Undecidable equivalences for basic parallel processes (original) (raw)
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.
Deciding Bisimulation-Like Equivalences with Finite-State Processes
1999
We show that characteristic formulae for nite-state systems up to bisimulation-like equivalences (e.g., strong and weak bisimilarity) can be given in the simple branching-time temporal logic EF. Since EF is a very weak fragment of the modal µ-calculus, model checking with EF is decidable for many more classes of infinite-state systems. This yields a general method for proving decidability of bisimulation-like equivalences between infinite-state processes and finite-state ones. We apply this method to the class of PAD processes, which strictly subsumes PA and pushdown (PDA) processes, showing that a large class of bisimulation-like equivalences (including, e.g., strong and weak bisimilarity) is decidable between PAD and finite-state processes. On the other hand, we also demonstrate that no `reasonable' bisimulation-like equivalence is decidable between state-extended PA processes and finite-state ones. Furthermore, weak bisimilarity with finite-state processes is shown to be und...
Comparing expressibility of normed BPA and normed BPP processes
Acta Informatica, 1999
We present an exact characterization of those transition systems which can be equivalently (up to bisimilarity) defined by the syntax of normed BPA τ and normed BPP τ processes. We give such a characterization for the subclasses of normed BPA and normed BPP processes as well.
Process Algebra as Abstract Data Types
arXiv (Cornell University), 2010
In this paper we introduced an algebraic semantics for process algebra in form of abstract data types. For that purpose, we developed a particular type of Σ algebra, the seed algebra, which describes exactly the behavior of a process within a labeled transition system. We have shown the possibility of characterizing the bisimulation of two processes with the isomorphism of their corresponding seed algebras. We pointed out that the traditional concept of isomorphism of algebra does not apply here, because there is even no one-one correspondence between the elements of two seed algebras. The lack of this one-one correspondence comes from the non-deterministic choice of transitions of a process. We introduce a technique of hidden operations to mask unwanted details of elements of a seed algebra, which only reflect non-determinism or other implicit control mechanism of process transition. Elements of a seed algebra are considered as indistinguishable if they show the same behavior after these unwanted details are masked. Each class of indistinguishable elements is called a non-hidden closure. We proved that bisimulation of two processes is equivalent to isomorphism of non-hidden closures of two seed algebras representing these two processes. We call this kind of isomorphism a deep isomorphism. We get different models of seed algebra by specifying different axiom systems for the same signature. Each model corresponds to a different kind of bisimulation. By proving the relations between these models we also established relations between 10 different bisimulations, which form a acyclic directed graph.
On the complexity of deciding behavioural equivalences and preorders
1996
Abstract This paper gives an overview of the computational complexity of all the equivalences in the linear/branching time hierarchy vG90a] and the preorders in the corresponding hierarchy of preorders. We consider nite state or regular processes as well as in nite-state BPA BK84b] processes. A distinction, which turns out to be important in the nite-state processes, is that of simulation-like equivalences/preorders vs. trace-like equivalences and preorders. Here we survey various known complexity results for these relations.
Undecidability of 2-Label BPP Equivalences and Behavioral Type Systems for the π-Calculus
Lecture Notes in Computer Science
The trace equivalence of BPP was shown to be undecidable by Hirshfeld. We show that the trace equivalence remains undecidable even if the number of labels is restricted to two. The undecidability result holds also for the simulation of two-label BPP processes. These results imply undecidability of some behavioral type systems for the π-calculus.