Lifting Non-Finite Axiomatizability Results to Extensions of Process Algebras (original) (raw)
Related papers
Finite equational bases in process algebra: Results and open questions
Processes, Terms and Cycles: Steps on the Road to Infinity, 2005
Van Glabbeek (1990) presented the linear time/branching time spectrum of behavioral equivalences for finitely branching, concrete, sequential processes. He studied these semantics in the setting of the basic process algebra BCCSP, and tried to give finite complete axiomatizations for them. Obtaining such axiomatizations in concurrency theory often turns out to be difficult, even in the setting of simple languages like BCCSP. This has raised a host of open questions that have been the subject of intensive research in recent years. Most of these questions have been settled over BCCSP, either positively by giving a finite complete axiomatization, or negatively by proving that such an axiomatization does not exist. Still some open questions remain. This paper reports on these results, and on the state-of-the-art in axiomatizations for richer process algebras with constructs like sequential and parallel composition.
On the axiomatizability of priority II
Theoretical Computer Science, 2011
This paper studies the equational theory of bisimulation equivalence over the process algebra BCCSP extended with the priority operator of Baeten, Bergstra and Klop. It is proven that, in the presence of an infinite set of actions, bisimulation equivalence has no finite, sound, ground-complete equational axiomatization over that language. This negative result applies even if the syntax is extended with an arbitrary collection of auxiliary operators, and motivates the study of axiomatizations using conditional equations. In the presence of an infinite set of actions, it is shown that, in general, bisimulation equivalence has no finite, sound, ground-complete axiomatization consisting of conditional equations over the language studied in this paper. Finally, sufficient conditions on the priority structure over actions are identified that lead to a finite, ground-complete axiomatization of bisimulation equivalence using conditional equations.
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.
Basic process algebra with iteration: Completeness of its equational axioms
The Computer Journal, 1994
Bergstra, Bethke and Ponse proposed an axiomatization for Basic Process Algebra extended with (binary) iteration. In this paper, we prove that this axiomatization is complete with respect to strong bisimulation equivalence. To obtain this result, we will set up a term rewriting system, based on the axioms, and prove that this term rewriting system is terminating, and that bisimilar normal forms are syntactically equal modulo commutativity and associativity of the +.
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.
A Complete Equational Axiomatization for BPAδε with Prefix Iteration
2016
Prefix iteration µ*x is added to Basic Process Algebra with deadlock and empty process. We present a finite equational axiomatization for this process algebra, and we prove that this axiomatization is complete with respect to strong bisimulation equivalence. This result is a mild generalization of a similar result in the setting of basic CCS in Fokkink (1994b). To obtain this completeness result, we set up a rewrite system, based on the axioms. In order to prove that this rewrite system is terminating modulo AC of the +, we generalize a termination theorem from Zantema and Geser (1994) to the setting of rewriting modulo equations. Finally, we show that bisimilar normal forms are syntactically equal modulo AC of the +.
On the Axiomatizability of Priority
2006
This paper studies the equational theory of bisimulation equivalence over the process algebra BCCSP extended with the priority operator of Baeten, Bergstra and Klop. It is proven that, in the presence of an infinite set of actions, bisimulation equivalence has no finite, sound, ground-complete equational axiomatization over that language. This negative result applies even if the syntax is extended with an arbitrary collection of auxiliary operators, and motivates the study of axiomatizations using conditional equations. In the presence of an infinite set of actions, it is shown that, in general, bisimulation equivalence has no finite, sound, ground-complete axiomatization consisting of conditional equations over BCCSP. Sufficient conditions on the priority structure over actions are identified that lead to a finite, ground-complete axiomatization of bisimulation equivalence using conditional equations.
Axiomatizations for the perpetual loop in process algebra
ICALP, 1997
Milner proposed an axiomatization for the Kleene star in basic process algebra, in the presence of deadlock and empty process, modulo bisimulation equivalence. In this paper, Milner's axioms are adapted to no-exit iteration x^w, which executes x infinitely many times in a row, and it is shown that this axiomatization is complete for no-exit iteration in basic process algebra with deadlock and empty process, modulo bisimulation.
Split-2 Bisimilarity has a Finite Axiomatization over CCS with
Hennessy's Merge
Logical Methods in Computer Science, 2005
This note shows that split-2 bisimulation equivalence (also known as timed equivalence) affords a finite equational axiomatization over the process algebra obtained by adding an auxiliary operation proposed by Hennessy in 1981 to the recursion free fragment of Milner's Calculus of Communicating Systems. Thus the addition of a single binary operation, viz. Hennessy's merge, is sufficient for the finite equational axiomatization of parallel composition modulo this non-interleaving equivalence. This result is in sharp * Partially supported by the Statens Naturvidenskabelige Forskningsråd (Danish Natural Science Research Council), project "The Equational Logic of Parallel Processes", nr. 21-03-0342.