The Complexity of Deciding Characteristic Formulae in Van Glabbeek’s Branching-Time Spectrum

Several notions of behavioural relations have been proposed in concurrency theory to describe when one process is a suitable implementation of another. Many such relations have been catalogued by van Glabbeek in his seminal linear-time/branching-time spectrum [22], together with a variety of alternative ways of describing them including testing scenarios and axiom systems. To our mind, modal characterizations of behavioural equivalences and preorders are some of the most classic and pleasing results in concurrency theory – see, for instance, [25] for the seminal Hennessy-Milner theorem and [12, 16, 17, 22] for similar results for other relations in van Glabbeek’s spectrum and other settings. By way of example, in their archetypal modal characterization of bisimilarity, Hennessy and Milner have shown in [25] that, under a mild finiteness condition, two processes are bisimilar if, and only if, they satisfy the same formulae in a multi-modal logic that is now often called Hennessy-Milner logic. Apart from its intrinsic theoretical interest, this seminal logical characterization of bisimilarity means that, when two processes are not bisimilar, there is always a formula that distinguishes between them. Such a formula describes a reason why the two processes are not bisimilar, provides useful debugging information and can be algorithmically constructed over finite processes – see, for instance, [8, 14] and [35], where Martens and Groote show that, in general, computing minimal distinguishing Hennessy-Milner formulae is $\mathrm{𝖭𝖯}$-hard.

On the other hand, the Hennessy-Milner theorem seems to be less useful to show that two processes are bisimilar, since that would involve verifying that they satisfy the same formulae, and there are infinitely many of those. However, as shown in works such as [3, 6, 12, 23, 39], the logics that underlie classic modal characterization theorems for equivalences and preorders over processes allow one to express characteristic formulae. Intuitively, a characteristic formula $\chi (p)$ for a process $p$ gives a complete logical characterization of the behaviour of $p$ modulo the behavioural semantics of interest $\lesssim$, in the sense that any process is related to $p$ with respect to $\lesssim$ if, and only if, it satisfies $\chi (p)$.¹¹1Formulae akin to characteristic ones first occurred in the study of equivalence of structures using first-order formulae up to some quantifier rank. See, for example, the survey paper [40] and the textbook [20]. The existence of formulae in first-order logic with counting that characterize graphs up to isomorphism has significantly contributed to the study of the complexity of the Graph Isomorphism problem – see, for instance, [13, 30]. Since the formula $\chi (p)$ can be constructed from $p$, characteristic formulae reduce the problem of checking whether a process $q$ is related to $p$ by $\lesssim$ to a model checking problem, viz. whether $q$ satisfies $\chi (p)$. See, for instance, the classic reference [15] for applications of this approach.

Characteristic formulae, thus, allow one to reduce equivalence and preorder checking to model checking. But what model checking problems can be reduced to equivalence/preorder checking ones? To the best of our knowledge, that question was first studied by Boudol and Larsen in [11] in the setting of modal refinement over modal transition systems. See [3, 4] for other contributions in that line of research. The aforementioned articles showed that characteristic formulae coincide with those that are satisfiable and prime. (A formula is prime if whenever it entails a disjunction ${\phi }_1\vee {\phi }_2$, then it must entail ${\phi }_1$ or ${\phi }_2$.) Moreover, characteristic formulae with respect to bisimilarity coincide with the formulae that are satisfiable and complete [7]. (A modal formula is complete if, for each formula $\phi$, it entails either $\phi$ or its negation.) The aforementioned results give semantic characterizations of the formulae that are characteristic within the logics that correspond to the behavioural semantics in van Glabbeek’s spectrum. Those characterizations tell us for what logical specifications model checking can be reduced to equivalence or preorder checking. However, given a specification expressed as a modal formula, can one decide whether that formula is characteristic and therefore can be model checked using algorithms for behavioural equivalences or preorders? And, if so, what is the complexity of checking whether a formula is characteristic? Perhaps surprisingly, those questions were not addressed in the literature until the recent papers [2, 7], where it is shown that, in the setting of the modal logics that characterize bisimilarity over natural classes of Kripke structures and labelled transition systems, the problem of checking whether a formula is characteristic for some process modulo bisimilarity is computationally hard and, typically, has the same complexity as validity checking, which is $\mathrm{𝖯𝖲𝖯𝖠𝖢𝖤}$-complete for Hennessy-Milner logic and $\mathrm{𝖤𝖷𝖯}$-complete for its extension with fixed-point operators [26, 33] and the $\mu$-calculus [31].

The aforementioned hardness results for the logics characterizing bisimilarity tell us that deciding whether a formula is characteristic in bisimulation semantics is computationally hard. But what about the less expressive logics that characterize the coarser semantics in van Glabbeek’s spectrum? And for what logics characterizing relations in the spectrum does computational hardness manifest itself? Finally, what is the complexity of computing a characteristic formula for a process?

The aim of this paper is to answer the aforementioned questions for some of the simulation-based semantics in the spectrum. In particular, we study the complexity of determining whether a formula is characteristic modulo the simulation [36], complete simulation and ready simulation preorders [10, 34], as well as the trace simulation and the $n$-nested simulation preorders [24]. Since characteristic formulae are exactly the satisfiable and prime ones for each behavioural relation in van Glabbeek’s spectrum [3], the above-mentioned tasks naturally break down into studying the complexity of satisfiability and primality checking for formulae in the fragments of Hennessy-Milner logic that characterize those preorders. By using a reduction to the, seemingly unrelated, reachability problem in alternating graphs, as defined by Immerman in [28, Definition 3.24], we discover that both those problems are decidable in polynomial time for the simulation and the complete simulation preorders, as well as for the ready simulation preorder when the set of actions has constant size. On the other hand, when the set of actions is unbounded (that is, it is an input of the algorithmic problem at hand), the problems of checking satisfiability and primality for formulae in the logic characterizing the ready simulation preorder are $\mathrm{𝖭𝖯}$-complete and $\mathrm{𝖼𝗈𝖭𝖯}$-complete respectively. We also show that deciding whether a formula is characteristic in that setting is $\mathrm{𝖴𝖲}$-hard [9] (that is, it is at least as hard as the problem of deciding whether a given Boolean formula has exactly one satisfying truth assignment) and belongs to $\mathrm{𝖣𝖯}$, which is the class of languages that are the intersection of one language in $\mathrm{𝖭𝖯}$ and of one in $\mathrm{𝖼𝗈𝖭𝖯}$ [38].²²2The class $\mathrm{𝖣𝖯}$ contains both $\mathrm{𝖭𝖯}$ and $\mathrm{𝖼𝗈𝖭𝖯}$, and is contained in the class of problems that can be solved in polynomial time with an $\mathrm{𝖭𝖯}$ oracle. These negative results are in stark contrast with the positive results for the simulation and the complete simulation preorder, and indicate that augmenting the logic characterizing the simulation preorder with formulae that state that a process cannot perform a given action suffices to make satisfiability and primality checking computationally hard. In passing, we also prove that, in the presence of at least two actions, (1) for the logics characterizing the trace simulation and 2-nested simulation preorders, satisfiability and primality checking are $\mathrm{𝖭𝖯}$-complete and $\mathrm{𝖼𝗈𝖭𝖯}$-hard respectively, and deciding whether a formula is characteristic is $\mathrm{𝖴𝖲}$-hard, (2) for the logic that characterizes the trace simulation preorder, deciding whether a formula is characteristic is fixed-parameter tractable [18], with the modal depth of the input formula as the parameter, when the size of the action set is a constant, and (3) deciding whether a formula is characteristic in the modal logic for the 3-nested simulation preorder [24] is $\mathrm{𝖯𝖲𝖯𝖠𝖢𝖤}$-hard. (The proof of the last result relies on “simulating” Ladner’s reduction proving the $\mathrm{𝖯𝖲𝖯𝖠𝖢𝖤}$-hardness of satisfiability for modal logic [32] using the limited alternations of modal operators allowed by the logic for the 3-nested simulation preorder.)

We also study the complexity of computing characteristic formulae for finite, loop-free processes modulo the above-mentioned simulation semantics. To do so, we consider two different representations for formulae, namely an explicit form, where formulae are given by strings of symbols generated by their respective grammars, and a declarative form, where formulae are described by systems of equations. We prove that, even for the coarsest semantics we consider, such as the simulation and complete simulation preorders, computing the characteristic formula in explicit form for a finite, loop-free process cannot be done in polynomial time, unless $𝖯=\mathrm{𝖭𝖯}$. On the other hand, the characteristic formula for a process modulo the preorders we study, apart from the trace simulation preorder, can be computed in polynomial time if the output is given in declarative form. Intuitively, this is due to the fact that, unlike the explicit form, systems of equations allow for sharing of subformulae and there are formulae for which this sharing leads to an exponentially more concise representation. Finally, in sharp contrast to that result, we prove that, modulo the trace simulation preorder, even if characteristic formulae are always of polynomial declaration size and polynomial equational length, they cannot be efficiently computed unless $𝖯=\mathrm{𝖭𝖯}$. In passing, we remark that all the aforementioned lower and upper bounds hold also for finite processes with loops, provided that, as done in [6, 29, 39], we add greatest fixed points or systems of equations interpreted as greatest fixed points to the modal logics characterizing the semantics we study in this article.

We summarize our results in Table 2. We provide their proofs in the technical appendices of the full version of the paper [1].

In this paper, we model processes as finite, loop-free labelled transition systems (LTS). A finite LTS is a triple $𝒮=(P,A,⟶)$, where $P$ is a finite set of states (or processes), $A$ is a finite, non-empty set of actions and $⟶\subseteq P\times A\times P$ is a transition relation. As usual, we use $p\stackrel{a}{⟶}q$ instead of $(p,a,q)\in ⟶$. For each $t\in A^{\ast }$, we write $p\stackrel{t}{⟶}q$ to mean that there is a sequence of transitions labelled with $t$ starting from $p$ and ending at $q$. An LTS is loop-free iff $p\stackrel{t}{⟶}p$ holds only when $t$ is the empty trace $\epsilon$. A process $q$ is reachable from $p$ if $p\stackrel{t}{⟶}q$, for some $t\in A^{\ast }$. We define the size of an LTS $𝒮=(P,A,⟶)$, denoted by $|𝒮|$, to be $|P|+|⟶|$. The size of a process $p\in P$, denoted by $|p|$, is the cardinality of $\mathrm{reach}(p)=\{q|q\text{ is reachable from }p\}$ plus the cardinality of the set $⟶$ restricted to $\mathrm{reach}(p)$. We define the set of initials of $p$, denoted $I(p)$, as the set $\{a\in A\mid p\stackrel{a}{⟶}{p}^{\prime }\text{ for some }{p}^{\prime }\in P\}$. We write $p\stackrel{a}{⟶}$ if $a\in I(p)$, $p\stackrel{a}{\to ̸}$ if $a\notin I(p)$, and $p\to ̸$ if $I(p)=\mathrm{\varnothing }$. A sequence of actions $t\in A^{\ast }$ is a trace of $p$ if there is a $q$ such that $p\stackrel{t}{⟶}q$. We denote the set of traces of $p$ by $\mathrm{traces}(p)$. The depth of a finite, loop-free process $p$, denoted by $0pt(p)$, is the length of a longest trace $t$ of $p$.

In what follows, we shall often describe finite, loop-free processes using the fragment of Milner’s CCS [37] given by the following grammar:

where $a\in A$. For each action $a$ and terms $p,p^{\prime }$, we write $p\stackrel{a}{⟶}{p}^{\prime }$ iff

1. (i)

   $p=a.p^{\prime }$ or

2. (ii)

   $p=p_1+p_2$, for some $p_1,p_2$, and $p_1\stackrel{a}{⟶}{p}^{\prime }$ or $p_2\stackrel{a}{⟶}{p}^{\prime }$ holds.

In this paper, we consider the following relations in van Glabbeek’s spectrum: simulation, complete simulation, ready simulation, trace simulation, 2-nested simulation, 3-nested simulation, and bisimilarity. Their definitions are given below.

It is well-known that bisimilarity is an equivalence relation and all the other relations are preorders [22, 37]. We sometimes write $p\sim q$ instead of $p{\lesssim }_{BS}$. Moreover, we have that $\sim ⊊{\lesssim }_{3S}⊊{\lesssim }_{2S}⊊{\lesssim }_{TS}⊊{\lesssim }_{RS}⊊{\lesssim }_{CS}⊊{\lesssim }_S$ – see [22].

Each relation ${\lesssim }_X$, where $X\in \{S,CS,RS,TS,2S,3S,BS\}$, is characterized by a fragment ${ℒ}_X$ of Hennessy-Milner logic, $\mathrm{𝐇𝐌𝐋}$, defined as follows [22, 3].

Note that the explicit use of negation in the grammar for ${ℒ}_{BS}$ is unnecessary. However, we included the negation operator explicitly so that ${ℒ}_{BS}$ extends syntactically each of the other modal logics presented in Definition 3.

Given a formula $\phi \in {ℒ}_{BS}$, the modal depth of $\phi$, denoted by $\mathrm{md}(\phi )$, is the maximum nesting of modal operators in $\phi$. (See [1, Appendix A] for the formal definition.)

Truth in an LTS $𝒮=(P,A,⟶)$ is defined via the satisfaction relation $\vDash$ as follows:

If $p\vDash \phi$, we say that $\phi$ is true, or satisfied, in $p$. If $\phi$ is satisfied in every process in every LTS, we say that $\phi$ is valid. Formula ${\phi }_1$ entails ${\phi }_2$, denoted by ${\phi }_1\vDash {\phi }_2$, if every process that satisfies ${\phi }_1$ also satisfies ${\phi }_2$. Moreover, ${\phi }_1$ and ${\phi }_2$ are logically equivalent, denoted by ${\phi }_1\equiv {\phi }_2$, if ${\phi }_1\vDash {\phi }_2$ and ${\phi }_2\vDash {\phi }_1$. A formula $\phi$ is satisfiable if there is a process that satisfies $\phi$. Finally, $\mathrm{Sub}(\phi )$ denotes the set of subformulae of formula $\phi$.

For $ℒ\subseteq {ℒ}_{BS}$, we define the dual fragment of $ℒ$ to be $\overline{ℒ}=\{\phi \mid \neg \phi \in ℒ\}$, where $\neg \mathrm{𝐭𝐭}=\mathrm{𝐟𝐟}$, $\neg \mathrm{𝐟𝐟}=\mathrm{𝐭𝐭}$, $\neg (\phi \wedge \psi )=\neg \phi \vee \neg \psi$, $\neg (\phi \vee \psi )=\neg \phi \wedge \neg \psi$, $\neg [a]\phi =⟨a⟩\neg \phi$, $\neg ⟨a⟩\phi =[a]\neg \phi$, and $\neg \neg \phi =\phi$. It is not hard to see that $p\vDash \neg \phi$ iff $p\vDash ̸\phi$, for every process $p$. Given a process $p$, we define $ℒ(p)=\{\phi \in ℒ\mid p\vDash \phi \}$. A simplification of the Hennessy-Milner theorem gives a modal characterization of bisimilarity over finite processes. An analogous result is true for every preorder examined in this paper.

When the logic $ℒ$ is clear from the context, we say that $\phi$ is prime. Note that every unsatisfiable formula is trivially prime in $ℒ$, for every $ℒ$.

The definition of a characteristic formula within logic $ℒ$ is given next.

In [3, Table 1 and Theorem 5], Aceto, Della Monica, Fabregas, and Ingólfsdóttir presented characteristic formulae for each of the semantics we consider in this paper, and showed that characteristic formulae are exactly the satisfiable and prime ones.

We can also consider characteristic formulae modulo equivalence relations as follows.

When studying the complexity of finding a characteristic formula for some process $p$ with respect to the behavioural relations we have introduced above, we will need some way of measuring the size of the resulting formula as a function of $|p|$. A formula in ${ℒ}_X$, where $X\in \{S,CS,RS,TS,2S,3S,BS\}$, can be given in explicit form as in Definition 3 or by means of a system of equations. In the latter case, we say that the formula is given in declarative form. For example, formula $\varphi =⟨a⟩(⟨a⟩\mathrm{𝐭𝐭}\wedge ⟨b⟩\mathrm{𝐭𝐭})\wedge ⟨b⟩(⟨a⟩\mathrm{𝐭𝐭}\wedge ⟨b⟩\mathrm{𝐭𝐭})$ can be represented by the equations $\varphi =⟨a⟩{\varphi }_1\wedge ⟨b⟩{\varphi }_1$ and ${\varphi }_1=⟨a⟩\mathrm{𝐭𝐭}\wedge ⟨b⟩\mathrm{𝐭𝐭}$. We define:

• $\mathrm{■}$

  the size of formula $\phi$, denoted by $|\phi |$, to be the number of symbols that appear in the explicit form of $\phi$,

• $\mathrm{■}$

  the declaration size of formula $\phi$, denoted by $\mathrm{decl}(\phi )$, to be the number of equations that are used in the declarative form of $\phi$, and

• $\mathrm{■}$

  the equational length of formula $\phi$, denoted by $\mathrm{eqlen}(\phi )$, to be the maximum number of symbols that appear in an equation in the declarative form of $\phi$.

For example, for the aforementioned formula $\varphi$, we have that $|\varphi |=13$, $\mathrm{decl}(\varphi )=2$, and $\mathrm{eqlen}(\varphi )=5$. Note that $\mathrm{decl}(\phi )\le |\mathrm{Sub}(\phi )|\le |\phi |$, for each $\phi$.

In this section, we address the complexity of deciding whether formulae in ${ℒ}_S$, ${ℒ}_{CS}$, ${ℒ}_{RS}$, ${ℒ}_{TS}$, ${ℒ}_{2S}$, and ${ℒ}_{3S}$ are characteristic. Since characteristic formulae in those logics are exactly the satisfiable and prime ones [3, Theorem 5], we study the complexity of checking satisfiability and primality separately in Subsections 3.1 and 3.2.

Let $X\in \{S,CS\}$ or $X=RS$ and $|A|$ is bounded by a constant. The complexity of finding characteristic formulae within ${ℒ}_X$ depends on the representation of the output. If the characteristic formula has to be given in explicit form, then the following result holds.