iii) and (iv) ensure that the Kleene iterates of $\lambda\vv{\bm{X}}^{\sharp}\ldotp\epsilon^{\sharp}\sqcup\operatorname{{Pre}}^{\sharp}(\vv{\bm{X}}^{\sharp})$ starting from $\vv{\bm{\bot_{D}}}$ are computable and finitely many and that it is decidable when the iterates converge for $\leq_{D}$, namely, reach the least fixpoint. Finally, hypothesis (v) ensures the decidability of the $\leq_{D}$-inclusion check of this least fixpoint in $\alpha(\vv{\bm{L_{2}}}^{{\!{\scriptstyle{I}}}})$. ∎

It is worth pointing out that, analogously to Theorem 4.6, the above Theorem 6.1 can be also stated in a symmetric version for right (rather than left) concatenation.

6.2. Antichains as a Galois Connection

Let $\mathcal{A}_{1}=\langle{Q_{1},\delta_{1},I_{1},F_{1},\Sigma}\rangle$ and $\mathcal{A}_{2}=\langle{Q_{2},\delta_{2},I_{2},F_{2},\Sigma}\rangle$ be two FAs and consider the state-based left ${\mathcal{L}(\mathcal{A}_{2})}$-consistent wqo $\mathord{\leqslant_{\mathcal{A}_{2}}^{l}}$ defined by (15). Theorem 5.3 shows that Algorithm $\mathtt{FAIncW}$ decides ${\mathcal{L}(\mathcal{A}_{1})}\subseteq{\mathcal{L}(\mathcal{A}_{2})}$ by computing vectors of finite sets of words. Since $u\leqslant_{\mathcal{A}_{2}}^{l}v\Leftrightarrow\operatorname{pre}^{\mathcal{A}_{2}}_{u}(F_{2})\subseteq\operatorname{pre}^{\mathcal{A}_{2}}_{v}(F_{2})$, we can equivalently consider the set of states $\operatorname{pre}^{\mathcal{A}_{2}}_{u}(F_{2})\in\wp(Q_{2})$ rather than a word $u\in\Sigma^{*}$. This observation suggests to design a version of Algorithm $\mathtt{FAIncW}$ that computes Kleene iterates on the poset $\langle{\operatorname{{AC}}_{\langle{\wp(Q_{2}),\subseteq}\rangle},\sqsubseteq}\rangle$ of antichains of sets of states of the complete lattice $\langle{\wp(Q_{2}),\subseteq}\rangle$. In order to do this, $\langle{\operatorname{{AC}}_{\langle{\wp(Q_{2}),\subseteq}\rangle},\sqsubseteq}\rangle$ is viewed as an abstract domain through the following maps $\alpha\colon\wp(\Sigma^{*})\rightarrow\operatorname{{AC}}_{\langle{\wp(Q_{2}),\subseteq}\rangle}$ and $\gamma\colon\operatorname{{AC}}_{\langle{\wp(Q_{2}),\subseteq}\rangle}\rightarrow\wp(\Sigma^{*})$. Moreover, we use the abstract function ${\operatorname{{Pre}}}_{\mathcal{A}_{1}}^{\mathcal{A}_{2}}:(\operatorname{{AC}}_{\langle{\wp(Q_{2}),\subseteq}\rangle})^{|Q_{1}|}\rightarrow(\operatorname{{AC}}_{\langle{\wp(Q_{2}),\subseteq}\rangle})^{|Q_{1}|}$ defined as follows:

[TABLE]

where $\lfloor{X}\rfloor$ is the unique minor set w.r.t. subset inclusion of $X\subseteq\wp(Q_{2})$. Observe that the functions $\alpha$ and ${\operatorname{{Pre}}}_{\mathcal{A}_{1}}^{\mathcal{A}_{2}}$ are well-defined because minors of finite subsets of $\wp(Q_{2})$ are uniquely defined antichains.

Lemma 6.2.

The following properties hold:

1. (a)

$\langle{\wp(\Sigma^{*}),\subseteq}\rangle\galois{\alpha}{\gamma}\langle{\operatorname{{AC}}_{\langle{\wp(Q_{2}),\subseteq}\rangle},\sqsubseteq}\rangle$* is a GC.* 2. (b)

$\gamma\comp\alpha=\rho_{\leqslant^{l}_{\mathcal{A}_{2}}}$. 3. (c)

$\operatorname{{Pre}}_{\mathcal{A}_{1}}^{\mathcal{A}_{2}}={\alpha\comp\operatorname{{Pre}}_{\mathcal{A}_{1}}\comp\gamma}$.

Proof.

1. (a)

Let us first observe that $\alpha$ is well-defined: in fact, $\alpha(X)$ is an antichain of $\langle{\wp(Q_{2}),\subseteq}\rangle$ since it is a minor for the well-quasiorder $\subseteq$ and, therefore, it is finite. Then, for all $X\in\wp(\Sigma^{*})$ and $Y\in\operatorname{{AC}}_{\langle{\wp(Q_{2}),\subseteq}\rangle}$, it turns out that:

[TABLE] 2. (b)

For all $X\in\wp(\Sigma^{*})$:

[TABLE] 3. (c)

For all $\vv{\bm{X}}\in(\operatorname{{AC}}_{\langle{\wp(Q_{2}),\subseteq}\rangle})^{|Q_{1}|}$:

[TABLE]

Thus, by Lemma 5.8 and Lemma 6.2, it turns out that the GC $\langle{\wp(\Sigma^{*}),\subseteq}\rangle\galois{\alpha}{\gamma}\langle{\operatorname{{AC}}_{\langle{\wp(Q_{2}),\subseteq}\rangle},\sqsubseteq}\rangle$ and the abstract function $\operatorname{{Pre}}_{\mathcal{A}_{1}}^{\mathcal{A}_{2}}$ satisfy the hypotheses (i)-(iv) of Theorem 6.1. In order to obtain an algorithm for deciding ${\mathcal{L}(\mathcal{A}_{1})}\subseteq{\mathcal{L}(\mathcal{A}_{2})}$ it remains to show that the hypothesis (v) of Theorem 6.1 holds, i.e., there is an algorithm to decide whether $\vv{\bm{Y}}\sqsubseteq\alpha(\vv{\bm{L_{2}}}^{{\!{\scriptstyle{I_{2}}}}})$ for every $\vv{\bm{Y}}\in\alpha(\wp(\Sigma^{*}))^{|Q_{1}|}$.

Notice that the Kleene iterates of $\lambda\vv{\bm{X}}^{\sharp}\ldotp\alpha(\vv{\bm{\epsilon}}^{{\!{\scriptstyle{F_{1}}}}})\sqcup\operatorname{{Pre}}_{\mathcal{A}_{1}}^{\mathcal{A}_{2}}(\vv{\bm{X}}^{\sharp})$ of Theorem 6.1 are vectors of antichains in $\langle{\operatorname{{AC}}_{\langle{\wp(Q_{2}),\subseteq}\rangle},\sqsubseteq}\rangle$, where each component is indexed by some $q\in Q_{1}$ and represents, through its minor, a set of sets of states that are predecessors of $F_{2}$ in $\mathcal{A}_{2}$ through a word $u$ generated by $\mathcal{A}_{1}$ from that state $q$, i.e., $\operatorname{pre}_{u}^{\mathcal{A}_{2}}(F_{2})$ with $u\in W^{\mathcal{A}_{1}}_{q,F_{1}}$. Since $\epsilon\in W_{q,F_{1}}^{\mathcal{A}_{1}}$ for all $q\in F_{1}$ and $\operatorname{pre}_{\epsilon}^{\mathcal{A}_{2}}(F_{2})=F_{2}$, the first iteration of $\operatorname{{\textsc{Kleene}}}$ gives the vector $\alpha(\vv{\bm{\epsilon}}^{{\!{\scriptstyle{F_{1}}}}})=\langle{{\psi^{F_{2}}_{\varnothing}(q\in^{\scaleto{?}{3.5pt}}F_{1})}}\rangle_{q\in Q_{1}}$. Let us also observe that by taking the minor of each vector component, we are considering smaller sets which still preserve the relation $\sqsubseteq$ since the following equivalences hold: $A\sqsubseteq B\Leftrightarrow\lfloor{A}\rfloor\sqsubseteq B\Leftrightarrow A\sqsubseteq\lfloor{B}\rfloor\Leftrightarrow\lfloor{A}\rfloor\sqsubseteq\lfloor{B}\rfloor$. Let $\langle{Y_{q}}\rangle_{q\in Q_{1}}$ be the output of $\operatorname{{\textsc{Kleene}}}(\sqsubseteq,\lambda\vv{\bm{X}}^{\sharp}\ldotp\alpha(\vv{\bm{\epsilon}}^{{\!{\scriptstyle{F_{1}}}}})\sqcup\operatorname{{Pre}}_{\mathcal{A}_{1}}^{\mathcal{A}_{2}}(\vv{\bm{X}}^{\sharp}),\vv{\bm{\varnothing}})$. Hence, we have that, for each component $q\in Q_{1}$, $Y_{q}=\lfloor{\{\operatorname{pre}_{u}^{\mathcal{A}_{2}}(F_{2})\mid u\in W_{q,F_{1}}^{\mathcal{A}_{1}}\}}\rfloor$ holds. Whenever the inclusion ${\mathcal{L}(\mathcal{A}_{1})}\subseteq{\mathcal{L}(\mathcal{A}_{2})}$ holds, all the sets of states in $Y_{q}$ for some initial state $q\in I_{1}$ are predecessors of $F_{2}$ in $\mathcal{A}_{2}$ through words in ${\mathcal{L}(\mathcal{A}_{2})}$, so that for each $q\in I_{1}$ and $S\in Y_{q}$, $S\cap I_{2}\neq\varnothing$ must hold. As a result, the following state-based algorithm $\mathtt{FAIncS}$ ($\mathtt{S}$ stands for state) decides the inclusion ${\mathcal{L}(\mathcal{A}_{1})}\subseteq{\mathcal{L}(\mathcal{A}_{2})}$ by computing on the abstract domain of antichains $\langle{\operatorname{{AC}}_{\langle{\wp(Q_{2}),\subseteq}\rangle},\sqsubseteq}\rangle$.

Theorem 6.3.

The algorithm $\mathtt{FAIncS}$ decides the inclusion problem ${\mathcal{L}(\mathcal{A}_{1})}\subseteq{\mathcal{L}(\mathcal{A}_{2})}$.

Proof.

We show that all the hypotheses (i)-(v) of Theorem 6.1 are satisfied for the abstract domain $\langle{D,\leq_{D}}\rangle=\langle{\operatorname{{AC}}_{\langle{\wp(Q_{2}),\subseteq}\rangle},\sqsubseteq}\rangle$ as defined by the GC of Lemma 6.2.

1. (i)

Since $\rho_{\leq^{l}_{\mathcal{A}_{2}}}(X)=\gamma(\alpha(X))$, it follows from Lemmata 5.2 and 5.8 that $L_{2}\in\gamma(D)$. Moreover, by Lemma 5.2 (b) with $\rho_{\leq^{l}_{\mathcal{A}_{2}}}=\gamma\alpha$, we have that for all $a\in\Sigma$, $X\in\wp(\Sigma^{*})$, $\gamma(\alpha(aX))=\gamma(\alpha(a\gamma(\alpha(X))))$. 2. (ii)

$\langle{\operatorname{{AC}}_{\langle{\wp(Q_{2}),\subseteq}\rangle},\sqsubseteq,\sqcup,\varnothing}\rangle$ is an effective domain because $Q_{2}$ is finite. 3. (iii)

By Lemma 6.2 (c) we have that $\alpha(\operatorname{{Pre}}_{\mathcal{A}_{1}}(\gamma(\vv{\bm{X}}^{\sharp})))={\operatorname{{Pre}}}_{\mathcal{A}_{1}}^{\mathcal{A}_{2}}(\vv{\bm{X}}^{\sharp})$ for all $\vv{\bm{X}}^{\sharp}\in\alpha(\wp(\Sigma^{*}))^{|Q_{1}|}$, and ${\operatorname{{Pre}}}_{\mathcal{A}_{1}}^{\mathcal{A}_{2}}$ is computable. 4. (iv)

$\alpha(\{\epsilon\})=\{F_{2}\}$ and $\alpha({\varnothing})=\varnothing$, hence $\alpha(\vv{\bm{\epsilon}}^{{\!{\scriptstyle{F_{1}}}}})$ is trivial to compute. 5. (v)

Since $\alpha(\vv{\bm{L_{2}}}^{{\!{\scriptstyle{I_{1}}}}})=\langle{\alpha({\psi^{L_{2}}_{\Sigma^{*}}(q\in^{\scaleto{?}{3.5pt}}I_{1})})}\rangle_{q\in Q_{1}}$, for all $\vv{\bm{Y}}\in\alpha(\wp(\Sigma^{*}))^{|Q_{1}|}$ the relation $\langle{Y_{q}}\rangle_{q\in Q_{1}}\sqsubseteq\alpha(\vv{\bm{L_{2}}}^{{\!{\scriptstyle{I_{1}}}}})$ trivially holds for all components $q\notin I_{1}$, since $\alpha(\Sigma^{*})$ is the greatest antichain. For the components $q\in I_{1}$, it suffices to show that $Y_{q}\sqsubseteq\alpha(L_{2})\Leftrightarrow\forall S\in Y_{q},\;S\cap I_{2}\neq\varnothing$, which is the check performed by lines 2-5 of algorithm $\mathtt{FAIncS}$:

[TABLE]

Thus, by Theorem 6.1, the algorithm $\mathtt{FAIncS}$ solves the inclusion problem ${\mathcal{L}(\mathcal{A}_{1})}\subseteq{\mathcal{L}(\mathcal{A}_{2})}$. ∎

6.3. Relationship to the Antichain Algorithm

De Wulf et al. (2006) introduced two so-called antichain algorithms, called forward and backward, for deciding the universality of the language accepted by a FA, i.e., whether the language is $\Sigma^{*}$ or not. Then, they extended the backward algorithm in order to decide inclusion of languages accepted by FAs. In what follows we show that the above algorithm $\mathtt{FAIncS}$ is equivalent to the corresponding extension of the forward antichain algorithm and, therefore, dual to the backward antichain algorithm for language inclusion put forward by De Wulf et al. ([)Theorem 6]DBLP:conf/cav/WulfDHR06. In order to do this, we first define the poset of antichains in which the forward antichain algorithm computes its fixpoint. Then, we give a formal definition of the forward antichain algorithm for deciding language inclusion and show that this algorithm coincides with $\mathtt{FAIncS}$ when applied to the reverse automata. Since language inclusion between the languages generated by two FAs holds iff inclusion holds between the languages generated by their reverse FAs, this entails that our algorithm $\mathtt{FAIncS}$ is equivalent to the forward antichain algorithm.

Consider a language inclusion problem ${\mathcal{L}(\mathcal{A}_{1})}\subseteq{\mathcal{L}(\mathcal{A}_{2})}$ where $\mathcal{A}_{1}=\langle{Q_{1},\delta_{1},I_{1},F_{1},\Sigma}\rangle$ and $\mathcal{A}_{2}=\langle{Q_{2},\delta_{2},I_{2},F_{2},\Sigma}\rangle$. Let us consider the following poset of antichains $\langle{\operatorname{{AC}}_{\langle{\wp(Q_{2}),\subseteq}\rangle},\operatorname{{\widetilde{\sqsubseteq}}}}\rangle$ where

[TABLE]

and notice that $\operatorname{{\widetilde{\sqsubseteq}}}$ coincides with the reverse $\sqsubseteq^{-1}$ of the relation defined by (1). As observed by De Wulf et al. ([)Lemma 1]DBLP:conf/cav/WulfDHR06, it turns out that $\langle{\operatorname{{AC}}_{\langle{\wp(Q_{2}),\subseteq}\rangle},\operatorname{{\widetilde{\sqsubseteq}}},\operatorname{{\widetilde{\sqcup}}},\operatorname{{\widetilde{\sqcap}}},\{\varnothing\},\varnothing}\rangle$ is a finite lattice, where $\operatorname{{\widetilde{\sqcup}}}$ and $\operatorname{{\widetilde{\sqcap}}}$ denote, resp., lub and glb, and $\{\varnothing\}$ and $\varnothing$ are, resp., the least and greatest elements. This lattice $\langle{\operatorname{{AC}}_{\langle{\wp(Q_{2}),\subseteq}\rangle},\operatorname{{\widetilde{\sqsubseteq}}}}\rangle$ is the domain in which the forward antichain algorithm computes on for deciding language universality (De Wulf et al., 2006, Theorem 3). The following result extends this forward algorithm in order to decide language inclusion.

Theorem 6.4 (**(**De

Wulf et al., 2006, Theorems 3 and 6)).

Let

[TABLE]

where $\operatorname{{Post}}_{\mathcal{A}_{1}}^{\mathcal{A}_{2}}(\langle{X_{q}}\rangle_{q\in Q_{1}})\triangleq\langle\lfloor{\{\operatorname{post}_{a}^{\mathcal{A}_{2}}(S)\in\wp(Q_{2})\mid\exists a\in\Sigma,q^{\prime}\in Q_{1},\,q\in\delta_{1}(q^{\prime},a)\wedge S\in X_{q^{\prime}}\}}\rfloor\rangle_{q\in Q_{1}}$. Then, ${\mathcal{L}(\mathcal{A}_{1})}\nsubseteq{\mathcal{L}(\mathcal{A}_{2})}$ if and only if there exists $q\in F_{1}$ such that $\vv{\bm{\mathcal{FP}}}_{q}\,\operatorname{{\widetilde{\sqsubseteq}}}\,\{F_{2}^{c}\}$.

Proof.

Let us first introduce some notation to describe the forward antichain algorithm by De Wulf et. al (2006) which decides ${\mathcal{L}(\mathcal{A}_{1})}\subseteq{\mathcal{L}(\mathcal{A}_{2})}$. Let us consider the poset $\langle{Q_{1}\times\wp(Q_{2}),\subseteq_{\times}}\rangle$ where (q_{1},S_{1})\subseteq_{\times}(q_{2},S_{2})\stackrel{{\scriptstyle{\mbox{\tiny\triangle}}}}{{\Leftrightarrow}}q_{1}=q_{2}\wedge S_{1}\subseteq S_{2}. Then, let $\langle{\operatorname{{AC}}_{\langle{Q_{1}\times\wp(Q_{2}),\subseteq_{\times}}\rangle},\operatorname{{\widetilde{\sqsubseteq}}}_{\times},\operatorname{{\widetilde{\sqcup}}}_{\times},\operatorname{{\widetilde{\sqcap}}}_{\times}}\rangle$ be the lattice of antichains over $\langle{Q_{1}\times\wp(Q_{2}),\subseteq_{\times}}\rangle$ where:

[TABLE]

Also, let $\operatorname{{Post}}:\operatorname{{AC}}_{\langle{Q_{1}\times\wp(Q_{2}),\subseteq_{\times}}\rangle}\rightarrow\operatorname{{AC}}_{\langle{Q_{1}\times\wp(Q_{2}),\subseteq_{\times}}\rangle}$ be defined as follows:

[TABLE]

Then, the dual of the backward antichain algorithm in (De Wulf et al., 2006, Theorem 6) states that ${\mathcal{L}(\mathcal{A}_{1})}\nsubseteq{\mathcal{L}(\mathcal{A}_{2})}$ iff there exists $q\in F_{1}$ such that $\mathcal{FP}\mathrel{\operatorname{{\widetilde{\sqsubseteq}}}_{\times}}\{(q,F_{2}^{c})\}$ where

[TABLE]

We observe that for some $X\in\operatorname{{AC}}_{\langle{Q_{1}\times\wp(Q_{2}),\subseteq_{\times}}\rangle}$, a pair $(q,S)\in Q_{1}\times\wp(Q_{2})$ such that $(q,S)\in X$ is used by (De Wulf et al., 2006, Theorem 6) simply as a way to associate states $q$ of $\mathcal{A}_{1}$ with sets $S$ of states of $\mathcal{A}_{2}$. In fact, an antichain $X\in\operatorname{{AC}}_{\langle{Q_{1}\times\wp(Q_{2}),\subseteq_{\times}}\rangle}$ can be equivalently formalized by a vector $\langle{\{S\in\wp(Q_{2})\mid(q,S)\in X\}}\rangle_{q\in Q_{1}}\in(\operatorname{{AC}}_{\langle{\wp(Q_{2}),\subseteq}\rangle})^{|Q_{1}|}$ whose components are indexed by states $q\in Q_{1}$ and are antichains of set of states in $\operatorname{{AC}}_{\langle{\wp(Q_{2}),\subseteq}\rangle}$. Correspondingly, we consider the lattice $\langle{\operatorname{{AC}}_{\langle{\wp(Q_{2}),\subseteq}\rangle},\operatorname{{\widetilde{\sqsubseteq}}}}\rangle$, where for all $X,Y\in\operatorname{{AC}}_{\langle{\wp(Q_{2}),\subseteq}\rangle}$:

[TABLE]

Then, these definitions allow us to replace $\operatorname{{Post}}$ by an equivalent function

[TABLE]

that transforms vectors of antichains as follows:

[TABLE]

In turn, the above $\mathcal{FP}\in\operatorname{{AC}}_{\langle{Q_{1}\times\wp(Q_{2}),\subseteq_{\times}}\rangle}$ is replaced by the following equivalent vector:

[TABLE]

Finally, the condition $\exists q\in F_{1},\mathcal{FP}\mathrel{\operatorname{{\widetilde{\sqsubseteq}}}_{\times}}\{(q,F_{2}^{c})\}$ is equivalent to $\exists q\in F_{1},\vv{\bm{\mathcal{FP}}}_{q}\mathrel{\operatorname{{\widetilde{\sqsubseteq}}}}\{F_{2}^{c}\}$. ∎

Let us recall that $\mathcal{A}^{R}$ denotes the reverse automaton of $\mathcal{A}$, where arrows are flipped and the initial/final states become final/initial. Note that language inclusion can be decided by considering the reverse automata since ${\mathcal{L}(\mathcal{A}_{1})}\subseteq{\mathcal{L}(\mathcal{A}_{2})}\Leftrightarrow{\mathcal{L}(\mathcal{A}_{1}^{R})}\subseteq{\mathcal{L}(\mathcal{A}_{2}^{R})}$ holds. Furthermore, let us observe that $\operatorname{{Post}}_{\mathcal{A}_{1}}^{\mathcal{A}_{2}}=\operatorname{{Pre}}_{\mathcal{A}_{1}^{R}}^{\mathcal{A}_{2}^{R}}$. We therefore obtain the following consequence of Theorem 6.4.

Corollary 6.5.

Let

[TABLE]

Then, ${\mathcal{L}(\mathcal{A}_{1})}\nsubseteq{\mathcal{L}(\mathcal{A}_{2})}$ iff $\exists q\in I_{1},\vv{\bm{\mathcal{FP}}}_{q}\,\operatorname{{\widetilde{\sqsubseteq}}}\,\{I_{2}^{c}\}$.

Since $\operatorname{{\widetilde{\sqsubseteq}}}=\mathord{\sqsubseteq^{-1}}$, we have that $\operatorname{{\widetilde{\sqcap}}}=\sqcup$, $\operatorname{{\widetilde{\sqcup}}}=\sqcap$ and the greatest element $\varnothing$ for $\operatorname{{\widetilde{\sqsubseteq}}}$ is the least element for $\mathord{\sqsubseteq}$. Moreover, by (20), $\alpha(\vv{\bm{\epsilon}}^{{\!{\scriptstyle{F_{1}}}}})=\langle{{\psi^{\{F_{2}\}}_{\varnothing}(q\in^{\scaleto{?}{3.5pt}}F_{1})}}\rangle_{q\in Q_{1}}$. Therefore, we can rewrite the vector $\vv{\bm{\mathcal{FP}}}$ of Corollary 6.5 as

[TABLE]

which is precisely the least fixpoint in $\langle{(\operatorname{{AC}}_{\langle{\wp(Q_{2}),\subseteq}\rangle})^{|Q_{1}|},\sqsubseteq}\rangle$ of $\operatorname{{Pre}}_{\mathcal{A}_{1}}^{\mathcal{A}_{2}}$ above $\alpha(\vv{\bm{\epsilon}}^{{\!{\scriptstyle{F_{1}}}}})$. Hence, it turns out that the Kleene iterates of the least fixpoint computation that converge to $\vv{\bm{\mathcal{FP}}}$ exactly coincide with the iterates computed by the $\operatorname{{\textsc{Kleene}}}$ procedure of the state-based algorithm $\mathtt{FAIncS}$. In particular, if $\vv{\bm{Y}}$ is the output vector of $\operatorname{{\textsc{Kleene}}}(\sqsubseteq,\lambda\vv{\bm{X}}\ldotp\alpha(\vv{\bm{\epsilon}}^{{\!{\scriptstyle{F_{1}}}}})\sqcup\operatorname{{Pre}}_{\mathcal{A}_{1}}^{\mathcal{A}_{2}}(\vv{\bm{X}}),\vv{\bm{\varnothing}})$ at line 1 of $\mathtt{FAIncS}$ then $\vv{\bm{Y}}=\vv{\bm{\mathcal{FP}}}$. Furthermore, $\exists q\in I_{1},\vv{\bm{\mathcal{FP}}}_{q}\>\operatorname{{\widetilde{\sqsubseteq}}}\>\{I_{2}^{c}\}\Leftrightarrow\exists q\in I_{1},\exists S\in\vv{\bm{\mathcal{FP}}}_{q},\;S\cap I_{2}=\varnothing$. Summing up, the $\sqsubseteq$-lfp algorithm $\mathtt{FAIncS}$ exactly coincides with the $\operatorname{{\widetilde{\sqsubseteq}}}$-gfp antichain algorithm as given by Corollary 6.5.

We can easily derive an antichain algorithm which is perfectly equivalent to $\mathtt{FAIncS}$ by considering the antichain lattice $\langle{\operatorname{{AC}}_{\langle{\wp(Q_{2}),\supseteq}\rangle},\sqsubseteq}\rangle$ for the dual lattice $\langle{\wp(Q_{2}),\supseteq}\rangle$ and by replacing the functions $\alpha$, $\gamma$ and $\operatorname{{Pre}}_{\mathcal{A}_{1}}^{\mathcal{A}_{2}}$ of Lemma 6.2, resp., with the following dual versions:

[TABLE]

where $\operatorname{{cpre}}_{u}^{\mathcal{A}_{2}}(S)\triangleq(\operatorname{pre}_{u}^{\mathcal{A}_{2}}(S^{c}))^{c}$ for $u\in\Sigma^{*}$. When using these functions, the corresponding algorithm computes on the abstract domain $\langle{\operatorname{{AC}}_{\langle{\wp(Q_{2}),\supseteq}\rangle},\sqsubseteq}\rangle$ and it turns out that ${\mathcal{L}(\mathcal{A}_{1})}\subseteq{\mathcal{L}(\mathcal{A}_{2})}$ iff $\operatorname{{\textsc{Kleene}}}(\sqsubseteq,\lambda\vv{\bm{X}}^{\sharp}\ldotp\alpha^{c}(\vv{\bm{\epsilon}}^{{\!{\scriptstyle{F_{1}}}}})\sqcup\operatorname{{CPre}}_{\mathcal{A}_{1}}^{\mathcal{A}_{2}}(\vv{\bm{X}}^{\sharp}),\vv{\bm{\varnothing}})\sqsubseteq\alpha^{c}(\vv{\bm{L_{2}}}^{{\!{\scriptstyle{I_{1}}}}})$. This language inclusion algorithm coincides with the backward antichain algorithm defined by De Wulf et al. ([)Theorem 6]DBLP:conf/cav/WulfDHR06 since both compute on the same lattice, $\lfloor{X}\rfloor$ corresponds to the maximal (w.r.t. set inclusion) elements of $X$, $\alpha^{c}(\{\epsilon\})=\{F_{2}^{c}\}$ and for all $X\in\alpha^{c}(\wp(\Sigma^{*}))$, we have that $X\sqsubseteq\alpha^{c}(L_{2})\Leftrightarrow\forall S\in X,\;I_{2}\nsubseteq S$.

We have thus shown that the two forward/backward antichain algorithms introduced by De Wulf et al. (2006) can be systematically derived by instantiating our framework. The original antichain algorithms were later improved by Abdulla et al. (2010) and, subsequently, by Bonchi and Pous (2013). Among their improvements, they showed how to exploit a precomputed binary relation between pairs of states of the input automata such that language inclusion holds for all the pairs in the relation. When that binary relation is a simulation relation, our framework allows to partially match their results by using the simulation-based quasiorder $\preceq^{r}_{\mathcal{A}}$ defined in Section 5.3.2. However, this relation $\preceq^{r}_{\mathcal{A}}$ does not consider pairs of states $Q_{2}\times Q_{2}$ whereas the aforementioned algorithms do.

7. Inclusion for Context Free Languages

A context-free grammar (CFG) is a tuple $\mathcal{G}=\langle{\mathcal{V},\Sigma,P}\rangle$ where $\mathcal{V}=\{X_{0},\ldots,X_{n}\}$ is a finite set of variables including a start symbol $X_{0}$, $\Sigma$ is a finite alphabet of terminals and $P$ is a finite set of productions $X_{i}\rightarrow\beta$ where $\beta\in(\mathcal{V}\cup\Sigma)^{*}$. We assume, for simplicity and without loss of generality, that CFGs are in Chomsky Normal Form (CNF), that is, every production $X_{i}\rightarrow\beta\in P$ is such that $\beta\in(\mathcal{V}\times\mathcal{V})\cup\Sigma\cup\{\epsilon\}$ and if $\beta=\epsilon$ then $i=0$ (Chomsky, 1959). We also assume that for all $X_{i}\in\mathcal{V}$ there exists a production $X_{i}\rightarrow\beta\in P$, otherwise $X_{i}$ can be safely removed from $\mathcal{V}$. Given two strings $w,w^{\prime}\in(\mathcal{V}\cup\Sigma)^{*}$ we write $w\rightarrow w^{\prime}$ iff there exists $u,v\in(\mathcal{V}\cup\Sigma)^{*}$ and $X\to\beta\in P$ such that $w=uXv$ and $w^{\prime}=u\beta v$. We denote by $\rightarrow^{*}$ the reflexive-transitive closure of $\rightarrow$. The language generated by a $\mathcal{G}$ is ${\mathcal{L}(\mathcal{G})}\triangleq\{w\in\Sigma^{*}\mid X_{0}\rightarrow^{*}w\}$.

7.1. Extending the Framework to CFGs

Similarly to the case of automata, a CFG $\mathcal{G}=(\mathcal{V},\Sigma,P)$ in CNF induces the following set of equations:

[TABLE]

Given a subset of variables $S\subseteq\mathcal{V}$ of a grammar, the set of words generated from some variable in $S$ is defined as

[TABLE]

When $S=\{X\}$ we slightly abuse the notation and write $W_{X}^{\mathcal{G}}$. Also, we drop the superscript $\mathcal{G}$ when the grammar is clear from the context. The language generated by $\mathcal{G}$ is therefore ${\mathcal{L}(\mathcal{G})}=W^{\mathcal{G}}_{X_{0}}$.

We define the vector $\vv{\bm{b}}\in\wp(\Sigma^{*})^{|\mathcal{V}|}$ and the function $\operatorname{{Fn}}_{\mathcal{G}}:\wp(\Sigma^{*})^{|\mathcal{V}|}\to\wp(\Sigma^{*})^{|\mathcal{V}|}$, which are used to formalize the fixpoint equations in $\operatorname{{Eqn}}(\mathcal{G})$, as follows:

[TABLE]

Notice that $\lambda\vv{\bm{X}}\ldotp\vv{\bm{b}}\mathrel{\cup}\operatorname{{Fn}}_{\mathcal{G}}(\vv{\bm{X}})$ is a well-defined monotonic function in $\wp(\Sigma^{*})^{|\mathcal{V}|}\rightarrow\wp(\Sigma^{*})^{|\mathcal{V}|}$, which therefore has the least fixpoint $\langle{Y_{i}}\rangle_{i\in[0,n]}=\operatorname{lfp}(\lambda\vv{\bm{X}}\ldotp\vv{\bm{b}}\cup\operatorname{{Fn}}_{\mathcal{G}}(\vv{\bm{X}}))$. It is known (Ginsburg and Rice, 1962) that the language ${\mathcal{L}(\mathcal{G})}$ accepted by $\mathcal{G}$ is such that ${\mathcal{L}(\mathcal{G})}=Y_{0}$.

Example 7.1.

Consider the CFG $\mathcal{G}=\langle{\{X_{0},X_{1}\},\{a,b\},\{X_{0}\rightarrow X_{0}X_{1}\mid X_{1}X_{0}\mid b,\>X_{1}\rightarrow a\}}\rangle$ in CNF. The corresponding equation system is

[TABLE]

so that

[TABLE]

Moreover, we have that $\vv{\bm{b}}\in\wp(\Sigma^{*})^{2}$ and $\operatorname{{Fn}}_{\mathcal{G}}:\wp(\Sigma^{*})^{2}\rightarrow\wp(\Sigma^{*})^{2}$ are given by

[TABLE]

It turns out that

[TABLE]

where $\vv{\bm{L_{2}}}^{{\!{\scriptstyle{X_{0}}}}}\triangleq\langle{{\psi^{L_{2}}_{\Sigma^{*}}(i=^{\scaleto{?}{3.5pt}}0)}}\rangle_{i\in[0,n]}$.

Theorem 7.2.

Let $\mathcal{G}=(\mathcal{V},\Sigma,P)$ be a CFG in CNF. If $\rho\in\operatorname{uco}(\wp(\Sigma^{*}))$ is backward complete for both $\lambda X.Xa$ and $\lambda X.aX$, for all $a\in\Sigma$ then $\rho$ is backward complete for $\lambda\vv{\bm{X}}\ldotp\vv{\bm{b}}\cup\operatorname{{Fn}}_{\mathcal{G}}(\vv{\bm{X}})$.

Proof.

Let us first show that backward completeness for left and right concatenation can be extended from letter to words. We give the proof for left concatenation, the right case is symmetric. We prove that $\rho(wX)=\rho(w\rho(X))$ for every $w\in\Sigma^{*}$. We proceed by induction on $|w|\geq 0$. The base case $|w|=0$ iff $w=\epsilon$ is trivial because $\rho$ is idempotent. For the inductive case $|w|>0$ let $w=au$ for some $u\in\Sigma^{*}$ and $a\in\Sigma$, so that:

[TABLE]

Next we turn to the binary concatenation case, i.e., we prove that $\rho(YZ)=\rho(\rho(Y)\rho(Z))$ for all $Y,Z\in\wp(\Sigma^{*})$:

[TABLE]

Then, the proof follows the same lines of the proof of Theorem 4.3. Indeed, it follows from the definition of $\operatorname{{Fn}}_{\mathcal{G}}(\langle{X_{i}}\rangle_{i\in[0,n]})$ that:

[TABLE]

Hence, by a straightforward componentwise application on vectors in $\wp(\Sigma^{*})^{|\mathcal{V}|}$, we obtain that $\rho$ is backward complete for $\operatorname{{Fn}}_{\mathcal{G}}$. Finally, $\rho$ is backward complete for $\lambda\vv{\bm{X}}\ldotp(\vv{\bm{b}}\cup\operatorname{{Fn}}_{\mathcal{G}}(\vv{\bm{X}}))$, because:

[TABLE]

The following result, which is an adaptation of Theorem 4.7 to grammars, relies on Theorem 7.2 for designing an algorithm that solves the inclusion problem ${\mathcal{L}(\mathcal{G})}\subseteq L_{2}$ by exploiting a language abstraction $\rho$ that satisfies some requirements of backward completeness and computability.

Theorem 7.3.

Let $\mathcal{G}=\langle{\mathcal{V},\Sigma,P}\rangle$ be a CFG in CNF, $L_{2}\in\wp(\Sigma^{*})$ and $\rho\in\operatorname{uco}(\Sigma^{*})$. Assume that the following properties hold:

1. (i)

The closure $\rho$ is backward complete for both $\lambda X\in\wp(\Sigma^{*})\ldotp aX$ and $\lambda X\in\wp(\Sigma^{*})\ldotp Xa$ for all $a\in\Sigma$ and satisfies $\rho(L_{2})=L_{2}$. 2. (ii)

$\rho(\wp(\Sigma^{*}))$* does not contain infinite ascending chains.* 3. (iii)

If $X,Y\in\wp(\Sigma^{*})$ are finite sets of words then the inclusion $\rho(X)\subseteq^{\scaleto{?}{3.5pt}}\rho(Y)$ is decidable. 4. (iv)

If $Y\in\wp(\Sigma^{*})$ is a finite set of words then the inclusion $\rho(Y)\subseteq^{\scaleto{?}{3.5pt}}L_{2}$ is decidable.

Then,

$\langle{Y_{i}}\rangle_{i\in[0,n]}:=\operatorname{{\textsc{Kleene}}}(\operatorname{{Incl}}_{\rho},\lambda\vv{\bm{X}}\ldotp\vv{\bm{b}}\cup\operatorname{{Fn}}_{\mathcal{G}}(\vv{\bm{X}}),\vv{\bm{\varnothing}})$;**

return* $\operatorname{{Incl}}_{\rho}(\langle{Y_{i}}\rangle_{i\in[0,n]},\vv{\bm{L_{2}}}^{{\!{\scriptstyle{X_{0}}}}})$*;**

is a decision algorithm for ${\mathcal{L}(\mathcal{G})}\subseteq L_{2}$.

Proof.

Analogous to the proof of Theorem 4.7. ∎

7.2. Instantiating the Framework

Let us instantiate the general algorithmic framework provided by Theorem 7.3 to the class of closure operators induced by quasiorder relations on words. As a consequence of Lemma 5.2, we have the following characterization of $L$-consistent quasiorders.

Lemma 7.4.

Let $L\in\wp(\Sigma^{*})$ and $\mathord{\leqslant_{L}}$ be a quasiorder on $\Sigma^{*}$. Then, $\mathord{\leqslant_{L}}$ is a $L$-consistent quasiorder on $\Sigma^{*}$ if and only if

1. (a)

$\rho_{\leqslant_{L}}(L)=L$, and 2. (b)

$\rho_{\leqslant_{L}}$* is backward complete for for $\lambda X\ldotp aX$ and $\lambda X\ldotp Xa$ for all $a\in\Sigma$.*

Analogously to Section 5.1 for automata, Theorem 7.3 induces an algorithm for deciding the language inclusion ${\mathcal{L}(\mathcal{G})}\subseteq L_{2}$ for any CFG $\mathcal{G}$ and regular language $L_{2}$. More in general, given a language $L_{2}\in\wp(\Sigma^{*})$ whose membership problem is decidable and a decidable $L_{2}$-consistent wqo, the following algorithm $\mathtt{CFGIncW}$ ($\mathtt{CFG}$ $\mathtt{Inc}$lusion based on $\mathtt{W}$ords) decides ${\mathcal{L}(\mathcal{G})}\subseteq L_{2}$.

Theorem 7.5.

*Let $\mathcal{G}=\langle{Q,\delta,I,F,\Sigma}\rangle$ be a CFG and let $L_{2}\in\wp(\Sigma^{*})$ be a language such that:

(i) membership $u\in^{\scaleto{?}{3.5pt}}L_{2}$ is decidable;

(ii) there exists a decidable $L_{2}$-consistent wqo on $\Sigma^{*}$.

Then, algorithm $\mathtt{CFGIncW}$ decides the inclusion ${\mathcal{L}(\mathcal{G})}\subseteq L_{2}$.*

Proof.

The proof is analogous to the proof of Theorem 5.3: it applies Theorem 7.3 and Lemma 7.4 in the same way of the proof of Theorem 5.3 where the role of a left $L_{2}$-consistent wqo on $\Sigma^{*}$ is replaced by a $L_{2}$-consistent wqo. ∎

7.2.1. Myhill and State-based Quasiorders

In the following, we will consider two quasiorders on $\Sigma^{*}$ and we will show that they fulfill the requirements of Theorem 7.5, so that they correspondingly yield algorithms for deciding the language inclusion ${\mathcal{L}(\mathcal{G})}\subseteq L_{2}$ for every CFG $\mathcal{G}$ and regular language $L_{2}$.

The context for a language $L\in\wp(\Sigma^{*})$ w.r.t. a given word $w\in\Sigma^{*}$ is defined as usual:

[TABLE]

Correspondingly, let us define the following quasiorder relation on $\mathord{\leqq_{L}}\subseteq\Sigma^{*}\times\Sigma^{*}$:

[TABLE]

De Luca and Varricchio (1994, Section 2) call $\leqq_{L}$ the Myhill quasiorder relative to $L$. The following result is the analogue of Lemma 5.6 for the Nerode quasiorder: it shows that the Myhill quasiorder is the weakest $L_{2}$-consistent quasiorder for which the above algorithm $\mathtt{CFGIncW}$ can be instantiated to decide a language inclusion ${\mathcal{L}(\mathcal{G})}\subseteq L_{2}$.

Lemma 7.6.

Let $L\in\wp(\Sigma^{*})$.

1. (a)

$\mathord{\leqq_{L}}$* is a $L$-consistent quasiorder. If $L$ is regular then, additionally, $\mathord{\leqq_{L}}$ is a decidable wqo.* 2. (b)

If $\mathord{\leqslant}$ is a $L$-consistent quasiorder on $\Sigma^{*}$ then $\rho_{\leqq_{L}}(\wp(\Sigma^{*}))\subseteq\rho_{\leqslant}(\wp(\Sigma^{*}))$.

Proof.

The proof follows the same lines of the proof of Lemma 5.6.

Let us consider (a). De Luca and Varricchio (1994, Section 2) observe that $\mathord{\leqq_{L}}$ is monotonic. Moreover, if $L$ is regular then $\mathord{\leqq_{L}}$ is a wqo (de Luca and Varricchio, 1994, Proposition 2.3). Let us observe that given $u\in L$ and $v\notin L$ we have that $(\epsilon,\epsilon)\in\operatorname{{ctx}}_{L}(u)$ while $(\epsilon,\epsilon)\notin\operatorname{{ctx}}_{L}(v)$. Hence, $\mathord{\leqq_{L}}$ is a $L$-consistent quasiorder. Finally, if $L$ is regular then $\leqq_{L}$ is clearly decidable.

Let us consider (b). By the characterization of $L$-consistent quasiorders of Lemma 7.4, De Luca and Varricchio (1994, Section 2, point 4) observe that $\mathord{\leqq_{L}}$ is maximum in the set of all $L$-consistent quasiorders, i.e. every $L$-consistent quasiorder $\leqslant$ is such that $x\leqslant y\Rightarrow x\leqq_{L}y$. As a consequence, $\rho_{\leqslant}(X)\subseteq\rho_{\leqq_{L}}(X)$ holds for all $X\in\wp(\Sigma^{*})$, namely, $\rho_{\leqq_{L}}(\wp(\Sigma^{*}))\subseteq\rho_{\leqslant}(\wp(\Sigma^{*}))$. ∎

Example 7.7.

Let us illustrate the use of the Myhill quasiorder $\leqq_{{\mathcal{L}(\mathcal{A})}}$ in Algorithm $\mathtt{CFGIncW}$ for solving the language inclusion ${\mathcal{L}(\mathcal{G})}\subseteq{\mathcal{L}(\mathcal{A})}$, where $\mathcal{G}$ is the CFG in Example 7.1 and $\mathcal{A}$ is the FA depicted in Figure 4. The equations for $\mathcal{G}$ are as follows:

[TABLE]

We write $\{(S,T)\}\cup\{(X,Y)\}$ to compactly denote a set $\{(u,v)\mid(u,v)\in S\times T\cup X\times Y\}$. Then, we have the following contexts (among others) for $L={\mathcal{L}(\mathcal{A})}=(b+ab^{*}a)(a+b)^{*}$:

[TABLE]

Notice that $a\leqq_{L}ba$ and $\operatorname{{ctx}}_{L}(ab)=\operatorname{{ctx}}_{L}(a)$ and $\operatorname{{ctx}}_{L}(ba)=\operatorname{{ctx}}_{L}(baa)=\operatorname{{ctx}}_{L}(aab)=\operatorname{{ctx}}_{L}(aba)$. Next, we show the computation of the Kleene iterates according to Algorithm $\mathtt{CFGIncW}$ using $\sqsubseteq_{\leqq_{L}}$ by recalling from Example 7.1 that $\vv{\bm{b}}=\langle{\{b\},\{a\}}\rangle$ and $\operatorname{{Fn}}_{\mathcal{G}}(\langle{X_{0},X_{1}}\rangle)=\langle{X_{0}X_{1}\cup X_{1}X_{0},\varnothing}\rangle$:

[TABLE]

It turns out that $\langle{\{baa,aba,ba,aab,ab,b\},\{a\}}\rangle\sqsubseteq_{\leqq_{L}}\langle{\{ba,ab,b\},\{a\}}\rangle$ because $a\leqq_{L}baa$, $a\leqq_{L}aba$, $a\leqq_{L}aab$ hold, so that $\operatorname{{\textsc{Kleene}}}(\sqsubseteq_{\leqq_{L}},\lambda\vv{\bm{X}}\ldotp\vv{\bm{b}}\cup\operatorname{{Fn}}_{\mathcal{G}}(\vv{\bm{X}}),\vv{\bm{\varnothing}})$ stops with $\vv{\bm{Y}}^{(3)}$ and outputs $\vv{\bm{Y}}=\langle{\{ba,ab,b\},\{a\}}\rangle$. Since $ab\in\vv{\bm{Y}}_{0}$ but $ab\notin{\mathcal{L}(\mathcal{A})}$, Algorithm $\mathtt{CFGIncW}$ correctly concludes that ${\mathcal{L}(\mathcal{G})}\subseteq{\mathcal{L}(\mathcal{A})}$ does not hold. $\Diamond$

Similarly to Section 5.3, next we consider a state-based quasiorder that can be used with Algorithm $\mathtt{CFGIncW}$. First, given a FA $\mathcal{A}=\langle{Q,\delta,I,F,\Sigma}\rangle$ we define the state-based equivalent of the context of a word $w\in\Sigma^{*}$ as follows:

[TABLE]

Then, the quasiorder $\leq_{\mathcal{A}}$ on $\Sigma^{*}$ induced by $\mathcal{A}$ is defined as follows: for all $u,v\in\Sigma^{*}$,

[TABLE]

The following result is the analogue of Lemma 5.8 and shows that $\mathord{\leq_{\mathcal{A}}}$ is a ${\mathcal{L}(\mathcal{A})}$-consistent well-quasiorder and, therefore, it can be used with Algorithm $\mathtt{CFGIncW}$ to solve a language inclusion ${\mathcal{L}(\mathcal{G})}\subseteq{\mathcal{L}(\mathcal{A})}$.

Lemma 7.8.

The relation $\mathord{\leq_{\mathcal{A}}}$ is a decidable ${\mathcal{L}(\mathcal{A})}$-consistent wqo.

Proof.

For every $u\in\Sigma^{*}$, $\operatorname{{ctx}}_{\mathcal{A}}(u)$ is a finite and computable set, so that $\mathord{\leq_{\mathcal{A}}}$ is a decidable wqo. Next, we show that $\mathord{\leq_{\mathcal{A}}}$ is ${\mathcal{L}(A)}$-consistent according to Definition 5.1 (a)-(b).

(a) By picking $u\in{\mathcal{L}(\mathcal{A})}$ and $v\notin{\mathcal{L}(\mathcal{A})}$ we have that $\operatorname{{ctx}}_{\mathcal{A}}(u)$ contains a pair $(q_{i},q_{f})$ with $q_{i}\in I$ and $q_{f}\in F$ while $\operatorname{{ctx}}_{\mathcal{A}}(v)$ does not, hence $u\nleq_{\mathcal{A}}v$.

(b) Let us check that $\leq_{\mathcal{A}}$ is monotonic. Observe that $\operatorname{{ctx}}_{\mathcal{A}}:\langle{\Sigma^{*},\leq_{\mathcal{A}}}\rangle\rightarrow\langle{\wp(Q^{2}),\subseteq}\rangle$ is monotonic. Therefore, for all $x_{1},x_{2}\in\Sigma^{*}$ and $a,b\in\Sigma$,

[TABLE]

For the Myhill wqo $\leqq_{{\mathcal{L}(\mathcal{A})}}$, it turns out that for all $u,v\in\Sigma^{*}$,

[TABLE]

Therefore, $u\leq_{\mathcal{A}}v\Rightarrow u\leqq_{{\mathcal{L}(\mathcal{A})}}v$ and, consequently, $\rho_{\leqq_{{\mathcal{L}(\mathcal{A})}}}(\wp(\Sigma^{*}))\subseteq\rho_{\leq^{l}_{\mathcal{A}_{2}}}(\wp(\Sigma^{*}))$ holds.

Example 7.9.

Let us illustrate the use of the state-based quasiorder $\leq_{\mathcal{A}}$ to solve the language inclusion ${\mathcal{L}(\mathcal{G})}\subseteq{\mathcal{L}(\mathcal{A})}$ of Example 7.7. Here, among others, we have the following contexts:

[TABLE]

Moreover, $\operatorname{{ctx}}_{\mathcal{A}}(ba)=\operatorname{{ctx}}_{\mathcal{A}}(baa)=\operatorname{{ctx}}_{\mathcal{A}}(aab)=\operatorname{{ctx}}_{\mathcal{A}}(aba)$. Recall from Example 7.7 that for the Myhill quasiorder we have that $a\leqq_{{\mathcal{L}(\mathcal{A})}}ba$, while for the state-based quasiorder $a\nleq_{\mathcal{A}}ba$. The Kleene iterates computed by Algorithm $\mathtt{CFGIncW}$ when using $\sqsubseteq_{\mathord{\leq_{\mathcal{A}}}}$ are exactly the same of Example 7.7. Here, it turns out that $\mathtt{CFGIncW}$ outputs $\vv{\bm{Y}}^{(2)}=\langle{\{ba,ab,b\},\{a\}}\rangle$ because $\vv{\bm{Y}}^{(3)}=\langle{\{baa,aba,ba,aab,ab,b\},\{a\}}\rangle\sqsubseteq_{\leqq_{L}}\langle{\{ba,ab,b\},\{a\}}\rangle=\vv{\bm{Y}}^{(2)}$ holds: in fact, we have that $ba\leq_{\mathcal{A}}baa$, $ba\leq_{\mathcal{A}}aba$, $ba\leq_{\mathcal{A}}aab$ hold. Since $ab\in\vv{\bm{Y}}^{(2)}_{0}$ but $ab\notin{\mathcal{L}(\mathcal{A})}$, Algorithm $\mathtt{CFGIncW}$ derives that ${\mathcal{L}(\mathcal{G})}\not\subseteq{\mathcal{L}(\mathcal{A})}$. $\Diamond$

7.3. An Antichain Inclusion Algorithm for CFGs

We can easily formulate an equivalent of Theorem 6.1 for context-free languages, therefore defining an algorithm for solving ${\mathcal{L}(\mathcal{G})}\subseteq L_{2}$ by computing on an abstract domain as defined by a Galois connection.

Theorem 7.10.

Let $\mathcal{G}=\langle{\mathcal{V},\Sigma,P}\rangle$ be a CFG in CNF and let $L_{2}\in\wp(\Sigma^{*})$. Let $\langle{D,\leq_{D}}\rangle$ be a poset and $\langle{\wp(\Sigma^{*}),\subseteq}\rangle\galois{\alpha}{\gamma}\langle{D,\sqsubseteq}\rangle$ be a GC. Assume that the following properties hold:

1. (i)

$L_{2}\in\gamma(D)$* and for every $a\in\Sigma$, $X\in\wp(\Sigma^{*})$, $\gamma(\alpha(aX))=\gamma(\alpha(a\gamma(\alpha(X))))$ and $\gamma(\alpha(Xa))=\gamma(\alpha(\gamma(\alpha(X))a))$.* 2. (ii)

$(D,\leq_{D},\sqcup,\bot_{D})$* is an effective domain, meaning that: $(D,\leq_{D},\sqcup,\bot_{D})$ is an ACC join-semilattice with bottom $\bot_{D}$, every element of $D$ has a finite representation, the binary relation $\leq_{D}$ is decidable and the binary lub $\sqcup$ is computable.* 3. (iii)

There is an algorithm, say $\operatorname{{Fn}}^{\sharp}(\vv{\bm{X}}^{\sharp})$, which computes $\alpha\comp\operatorname{{Fn}}_{\mathcal{G}}\comp\gamma$. 4. (iv)

There is an algorithm, say $\operatorname{{\textit{b}}}^{\sharp}$, which computes $\alpha(\vv{\bm{b}})$. 5. (v)

There is an algorithm, say $\operatorname{{Incl^{\sharp}}}$, which decides $\vv{\bm{X}}^{\sharp}\leq_{D}\alpha(\vv{\bm{L_{2}}}^{{\!{\scriptstyle{X_{0}}}}})$, for all $\vv{\bm{X}}^{\sharp}\in\alpha(\wp(\Sigma^{*}))^{|\mathcal{V}|}$.

Then,

$\langle{Y_{i}^{\sharp}}\rangle_{i\in[0,n]}:=\operatorname{{\textsc{Kleene}}}(\leq_{D},\lambda\vv{\bm{X}}^{\sharp}\ldotp\operatorname{{\textit{b}}}^{\sharp}\sqcup\operatorname{{Fn}}^{\sharp}(\vv{\bm{X}}^{\sharp}),\vv{\bm{\bot_{D}}})$;**

return* $\operatorname{{Incl^{\sharp}}}(\langle{Y_{i}^{\sharp}}\rangle_{i\in[0,n]})$*;**

is a decision algorithm for ${\mathcal{L}(\mathcal{G})}\subseteq L_{2}$.

Proof.

Analogous to the proof of Theorem 6.1. ∎

Similarly to what is done in Section 6.1, in order to solve an inclusion problem $\mathcal{L}(\mathcal{G})\subseteq\mathcal{L}(\mathcal{A})$, where $\mathcal{A}=\langle{Q,\delta,I,F,\Sigma}\rangle$ is a FA, we leverage Theorem 7.10 to systematically design a “state-based” algorithm that computes Kleene iterates on the antichain poset $\langle{\operatorname{{AC}}_{\langle{\wp(Q\times Q),\subseteq}\rangle},\sqsubseteq}\rangle$ viewed as an abstraction of $\langle{\wp(\Sigma^{*}),\subseteq}\rangle$. Here, the abstraction and concretization maps $\alpha\colon\wp(\Sigma^{*})\rightarrow\operatorname{{AC}}_{\langle{\wp(Q\times Q),\subseteq}\rangle}$ and $\gamma\colon\operatorname{{AC}}_{\langle{\wp(Q\times Q),\subseteq}\rangle}\rightarrow\wp(\Sigma^{*})$ and the function ${\operatorname{{Fn}}}_{\mathcal{G}}^{\mathcal{A}}:(\operatorname{{AC}}_{\langle{\wp(Q\times Q),\subseteq}\rangle})^{|\mathcal{V}|}\rightarrow(\operatorname{{AC}}_{\langle{\wp(Q\times Q),\subseteq}\rangle})^{|\mathcal{V}|}$ are defined as follows:

[TABLE]

where $\lfloor{X}\rfloor$ is the unique minor set w.r.t. subset inclusion of some $X\subseteq\wp(Q\times Q)$ and $X\comp Y\triangleq\{(q,q^{\prime})\in Q\times Q\mid(q,q^{\prime\prime})\in X,\,(q^{\prime\prime},q^{\prime})\in Y\}$ denotes the standard composition of two relations $X,Y\subseteq Q\times Q$. By the analogue of Lemma 6.2 (the proof follows the same pattern and is therefore omitted), it turns out that:

1. (a)

$\langle{\wp(\Sigma^{*}),\subseteq}\rangle\galois{\alpha}{\gamma}\langle{\operatorname{{AC}}_{\langle{\wp(Q\times Q),\subseteq}\rangle},\sqsubseteq}\rangle$ is a GC, 2. (b)

$\gamma\comp\alpha=\rho_{\leqslant_{\mathcal{A}}}$, 3. (c)

$\operatorname{{Fn}}_{\mathcal{G}}^{\mathcal{A}}=\alpha\comp\operatorname{{Fn}}_{\mathcal{G}}\comp\gamma$.

Thus, the GC $\langle{\wp(\Sigma^{*}),\subseteq}\rangle\galois{\alpha}{\gamma}\langle{\operatorname{{AC}}_{\langle{\wp(Q\times Q),\subseteq}\rangle},\sqsubseteq}\rangle$ and the abstract function $\operatorname{{Fn}}_{\mathcal{G}}^{\mathcal{A}}$ satisfy the hypotheses (i)-(iv) of Theorem 7.10. Here, the inclusion check $\vv{\bm{X}}^{\sharp}\leq_{D}\alpha(\vv{\bm{\mathcal{L}(\mathcal{A})}}^{{\!{\scriptstyle{X_{0}}}}})$ boils down to verify that for the start component $Y_{0}$ of the output $\langle{Y_{i}}\rangle_{i\in[0,n]}$ of $\operatorname{{\textsc{Kleene}}}(\sqsubseteq,\lambda\vv{\bm{X}}^{\sharp}\ldotp\alpha(\vv{\bm{b}})\sqcup\operatorname{{Fn}}_{\mathcal{G}}^{\mathcal{A}}(\vv{\bm{X}}^{\sharp}),\vv{\bm{\varnothing}})$, for all $R\in Y_{0}$, $R$ does not contain a pair $(q_{i},q_{f})\in I\times F$. We therefore derive the following state-based algorithm $\mathtt{CFGIncS}$ ($\mathtt{S}$ stands for state) that decides an inclusion $L(\mathcal{G})\subseteq L(\mathcal{A})$ on the abstract domain of antichains $\operatorname{{AC}}_{\langle{\wp(Q\times Q),\subseteq}\rangle}$.

Theorem 7.11.

The algorithm $\mathtt{CFGIncS}$ decides the inclusion problem $L(\mathcal{G})\subseteq L(\mathcal{A})$.

Proof.

The proof follows the same pattern of the proof of Theorem 6.3. We just focus on the inclusion check at lines 2-4, which is slightly different from the check at lines 2-5 of Algorithm $\mathtt{FAIncS}$. Let $L_{2}=\mathcal{L}(\mathcal{A})$. Since $\alpha(\vv{\bm{L_{2}}}^{{\!{\scriptstyle{X_{0}}}}})=\langle{\alpha({\psi^{L_{2}}_{\Sigma^{*}}(i=^{\scaleto{?}{3.5pt}}0)})}\rangle_{i\in[0,n]}$, for all $\vv{\bm{Y}}\in\alpha(\wp(\Sigma^{*}))^{|\mathcal{V}|}$ the relation $\vv{\bm{Y}}\sqsubseteq\alpha(\vv{\bm{L_{2}}}^{{\!{\scriptstyle{X_{0}}}}})$ trivially holds for all components $Y_{i}$ with $i\neq 0$. For $Y_{0}$, it is enough to prove that $Y_{0}\sqsubseteq\alpha(L_{2})\Leftrightarrow\forall R\in Y_{q},\;R\cap(I\times F)\neq\varnothing$:

[TABLE]

Hence, Theorem 7.10 entails that Algorithm $\mathtt{CFGIncS}$ decides ${\mathcal{L}(\mathcal{G})}\subseteq{\mathcal{L}(\mathcal{A})}$. ∎

The resulting algorithm $\mathtt{CFGIncS}$ shares some features with two previous related works. On the one hand, it is related to the work of Hofmann and Chen (2014) which defines an abstract interpretation-based language inclusion decision procedure similar to ours. Even though Hofmann and Chen’s algorithm and ours both manipulate sets of pairs of states of an automaton, their abstraction is based on equivalence relations and not quasiorders. Since quasiorders are strictly more general than equivalences our framework can be instantiated to a larger class of abstractions, most importantly coarser ones. Finally, it is worth pointing out that Hofmann and Chen (2014) approach aims at including languages of finite and also infinite words.

A second related work is that of Holík and Meyer (2015) who define an antichain-based algorithm manipulating sets of pairs of states. However, they tackle the inclusion problem ${\mathcal{L}(\mathcal{G})}\subseteq{\mathcal{L}(\mathcal{A})}$, where $\mathcal{G}$ is a grammar and $\mathcal{A}$ and automaton, by rephrasing it as a data flow analysis problem over a relational domain. In this scenario, the solution of the problem requires the computation of a least fixpoint on the relational domain, followed by an inclusion check between sets of relations. Then, they use the “antichain principle” to improve the performance of the fixpoint computation and, finally, they move from manipulating relations to manipulating pairs of states. As a result, Holík and Meyer (2015) devise an antichain algorithm for checking the inclusion ${\mathcal{L}(\mathcal{G})}\subseteq{\mathcal{L}(\mathcal{A})}$.

By contrast to these two approaches, our design technique is direct and systematic, since the algorithm $\mathtt{CFGIncS}$ is derived from the known Myhill quasiorder. We believe that our approach reveals the relationship between the original antichain algorithm by De Wulf et al. (2006) for regular languages and the one by Holík and Meyer (2015) for context-free languages, which is the relation between our algorithms $\mathtt{FAIncS}$ and $\mathtt{CFGIncS}$. Specifically, we have shown that these two algorithms are conceptually identical and just differ in the well-quasiorder used to define the abstract domain where computations take place.

8. An Equivalent Greatest Fixpoint Algorithm

Let us assume that $g\colon C\rightarrow C$ is a monotonic function on a complete lattice $\langle{C,\leq,\vee,\wedge}\rangle$ which admits its unique right-adjoint $\widetilde{g}\colon C\rightarrow C$, i.e., $\forall c,c^{\prime}\in C,\,g(c)\leq c^{\prime}\Leftrightarrow c\leq\widetilde{g}(c^{\prime})$ holds. Then, Cousot (2000, Theorem 4) shows that the following equivalence holds: for all $c,c^{\prime}\in C$,

[TABLE]

This property has been used in (Cousot, 2000) to derive equivalent least/greatest fixpoint-based invariance proof methods for programs. In the following, we use (23) to derive an algorithm for deciding the inclusion ${\mathcal{L}(\mathcal{A}_{1})}\subseteq{\mathcal{L}(\mathcal{A}_{2})}$, which relies on the computation of a greatest fixpoint rather than a least fixpoint. This can be achieved by exploiting the following simple observation, which defines an adjunction between concatenation and quotients of sets of words.

Lemma 8.1.

For all $X,Y\in\wp(\Sigma^{*})$ and $w\in\Sigma^{*}$, $wY\subseteq Z\Leftrightarrow Y\subseteq w^{-1}Z$ and $Yw\subseteq Z\Leftrightarrow Y\subseteq Zw^{-1}$.

Proof.

By definition, for all $u\in\Sigma^{*}$, $u\in w^{-1}Z$ iff $wu\in Z$. Hence, $Y\subseteq w^{-1}Z\Leftrightarrow\forall u\in Y,\>wu\in Z\Leftrightarrow wY\subseteq Z$. Symmetrically, $Yw\subseteq Z$ $\Leftrightarrow$ $Y\subseteq Zw^{-1}$ holds. ∎

Given a FA $\mathcal{A}=\langle{Q,\delta,I,F,\Sigma}\rangle$, we define the function $\widetilde{\operatorname{{Pre}}}_{\mathcal{A}}:\wp(\Sigma^{*})^{|Q|}\rightarrow\wp(\Sigma^{*})^{|Q|}$ on $Q$-indexed vectors of sets of words as follows:

[TABLE]

where, as usual, $\bigcap\varnothing=\Sigma^{*}$. It turns out that $\widetilde{\operatorname{{Pre}}}_{\mathcal{A}}$ is the usual weakest liberal precondition which is right-adjoint of $\operatorname{{Pre}}_{\mathcal{A}}$.

Lemma 8.2.

For all $\vv{\bm{X}},\vv{\bm{Y}}\in\wp(\Sigma^{*})^{|Q|}$, $\operatorname{{Pre}}_{\mathcal{A}}(\vv{\bm{X}})\subseteq\vv{\bm{Y}}\Leftrightarrow\vv{\bm{X}}\subseteq\widetilde{\operatorname{{Pre}}}_{\mathcal{A}}(\vv{\bm{Y}})$.

Proof.

[TABLE]

∎

Hence, from equivalences (9) and (23) we obtain that for all FAs $\mathcal{A}_{1}$ and $L_{2}\in\wp(\Sigma^{*})$:

[TABLE]

The following algorithm $\mathtt{FAIncGfp}$ decides the inclusion ${\mathcal{L}(\mathcal{A}_{1})}\subseteq L_{2}$ when $L_{2}$ is regular by implementing the greatest fixpoint computation in equivalence (24).

The intuition behind Algorithm $\mathtt{FAIncGfp}$ is that

[TABLE]

Therefore, $\mathtt{FAIncGfp}$ computes the set $\textstyle\bigcap\{w^{-1}L_{2}\mid w\in{\mathcal{L}(\mathcal{A}_{1})}\}$. by using the automaton $\mathcal{A}_{1}$ and by considering prefixes of ${\mathcal{L}(\mathcal{A}_{1})}$ of increasing lengths. This means that after $n$ iterations of $\operatorname{{\textsc{Kleene}}}$, the algorithm $\mathtt{FAIncGfp}$ has computed

[TABLE]

for every state $q\in Q_{1}$. The regularity of $L_{2}$ and the property of regular languages of being closed under intersections and quotients entail that each Kleene iterate of $\operatorname{{\textsc{Kleene}}}(\supseteq,\lambda\vv{\bm{X}}\ldotp\vv{\bm{L_{2}}}^{{\!{\scriptstyle{I_{1}}}}}\cap\widetilde{\operatorname{{Pre}}}_{\mathcal{A}_{1}}(\vv{\bm{X}}),\vv{\bm{{\Sigma^{*}}}})$ is a (computable) regular language. To the best of our knowledge, this gfp-based language inclusion algorithm $\mathtt{FAIncGfp}$ has never been described in the literature before.

Next, we discharge the fundamental assumption guaranteeing the correctness of this algorithm $\mathtt{FAIncGfp}$: the Kleene iterates computed by $\mathtt{FAIncGfp}$ are finitely many. In order to do that, we consider an abstract version of the greatest fixpoint computation exploiting a closure operator which ensures that the abstract Kleene iterates are finitely many. This closure operator $\rho_{\leq_{\mathcal{A}_{2}}}$ will be defined by using an ordering relation $\leq_{\mathcal{A}_{2}}$ induced by a FA $\mathcal{A}_{2}$ such that $L_{2}={\mathcal{L}(\mathcal{A}_{2})}$ and will be shown to be forward complete for the function $\lambda\vv{\bm{X}}\ldotp\vv{\bm{L_{2}}}^{{\!{\scriptstyle{I_{1}}}}}\cap\widetilde{\operatorname{{Pre}}}_{\mathcal{A}_{1}}(\vv{\bm{X}})$ used by $\mathtt{FAIncGfp}$.

Forward completeness of abstract interpretations (Giacobazzi and Quintarelli, 2001), also called exactness (Miné, 2017, Definition 2.15), is different from and orthogonal to backward completeness introduced in Section 3 and crucially used throughout Sections 4–7. In particular, a remarkable consequence of exploiting a forward complete abstraction is that the Kleene iterates of the concrete and abstract greatest fixpoint computations coincide. The intuition here is that this forward complete closure $\rho_{\leq_{\mathcal{A}_{2}}}$ allows us to establish that all the Kleene iterates of $\vv{\bm{X}}\ldotp\vv{\bm{L_{2}}}^{{\!{\scriptstyle{I_{1}}}}}\cap\widetilde{\operatorname{{Pre}}}_{\mathcal{A}_{1}}(\vv{\bm{X}})$ belong to the image of the closure $\rho_{\leq_{\mathcal{A}_{2}}}$, more precisely that every Kleene iterate is a language which is upward closed for $\leq_{\mathcal{A}_{2}}$. Interestingly, a similar phenomenon occurs in well-structured transition systems (Abdulla et al., 1996; Finkel and Schnoebelen, 2001).

Let us now describe in detail this abstraction. A closure $\rho\in\operatorname{uco}(C)$ on a concrete domain $C$ is forward complete for a monotonic function $f:C\rightarrow C$ if $\rho f\rho=f\rho$ holds. The intuition here is that forward completeness means that no loss of precision is accumulated when the output of a computation of $f\rho$ is approximated by $\rho$, or, equivalently, the concrete function $f$ maps abstract elements of $\rho$ into abstract elements of $\rho$. Dually to the case of backward completeness, forward completeness implies that $\operatorname{gfp}(f)=\operatorname{gfp}(f\rho)=\operatorname{gfp}(\rho f\rho)$ holds, when these greatest fixpoints exist (this is the case, e.g., when $C$ is a complete lattice). When the function $f\colon C\rightarrow C$ admits the right-adjoint $\widetilde{f}\colon C\rightarrow C$, i.e., $f(c)\leq c^{\prime}\Leftrightarrow c\leq\widetilde{f}(c^{\prime})$ holds, it turns out that forward and backward completeness are related by the following duality (Giacobazzi and Quintarelli, 2001, Corollary 1):

[TABLE]

Thus, by (25), in the following result instead of assuming the hypotheses implying that a closure $\rho$ is forward complete for the right-adjoint $\widetilde{\operatorname{{Pre}}}_{\mathcal{A}_{1}}$ we state some hypotheses which guarantee that $\rho$ is backward complete for its left-adjoint, which, by Lemma 8.2, is ${\operatorname{{Pre}}}_{\mathcal{A}_{1}}$.

Theorem 8.3.

Let $\mathcal{A}_{1}=\langle{Q_{1},\delta_{1},I_{1},F_{1},\Sigma}\rangle$ be a FA , $L_{2}$ be a regular language and $\rho\in\operatorname{uco}(\wp(\Sigma^{*}))$. Let us assume that:

1. (1)

$\rho(L_{2})=L_{2}$; 2. (2)

$\rho$* is backward complete for $\lambda X\ldotp aX$ for all $a\in\Sigma$.*

Then, ${\mathcal{L}(\mathcal{A}_{1})}\subseteq L_{2}$ iff $\vv{\bm{\epsilon}}^{{\!{\scriptstyle{F_{1}}}}}\subseteq\operatorname{gfp}(\lambda\vv{\bm{X}}.\rho(\vv{\bm{L_{2}}}^{{\!{\scriptstyle{I_{1}}}}}\cap\widetilde{\operatorname{{Pre}}}_{\mathcal{A}_{1}}(\rho(\vv{\bm{X}}))))$. Moreover, the Kleene iterates of $\lambda\vv{\bm{X}}\ldotp\rho(\vv{\bm{L_{2}}}^{{\!{\scriptstyle{I_{1}}}}}\cap\widetilde{\operatorname{{Pre}}}_{\mathcal{A}_{1}}(\rho(\vv{\bm{X}})))$ and $\lambda\vv{\bm{X}}\ldotp\vv{\bm{L_{2}}}^{{\!{\scriptstyle{I_{1}}}}}\cap\widetilde{\operatorname{{Pre}}}_{\mathcal{A}_{1}}(\vv{\bm{X}})$ from the initial value $\vv{\bm{{\Sigma^{*}}}}$ coincide in lockstep.

Proof.

Theorem 4.3 shows that if $\rho$ is backward complete for $\lambda X\ldotp aX$ for every $a\in\Sigma$ then it is backward complete for $\operatorname{{Pre}}_{{\mathcal{A}_{1}}}$. Thus, by (25), $\rho$ is forward complete for $\widetilde{\operatorname{{Pre}}}_{\mathcal{A}_{1}}$. Then, it turns out that $\rho$ is forward complete for $\lambda\vv{\bm{X}}\ldotp\vv{\bm{L_{2}}}^{{\!{\scriptstyle{I_{1}}}}}\cap\widetilde{\operatorname{{Pre}}}_{\mathcal{A}_{1}}(\vv{\bm{X}})$, because:

[TABLE]

Since, by forward completeness, $\operatorname{gfp}(\lambda\vv{\bm{X}}\ldotp\vv{\bm{L_{2}}}^{{\!{\scriptstyle{I_{1}}}}}\cap\widetilde{\operatorname{{Pre}}}_{\mathcal{A}_{1}}(\vv{\bm{X}}))=\operatorname{gfp}(\lambda\vv{\bm{X}}\ldotp\rho(\vv{\bm{L_{2}}}^{{\!{\scriptstyle{I_{1}}}}}\cap\widetilde{\operatorname{{Pre}}}_{\mathcal{A}_{1}}(\rho(\vv{\bm{X}}))))$, by equivalence (24), we conclude that ${\mathcal{L}(\mathcal{A}_{1})}\subseteq L_{2}$ iff $\vv{\bm{\epsilon}}^{{\!{\scriptstyle{F_{1}}}}}\subseteq\operatorname{gfp}(\lambda\vv{\bm{X}}\ldotp\rho(\vv{\bm{L_{2}}}^{{\!{\scriptstyle{I_{1}}}}}\cap\widetilde{\operatorname{{Pre}}}_{\mathcal{A}_{1}}(\rho(\vv{\bm{X}}))))$.

Finally, we observe that the Kleene iterates of $\lambda\vv{\bm{X}}\ldotp\vv{\bm{L_{2}}}^{{\!{\scriptstyle{I_{1}}}}}\cap\widetilde{\operatorname{{Pre}}}_{\mathcal{A}_{1}}(\vv{\bm{X}})$ and $\lambda\vv{\bm{X}}\ldotp\rho(\vv{\bm{L_{2}}}^{{\!{\scriptstyle{I_{1}}}}}\cap\widetilde{\operatorname{{Pre}}}_{\mathcal{A}_{1}}(\rho(\vv{\bm{X}})))$ starting from $\vv{\bm{{\Sigma^{*}}}}$ coincide in lockstep since $\rho(\vv{\bm{L_{2}}}^{{\!{\scriptstyle{I_{1}}}}}\cap\widetilde{\operatorname{{Pre}}}_{\mathcal{A}_{1}}(\rho(\vv{\bm{X}})))=\vv{\bm{L_{2}}}^{{\!{\scriptstyle{I_{1}}}}}\cap\widetilde{\operatorname{{Pre}}}_{\mathcal{A}_{1}}(\rho(\vv{\bm{X}}))$ and $\rho(\vv{\bm{L_{2}}}^{{\!{\scriptstyle{I_{1}}}}})=\vv{\bm{L_{2}}}^{{\!{\scriptstyle{I_{1}}}}}$. ∎

We can now establish that the Kleene iterates of $\operatorname{{\textsc{Kleene}}}(\supseteq,\lambda\vv{\bm{X}}\ldotp\vv{\bm{L_{2}}}^{{\!{\scriptstyle{I_{1}}}}}\cap\widetilde{\operatorname{{Pre}}}_{\mathcal{A}_{1}}(\vv{\bm{X}}),\vv{\bm{{\Sigma^{*}}}})$ are finitely many. Let $L_{2}={\mathcal{L}(\mathcal{A}_{2})}$, for some FA $\mathcal{A}_{2}$, and consider the corresponding left state-based quasiorder $\mathord{\leq_{\mathcal{A}_{2}}^{l}}$ on $\Sigma^{*}$ as defined by (15). By Lemma 5.8, $\mathord{\leq_{\mathcal{A}_{2}}^{l}}$ is a left $L_{2}$-consistent wqo. Furthermore, since $Q_{2}$ is finite we have that both $\mathord{\leq_{\mathcal{A}_{2}}^{l}}$ and $(\mathord{\leq_{\mathcal{A}_{2}}^{l}})^{-1}$ are wqos, so that, in turn, $\langle{\rho_{\leq_{\mathcal{A}_{2}}^{l}}(\wp(\Sigma^{*})),\subseteq}\rangle$ is a poset which is both ACC and DCC. In particular, the definition of $\mathord{\leq_{\mathcal{A}_{2}}^{l}}$ implies that every chain in $\langle{\rho_{\leq_{\mathcal{A}_{2}}^{l}}(\wp(\Sigma^{*})),\subseteq}\rangle$ has at most $2^{|Q_{2}|}$ elements, so that if we compute $2^{|Q_{2}|}$ Kleene iterates then we surely converge to the greatest fixpoint. Moreover, as a consequence of the DCC we have that $\operatorname{{\textsc{Kleene}}}(\supseteq,\lambda\vv{\bm{X}}\ldotp\rho_{\leq_{\mathcal{A}_{2}}}(\vv{\bm{L_{2}}}^{{\!{\scriptstyle{I_{1}}}}}\cap\widetilde{\operatorname{{Pre}}}_{\mathcal{A}_{1}}(\rho_{\leq_{\mathcal{A}_{2}}}(\vv{\bm{X}}))),\vv{\bm{{\Sigma^{*}}}})$ always terminates, thus implying that $\operatorname{{\textsc{Kleene}}}(\supseteq,\lambda\vv{\bm{X}}\ldotp\vv{\bm{L_{2}}}^{{\!{\scriptstyle{I_{1}}}}}\cap\widetilde{\operatorname{{Pre}}}_{\mathcal{A}_{1}}(\vv{\bm{X}}),\vv{\bm{{\Sigma^{*}}}})$ terminates as well, because their Kleene iterates go in lockstep as stated by Theorem 8.3. We have therefore shown the correctness of $\mathtt{FAIncGfp}$.

Corollary 8.4.

The algorithm $\mathtt{FAIncGfp}$ decides the inclusion ${\mathcal{L}(\mathcal{A}_{1})}\subseteq L_{2}$

Example 8.5.

Let us illustrate the greatest fixpoint algorithm $\mathtt{FAIncGfp}$ on the inclusion check $L(\mathcal{B})\subseteq L(\mathcal{A})$ where $\mathcal{A}$ is the FA in Fig. 1 and $\mathcal{B}$ is the following FA:

q_{3}$$q_{4}$$b$$a$$a

By Corollary 8.4, the Kleene iterates of $\lambda\vv{\bm{Y}}\ldotp\vv{\bm{L(\mathcal{A})}}^{{\!{\scriptstyle{\{q_{3}\}}}}}\cap\widetilde{\operatorname{{Pre}}}_{\mathcal{B}}(\vv{\bm{Y}})$ are guaranteed to converge in finitely many steps. We have that

[TABLE]

Then, the Kleene iterates are as follows (we automatically checked them by the FAdo tool (Almeida et al., 2009)):

[TABLE]

Thus, $\operatorname{{\textsc{Kleene}}}$ outputs the vector $\langle{Y_{3},Y_{4}}\rangle=\langle{L(\mathcal{A}),(b^{*}a)^{+}}\rangle$. Since $\epsilon\in L(\mathcal{A})$, $\mathtt{FAIncGfp}$ concludes that $L(\mathcal{B})\subseteq L(\mathcal{A})$ holds. $\Diamond$

Finally, it is worth citing that Fiedor et al. (2019) put forward an algorithm for deciding WS1S formulae which relies on the same lfp computation used in $\mathtt{FAIncS}$. Then, they derive a dual gfp computation by relying on Park’s duality (Park, 1969): $\operatorname{lfp}(\lambda X\ldotp f(X))=(\operatorname{gfp}(\lambda X\ldotp(f(X^{c}))^{c}))^{c}$. Their approach differs from ours since we use the equivalence (23) to compute a gfp, different from the lfp, which still allows us to decide the inclusion problem. Furthermore, their algorithm decides whether a given automaton accepts $\epsilon$ and it is not clear how their algorithm could be extended for deciding language inclusion.

9. Future Work

We believe that this work only scratched the surface of the use of well-quasiorders on words for solving language inclusion problems. In particular, our approach based on complete abstract interpretations allowed us to systematically derive well-known algorithms , such as the antichain algorithms by De Wulf et al. (2006), as well as novel algorithms, such as $\mathtt{FAIncGfp}$, for deciding the inclusion of regular languages.

Future directions include leveraging well-quasiorders for infinite words (Ogawa, 2004) to shed new light on the inclusion problem between $\omega$-languages. Our results could also be extended to inclusion of tree languages by relying on the extensions of Myhill-Nerode theorems for tree languages (Kozen, 1992). Another interesting topic for future work is the enhancement of quasiorders using simulation relations. Even though we already showed in this paper that simulations can be used to refine our language inclusion algorithms, we are not on par with the thoughtful use of simulation relations made by Abdulla et al. (2010) and Bonchi and Pous (2013). Finally, let us mention that the correspondence between least and greatest fixpoint-based inclusion checks assuming complete abstractions was studied by Bonchi et al. (2018) with the aim of formally connecting sound up-to techniques and complete abstract interpretations. Further possible developments include the study of our abstract interpretation-based algorithms for language inclusion from the viewpoint of sound up-to techniques.