Undecidable equivalences for basic parallel processes (original) (raw)
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Bisimulation equivalence is decidable for basic parallel processes
Lecture Notes in Computer Science, 1993
In a previous paper the authors proved the decidability of bisimulation equivalence over two subclasses of recurslve processes involving a parallel composition operator, namely the so-caUed norrned and live processes. In this paper, we extend this result to the whole class. The decidability proof permits us further to present a complete axiomatisation for this class of basic parallel processes. This result can be viewed as a proper extension of Miiner's complete axiomatisation of bisimulation equivalence on regular processes.
The Complexity of Deciding Behavioural Equivalences and Preorders
1996
We give an overview of the computational complexity of all the equivalences in van Glabbeek's linear/branching time hierarchy and the preorders in the corresponding hierarchy of preorders. We consider nite state or regular processes as well as in nite-state BPA and BPP processes. A distinction, which turns out to be important in the nite-state processes, is that of simulationlike equivalences/preorders vs. trace-like equivalences and preorders. Here we survey various known complexity results for these relations. For regular processes, all simulation-like equivalences and preorders are decidable in polynomial time whereas all trace-like equivalences and preorders are PSPACE-Complete. We also consider interesting special classes of regular processes such as deterministic, determinate, unary, locally unary, and tree-like processes and survey the known complexity results in these special cases. For in nite-state processes the results are quite di erent. For the class of context-free processes or BPA processes any preorder or equivalence beyond bisimulation is undecidable but bisimulation equivalence is polynomial time decidable for normed BPA processes and is known to be elementarily decidable in the general case. For the class of BPP processes, all preorders and equivalences apart from bisimilarity are undecidable. However, bisimilarity is decidable in this case and is known to be decidable in polynomial time for normed BPP processes.
Mathematical Structures in Computer Science, 1998
Fokkink and Zantema ((1994) Computer Journal 37:259-267) have shown that bisimulation equivalence has a finite equational axiomatization over the language of Basic Process Algebra with the binary Kleene star operation (BPA * ). In the light of this positive result on the mathematical tractability of bisimulation equivalence over BPA * , a natural question to ask is whether any other (pre)congruence relation in van Glabbeek's linear time/branching time spectrum is finitely (in)equationally axiomatizable over it. In this paper, we prove that, unlike bisimulation equivalence, none of the preorders and equivalences in van Glabbeek's linear time/branching time spectrum, whose discriminating power lies in between that of ready simulation and that of completed traces, has a finite equational axiomatization. This we achieve by exhibiting a family of (in)equivalences that holds in ready simulation semantics, the finest semantics that we consider, whose instances cannot all be proven by means of any finite set of (in)equations that is sound in completed trace semantics, which is the coarsest semantics that is appropriate for the language BPA * . To this end, for every finite collection of (in)equations that are sound in completed trace semantics, we build a model in which some of the (in)equivalences of the family under consideration fail. The construction of the model mimics the one used by Conway ((1971) Regular Algebra and Finite Machines, page 105) in his proof of a result, originally due to Redko, to the effect that infinitely many equations are needed to axiomatize equality of regular expressions.
Decidability of bisimulation equivalence for process generating context-free languages
Journal of the ACM, 1993
A context-free grammar (CFG) in Greibach Normal Form coincides, in another notation, with a system of guarded recursion equations in Basic Process Algebra. Hence to each CFG a process can be assigned as solution, which has as its set of finite traces the context-free language (CFL) determined by that CFG. While the equality problem for CFL's is unsolvable, the equality problem for the processes determined by CFG's turns out to be solvable. Here equality on processes is given by a model of process graphs modulo bisimulation equivalence. The proof is given by displaying a periodic structure of the process graphs determined by CFG's. As a corollary of the periodicity a short proof of the solvability of the equivalence problem for simple context-free languages is given.
On the decidability of process equivalences for the π-calculus
Theoretical Computer Science, 1997
We present general results for showing process equivalences applied to the finite control fragment of the z-calculus decidable. Firstly, a Finite Reachability Theorem states that up to finite name spaces and up to a static normalisation procedure, the set of reachable agent expressions is finite. Secondly, a Boundedness Lemma shows that no potential computations are missed when name spaces are chosen large enough, but finite. We show how these results lead to decidability for a number of n-calculus equivalences such as strong or weak, late or early bismulation equivalence. Furthermore, for strong late equivalence we show how our techniques can be used to adapt the well-known PaigeTarjan algorithm. Strikingly, this results in a single exponential running time not much worse than the running time for the case of for instance CCS. Our results considerably strengthens previous results on decidable equivalences for parameter-passing process calculi.
On the Complexity of Deciding Behavioural Equivalences and Preorders. A Survey
BRICS Report Series, 1996
This paper gives an overview of the computational complexity of all the equivalences in the linear/branching time hierarchy [vG90a] and the preorders<br />in the corresponding hierarchy of preorders. We consider finite state or regular processes as well as infinite-state BPA [BK84b] processes. <br />A distinction, which turns out to be important in the finite-state processes, is that of simulation-like equivalences/preorders vs. trace-like equivalences<br />and preorders. Here we survey various known complexity results for these relations. For regular processes, all simulation-like equivalences and preorders are decidable in polynomial time whereas all trace-like equivalences and preorders are PSPACE-Complete. We also consider interesting special<br />classes of regular processes such as deterministic, determinate, unary, locally unary, and tree-like processes and survey the known complexity results in<br />these special cases. For infinite-state proces...
Decidability of bisimulation equivalences for parallel timer processes
Lecture Notes in Computer Science, 1993
In this paper an abstract model of parallel timer processes (PTPs), allowing specification of temporal quantitative constraints on the behaviour of real time systems, is introduced. The parallel timer processes are defined in a dense time domain and are able to model both concurrent (with delay intervals overlapping on the time axis) and infinite behaviour. Both the strong and weak (abstracted from internal actions) bisimulation equivalence problems for PTPs are proved decidable. It is proved also that, if one provides the PTP model additionally with memory cells for moving timer value information along the time axis, the bisimulation equivalence (and even the vertex teachability) problems become undecidable.
On the Complexity of Semantic Equivalences for Pushdown Automata and BPA
Lecture Notes in Computer Science, 2002
We study the complexity of comparing pushdown automata (PDA) and context-free processes (BPA) to finite-state systems, w.r.t. strong and weak simulation preorder/equivalence and strong and weak bisimulation equivalence. We present a complete picture of the complexity of all these problems. In particular, we show that strong and weak simulation preorder (and hence simulation equivalence) is EXPTIME-complete between PDA/BPA and finite-state systems in both directions. For PDA the lower bound even holds if the finite-state system is fixed, while simulation-checking between BPA and any fixed finitestate system is already polynomial. Furthermore, we show that weak (and strong) bisimilarity between PDA and finite-state systems is PSPACE-complete, while strong (and weak) bisimilarity between two PDAs is EXPTIME-hard.
Information and Computation, 2010
Simulation preorder/equivalence and bisimulation equivalence are the most commonly used equivalences in concurrency theory. Their standard definitions are often called strong simulation/bisimulation, while weak simulation/bisimulation abstracts from internal τ-actions. We study the computational complexity of checking these strong and weak semantic preorders/equivalences between pushdown processes and finite-state processes. We present a complete picture of the computational complexity of these problems and also study fixed-parameter tractability in two important input parameters: x, the size of the finite control of the pushdown process, and y, the size of the finite-state process. All simulation problems are generally EXPTIME-complete and only become polynomial if both parameters x and y are fixed. Weak bisimulation equivalence is PSPACE-complete, but becomes polynomial if and only if parameter x is fixed. Strong bisimulation equivalence is PSPACE-complete, but becomes polynomial if either parameter x or y is fixed.