Complexity of Weak Bisimilarity and Regularity for BPA and BPP

It is an open problem whether weak bisimilarity is decidable for Basic Process Algebra (BPA) and Basic Parallel Processes (BPP). A PSPACE lower bound for BPA and NP lower bound for BPP have been demonstrated by Stribrna. Mayr achieved recently a result, saying that weak bisimilarity for BPP is Π^p₂-hard. We improve this lower bound to PSPACE, moreover for the restricted class of normed BPP.Weak regularity (finiteness) of BPA and BPP is not known to be decidable either. In the case of BPP there is a Π^p₂-hardness result by Mayr, which we improve to PSPACE. No lower bound has previously been established for BPA. We demonstrate DP-hardness, which in particular implies both NP and co-NP-hardness.In each of the bisimulation/regularity problems we consider also the classes of normed processes.Note: full version of the paper appears as [18].     

Deciding probabilistic bisimilarity over infinite-state probabilistic systems

We prove that probabilistic bisimilarity is decidable over probabilistic extensions of BPA and BPP processes. For normed subclasses of probabilistic BPA and BPP processes we obtain polynomial-time algorithms. Further, we show that probabilistic bisimilarity between probabilistic pushdown automata and finite-state systems is decidable in exponential time. If the number of control states in PDA is bounded by a fixed constant, then the algorithm needs only polynomial time. The work has been supported by the research centre Institute for Theoretical Computer Science (ITI), project No. 1M0545.     

Weak bisimilarity between finite-state systems and BPA or normed BPP is decidable in polynomial time

We prove that weak bisimilarity is decidable in polynomial time between finite-state systems and several classes of infinite-state systems: context-free processes and normed basic parallel processes (normed BPP). To the best of our knowledge, these are the first polynomial algorithms for weak bisimilarity problems involving infinite-state systems.     

Refining the Undecidability Border of Weak Bisimilarity

Weak bisimilarity is one of the most studied behavioural equivalences. This equivalence is undecidable for pushdown processes (PDA), process algebras (PA), and multiset automata (MSA, also known as parallel pushdown processes, PPDA). Its decidability is an open question for basic process algebras (BPA) and basic parallel processes (BPP). We move the undecidability border towards these classes by showing that the equivalence remains undecidable for weakly extended versions of BPA and BPP. In fact, we show that the weak bisimulation equivalence problem is undecidable even for normed subclasses of BPA and BPP extended with a finite constraint system.     

Partially-commutative context-free processes: Expressibility and tractability

Bisimulation equivalence is decidable in polynomial time for both sequential and commutative normed context-free processes, known as BPA and BPP, respectively. Despite apparent similarity between the two classes, different algorithmic techniques were used in each case. We provide one polynomial-time algorithm that works in a superclass of both normed BPA and BPP. It is derived in the setting of partially-commutative context-free processes, a new process class introduced in the paper. It subsumes both BPA and BPP and seems to be of independent interest. Expressibility issue of the new class, in comparison with the normed PA class, is also tackled in the paper.     

Decidability of Branching Bisimulation on Normed Commutative Context-Free Processes

We investigate normed commutative context-free processes (Basic Parallel Processes). We show that branching bisimilarity admits the bounded response property: in the Bisimulation Game, Duplicator always has a response leading to a process of size linearly bounded with respect to the Spoiler’s process. The linear bound is effective, which leads to decidability of branching bisimilarity. For weak bisimilarity, we are able merely to show existence of some linear bound, which is not sufficient for decidability. We conjecture however that the same effective bound holds for weak bisimilarity as well. We suppose that further elaboration of novel techniques developed in this paper may be sufficient to demonstrate decidability.     

Non-determinism and weak constraints in Datalog

This paper introduces a simple and powerful extension of stratified DATALOG which permits to express various DB-complexity classes. The new language, called DATALOG^¬s,c,p , extends DATALOG with stratified negation, a non-deterministic construct, calledchoice, and a weak form of constraints, calledpreference rules, that is, constraints that should be respected but, if they cannot be eventually enforced, they only invalidate the portions of the program which they are concerned with. Although DATALOG with stratified negation is not able to express all polynomial time queries,²⁰⁾ the introduction of the non-deterministic constructchoice permits to express, exactly, the ‘deterministic fragment’ of the class of DB-queriesP, under the non-deterministic semantics,NP, under the possible semantics, and coNP, under the certain semantics. The introduction of preference rules, further increases the expressive power of the language, and permits to express the complexity classes Σ ₂ ^p , under the possibility semantics, and Π ₂ ^p , under the certainty semantics.     

Comparing expressibility of normed BPA and normed BPP processes

We present an exact characterization of those transition systems which can be equivalently (up to bisimilarity) defined by the syntax of normed BPA and normed BPP processes. We give such a characterization for the subclasses of normed BPA and normed BPP processes as well. Next we demonstrate the decidability of the problem whether for a given normed BPA process there is some unspecified normed BPP process such that and are bisimilar. The algorithm is polynomial. Furthermore, we show that if the answer to the previous question is positive, then (an example of) the process is effectively constructible. Analogous algorithms are provided for normed BPP processes. Simplified versions of the mentioned algorithms which work for normed BPA and normed BPP are given too. As a simple consequence we obtain the decidability of bisimilarity in the union of normed BPA and normed BPP processes. Received: 3 June 1997     

Weak bisimilarity and regularity of context-free processes is EXPTIME-hard

We show that checking weak bisimulation equivalence of two context-free processes (also called BPA-processes) is EXPTIME-hard, even under the condition that the processes are normed. Furthermore, checking weak regularity (finiteness up to weak bisimilarity) for context-free processes is EXPTIME-hard as well. Adding a finite control of the minimal non-trivial size of 2 to the BPA process already makes weak bisimilarity undecidable.     

Some more independence results in complexity theory

A theorem of Hartmanis and Eopcroft is extended to obtain further independence results related to $\text{‘P = ?NP’}$. Alternative proofs are give for some of these results, following Hajek, obtaining Π₂⁰ and σ₂⁰ complete sets. It is pointed out however, that these theorems neither support the evidence for nor against the independence of P = NP.     

Petri nets and regular languages

It is shown that the regularity problem for firing sequence sets of Petri nets is decidable. For the proof, new techniques to characterize unbounded places are introduced. In the class $\text{L}$₀ of terminal languages of labelled Petri nets the regularity problem in undecidable. In addition some lower bounds for the undecidability of the equality problems in $\text{L}$₀ and $\text{L}$ are given. L₀^λ is shown to be not closed under complementation without reference to the reachability problem.     

On the expressiveness and decidability of higher-order process calculi

In higher-order process calculi, the values exchanged in communications may contain processes. A core calculus of higher-order concurrency is studied; it has only the operators necessary to express higher-order communications: input prefix, process output, and parallel composition. By exhibiting a deterministic encoding of Minsky machines, the calculus is shown to be Turing complete. Therefore its termination problem is undecidable. Strong bisimilarity, however, is shown to be decidable. Furthermore, the main forms of strong bisimilarity for higher-order processes (higher-order bisimilarity, context bisimilarity, normal bisimilarity, barbed congruence) coincide. They also coincide with their asynchronous versions. A sound and complete axiomatization of bisimilarity is given. Finally, bisimilarity is shown to become undecidable if at least four static (i.e., top-level) restrictions are added to the calculus.     

Some observations on NP real numbers and P-selective sets

We show that each standard left cut of a real number is a p-selective set. Classification results about NP real numbers and recursively enumerable real numbers follow from similar results about p-selective and semirecursive sets. In particular, it is proved that no left cut can be NP-hard unless the polynomial hierarchy collapses to ?₂^P. This result is surprising because it shows that the McLaughlin-Martin construction of a ?_tt-complete r.e. semirecursive set fails at the polynomial time complexity level.     

Deciding Branching Bisimilarity of Normed Context-Free Processes Is in Σ2

We show that the problem of deciding branching bisimulation equivalence for normed context-free processes is in Σ^p₂, the second level of the polynomial-time hierarchy, and hence in PSPACE. We also show that minimization of normed context-free process graphs is in PSPACE.     

Simulation Preorder over Simple Process Algebras

We consider the problem of simulation preorder/equivalence between infinite-state processes and finite-state ones. First, we describe a general method how to utilize the decidability of bisimulation problems to solve (certain instances of) the corresponding simulation problems. For certain process classes, the method allows us to design effective reductions of simulation problems to their bisimulation counterparts and some new decidability results for simulation have already been obtained in this way. Then we establish the decidability border for the problem of simulation preorder/equivalence between infinite-state processes and finite-state ones w.r.t. the hierarchy of process rewrite systems. In particular, we show that simulation preorder (in both directions) and simulation equivalence are decidable in EXPTIME between pushdown processes and finite-state ones. On the other hand, simulation preorder is undecidable between PA and finite-state processes in both directions. These results also hold for those PA and finite-state processes which are deterministic and normed, and thus immediately extend to trace preorder. Regularity (finiteness) w.r.t. simulation and trace equivalence is also shown to be undecidable for PA. Finally, we prove that simulation preorder (in both directions) and simulation equivalence are intractable between all classes of infinite-state systems (in the hierarchy of process rewrite systems) and finite-state ones. This result is obtained by showing that the problem whether a BPA (or BPP) process simulates a finite-state one is PSPACE-hard and the other direction is co -hard; consequently, simulation equivalence between BPA (or BPP) and finite-state processes is also co -hard.     

BPP and the polynomial hierarchy

BPP is the class of all sets that can be divided by a probabilistic Turing machine with bounded error probability within a polynomial time bound. Sipser (1983) showed that BPP is contained in the polynomial hierarchy of Meyer and Stockmeyer.In this paper it is shown by pure counting arguments that BPP is contained in Σ^P₂, the second level of the hierarchy.     

Isomorphic Implication

We study the isomorphic implication problem for Boolean constraints. We show that this is a natural analog of the subgraph isomorphism problem. We prove that, depending on the set of constraints, this problem is in P, or is NP-complete, or is NP-hard, coNP-hard, and in P_‖^NP. We show how to extend the NP-hardness and coNP-hardness to P_‖^NP-hardness for some cases, and conjecture that this can be done in all cases. Supported in part by grants NSF-CCR-0311021 and DFG VO 630/5-1 and VO 630/5-2. An extended abstract of this paper appears in Proceedings of the 30th International Symposium on Mathematical Foundations of Computer Science (MFCS 2005), pp. 119–130, Springer-Verlag Lecture Notes in Computer Science #3618, August 2005. Work of M. Bauland done in part while visiting CASCI’s Laboratory for Complexity at Rochester Institute of Technology. Work of E. Hemaspaandra done in part while on sabbatical at the University of Rochester.     

Decidability of performance equivalence for basic parallel processes

We study an extension of the class of Basic Parallel Processes (BPP), in which actions are durational and urgent and parallel components have independent local clocks. The main result is decidability of strong bisimilarity, known also as performance equivalence, in this class. This extends the earlier decidability result for plain BPP by Christensen et al. Our decision procedure is based on decidability of the validity problem for Presburger arithmetic. We prove also polynomial complexity in positive-duration fragment, thus properly extending a previous result by Bérard et al. Both ill-timed and well-timed semantics are treated.     

Characterizing the NP-PSPACE Gap in the Satisfiability Problem for Modal Logic

There has been a great deal of work on characterizing the complexityof the satisfiability and validity problem for modal logics.In particular, Ladner showed that the satisfiability problemfor all logics between K and S4 is PSPACE-hard, while for S5it is NP-complete. We show that it is negative introspection,the axiom ¬K_p K¬K_p, that causes the gap: if we addthis axiom to any modal logic between K and S4, then the satisfiabilityproblem becomes NP-complete. Indeed, the satisfiability problemis NP-complete for any modal logic that includes the negativeintrospection axiom.     

The Difference between Polynomial-Time Many-One and Truth-Table Reducibilities on Distributional Problems

In this paper we separate many-one reducibility from truth-table reducibility for distributional problems in DistNP under the hypothesis that P ≠ NP . As a first example we consider the 3-Satisfiability problem (3SAT) with two different distributions on 3CNF formulas. We show that 3SAT with a version of the standard distribution is truth-table reducible but not many-one reducible to 3SAT with a less redundant distribution unless P = NP . We extend this separation result and define a distributional complexity class C with the following properties: (1) C is a subclass of DistNP, this relation is proper unless P = NP. (2) C contains DistP, but it is not contained in AveP unless DistNP \subseteq AveZPP. (3) C has a ≤ ^p _m -complete set. (4) C has a ≤ ^p _tt -complete set that is not ≤ ^p _m -complete unless P = NP. This shows that under the assumption that P ≠ NP, the two completeness notions differ on some nontrivial subclass of DistNP.