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.     

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.     

On decidability of LTL model checking for process rewrite systems

We establish a decidability boundary of the model checking problem for infinite-state systems defined by Process Rewrite Systems (PRS) or weakly extended Process Rewrite Systems (wPRS), and properties described by basic fragments of action-based Linear Temporal Logic (LTL) with both future and past operators. It is known that the problem for general LTL properties is decidable for Petri nets and for pushdown processes, while it is undecidable for PA processes.We show that the problem is decidable for wPRS if we consider properties defined by LTL formulae with only modalities strict eventually, strict always, and their past counterparts. Moreover, we show that the problem remains undecidable for PA processes even with respect to the LTL fragment with the only modality until or the fragment with modalities next and infinitely often.     

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.     

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].     

Undecidable equivalences for basic parallel processes

The trace equivalence of BPP was shown to be undecidable by Hirshfeld. We show that all the preorders and equivalences except bisimulation in Glabbeek’s linear time-branching time spectrum are undecidable for BPP. The results are obtained by extending Hirshfeld’s encoding of Minsky machines into BPP. We also show that those preorders and equivalences are undecidable even for a restriction of BPP to 2-labels.     

The inclusion problem for simple languages

A deterministic pushdown acceptor is called a simple machine when it is restricted to have only one state, operate in real-time, and accept by empty store. While the existence of an effective procedure for deciding equivalence of languages accepted by these simple machines is well-known, it is shown that this family is powerful enough to have an undecidable inclusion problem. It follows that the inclusion problems for the LL(k) languages and the free monadic recursion schemes that do not use an identity function are also undecidable.     

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].     

Decision problems for pushdown threads

Threads as contained in a thread algebra emerge from the behavioral abstraction from programs in an appropriate program algebra. Threads may make use of services such as stacks, and a thread using a single stack is called a pushdown thread. Equivalence of pushdown threads is shown decidable whereas pushdown thread inclusion is undecidable. This is again an example of a borderline crossing where the equivalence problem is decidable, whereas the inclusion problem is not.     

An Equational Axiomatization for Multi-exit Iteration

This paper presents an equational axiomatization of bisimulation equivalence over the language of Basic Process Algebra (BPA) with multi-exit iteration. Multi-exit iteration is a generalization of the standard binary Kleene star operation that allows for the specification of agents that, up to bisimulation equivalence, are solutions of systems of recursion equations of the form wherenis a positive integer and theP_iand theQ_iare process terms. The addition of multi-exit iteration to BPA yields a more expressive language than that obtained by augmenting BPA with the standard binary Kleene star (BPA*). As a consequence, the proof of completeness of the proposed equational axiomatization for this language, although standard in its general structure, is much more involved than that for BPA*. An expressiveness hierarchy for the family ofk-exit iteration operators proposed by Bergstra, Bethke, and Ponse is also offered.     

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     

Verifying parallel programs with dynamic communication structures

We address the verification problem of networks of communicating pushdown systems modeling communicating parallel programs with procedure calls. Processes in such networks can read the control state of the other processes according to a given communication structure (specifying the observability rights between processes). The reachability problem of such models is undecidable in general. First, we define a class of networks that effectively preserves recognizability (hence, its reachability problem is decidable). Then, we consider networks where the communication structure can change dynamically during the execution according to a phase graph. The reachability problem for these dynamic networks being undecidable in general, we define a subclass for which it becomes decidable. Then, we consider reachability when the switches in the communication structures are bounded. We show that this problem is undecidable even for one switch. We define a natural class of models for which this problem is decidable. This class can be used in the definition of an efficient semi-decision procedure for the analysis of the general model of dynamic networks. Our techniques allowed to find bugs in two versions of a Windows NT Bluetooth driver.     

The extended equivalence problem for a class of non-real-time deterministic pushdown automata

We propose a new class of non-real-time deterministic pushdown automata (dpda's), named dpda's having the weak segmental property (WSP), and show that the equivalence problem is solvable for two dpda's, one of which is in this class. The equivalence checking algorithm to prove this problem is a further extended direct branching algorithm of Tomita; and with the new skipping step combined with the type B replacement of Oyamaguchi, Inagaki and Honda. The algorithm is still relatively simple. The class of dpda's given above is one of the widest known subclasses of proper dpda's (introduced by Ukkonen); with the decidable extended equivalence problem.This work was supported in part by Grants-in-Aid for Scientific Research Nos. 58550240 and 62550259 from the Ministry of Education, Science and Culture, Japan.Presently with the Ome Works, Toshiba Corporation, Ome, Tokyo 198, Japan.     

On subsumption and semiunification in feature algebras

We consider a generalization of term subsumption, or matching, to a class of mathematical structures which we call feature algebras. We show how these generalize both first-order terms and the feature structures used in computational linguistics. The notion of term subsumption generalizes to a natural notion of algebra homomorphism. In the setting of feature algebras, unification, corresponds naturally to solving constraints involving equalities between strings of unary function symbols, and semiunification also allows inequalities representing subsumption constraints. Our generalization allows us to show that the semiunification problem for finite feature algebras is undecidable. This implies that the corresponding problem for rational trees (cyclic terms) is also undecidable.     

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 results on subclass containment problems for special classes of DPDA'S related to nonsingular machines

A content-free language is said to be weakly (w-)nonsingular if it is accepted by a nonsingular deterministic pushdown automaton in the sense of Oyamaguchi, Inagaki and Honda (1980). It is undecidable whether a deterministic pushdown automaton (dpda) accepts a w-nonsingular language and whether a dpda is nonsingular in the sense of Valiant (1973). Next, the class of super-nonsingular dpda's (which is a subclass of w-nonsingular dpda's) is introduced. It is decidable whether a dpda is super-nonsingular and whether a dpda accepts a super-nonsingular language. As a consequence, the problem of deciding whether a dpda accepts an LL(k) language reduces to the problem of deciding whether a super-nonsingular dpda accepts an LL(k) language.     

The equivalence problem for two dpda's,one of which is a finite-turn or one-counter machine

For some non-real-time subclasses $\text{C}$ of deterministic pushdown automata (dpda), we give a general scheme to extend a decision procedure of the equivalence to that for two dpda's, one of which is in $\text{C}$. Using this scheme, we prove that the equivalence problem is decidable for two dpda's, one of which is a finite-turn or one-counter machine.     

On First-Order Logic and CPDA Graphs

Higher-order pushdown automata (n-PDA) are abstract machines equipped with a nested ‘stack of stacks of stacks’. Collapsible pushdown automata (n-CPDA) extend these devices by adding ‘links’ to the stack and are equi-expressive for tree generation with simply typed λY terms. Whilst the configuration graphs of HOPDA are well understood, relatively little is known about the CPDA graphs. The order-2 CPDA graphs already have undecidable MSO theories but it was only recently shown by Kartzow (Log. Methods Comput. Sci. 9(1), 2013) that first-order logic is decidable at the second level. In this paper we show the surprising result that first-order logic ceases to be decidable at order-3 and above. We delimit the fragments of the decision problem to which our undecidability result applies in terms of quantifer alternation and the orders of CPDA links used. Additionally we exhibit a natural sub-hierarchy enjoying limited decidability.     

Bisimulation equivalence of a BPP and a finite-state system can be decided in polynomial time

In this paper we consider the problem of deciding bisimulation equivalence of a BPP and a finite-state system. We show that the problem can be solved in polynomial time and we present an algorithm deciding the problem in time O(n⁴). The algorithm also constructs for each state of the finite-state system a ‘symbolic’ semilinear representation of the set of all states of the BPP system which are bisimilar with this state.