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 Proceedings of CALCO’07, volume 4624 of LNCS
, 2007
"... Abstract. This paper contributes to the study of the equational theory of the semantics in van Glabbeek’s linear time branching time spectrum over the language BCCSP, a basic process algebra for the description of finite synchronization trees. It offers an algorithm for producing a complete (respec ..."
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Abstract. This paper contributes to the study of the equational theory of the semantics in van Glabbeek’s linear time branching time spectrum over the language BCCSP, a basic process algebra for the description of finite synchronization trees. It offers an algorithm for producing a complete (respectively, groundcomplete) equational axiomatization of any behavioral congruence lying between ready simulation equivalence and partial traces equivalence from a complete (respectively, groundcomplete) inequational axiomatization of its underlying precongruence—that is, of the precongruence whose kernel is the equivalence. The algorithm preserves finiteness of the axiomatization when the set of actions is finite. 1
On Finite Alphabets and Infinite Bases
, 2007
"... Van Glabbeek (1990) presented the linear time – branching time spectrum of behavioral semantics. He studied these semantics in the setting of the basic process algebra BCCSP, and gave finite, sound and groundcomplete, axiomatizations for most of these semantics. Groote (1990) proved for some of van ..."
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Cited by 8 (4 self)
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Van Glabbeek (1990) presented the linear time – branching time spectrum of behavioral semantics. He studied these semantics in the setting of the basic process algebra BCCSP, and gave finite, sound and groundcomplete, axiomatizations for most of these semantics. Groote (1990) proved for some of van Glabbeek’s axiomatizations that they are ωcomplete, meaning that an equation can be derived if (and only if) all of its closed instantiations can be derived. In this paper we settle the remaining open questions for all the semantics in the linear time – branching time spectrum, either positively by giving a finite sound and groundcomplete axiomatization that is ωcomplete, or negatively by proving that such a finite basis for the equational theory does not exist. We prove that in case of a finite alphabet with at least two actions, failure semantics affords a finite basis, while for ready simulation, completed simulation, simulation, possible worlds, ready trace, failure trace and ready semantics, such a finite basis does not exist. Completed simulation semantics also lacks a finite basis in case of an infinite alphabet of actions.
Is Observational Congruence on µExpressions Axiomatisable in Equational Horn Logic?
, 2007
"... It is well known that bisimulation on µexpressions cannot be finitely axiomatised in equational logic. Complete axiomatisations such as those of Milner and Bloom/Ésik necessarily involve implicational rules. However, both systems rely on features which go beyond pure equational Horn logic: either t ..."
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Cited by 1 (1 self)
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It is well known that bisimulation on µexpressions cannot be finitely axiomatised in equational logic. Complete axiomatisations such as those of Milner and Bloom/Ésik necessarily involve implicational rules. However, both systems rely on features which go beyond pure equational Horn logic: either the rules are impure by involving nonequational sideconditions, or they are schematically infinitary like the congruence rule which is not Horn. It is an open question whether these complications cannot be avoided in the prooftheoretically and computationally clean and powerful setting of secondorder equational Horn logic. This paper presents a positive and a negative result regarding axiomatisability of observational congruence in equational Horn logic. Firstly, we show how Milner’s impure rule system can be reworked into a pure Horn axiomatisation that is complete for guarded processes. Secondly, we prove that for unguarded processes, both Milner’s and Bloom/Ésik’s axiomatisations are incomplete without the congruence rule, and neither system has a complete extension in rank 1 equational axioms. It remains open whether there are higherrank equational axioms or Horn rules which would render Milner’s or Bloom / Ésik’s axiomatisations complete for unguarded processes.
Is Observational Congruence Axiomatisable in Equational Horn Logic?
"... It is well known that bisimulation on µexpressions cannot be finitely axiomatised in equational logic. Complete axiomatisations such as those of Milner and Bloom / Ésik necessarily involve implicational rules. However, both systems rely on features which go beyond pure equational Horn logic: either ..."
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It is well known that bisimulation on µexpressions cannot be finitely axiomatised in equational logic. Complete axiomatisations such as those of Milner and Bloom / Ésik necessarily involve implicational rules. However, both systems rely on features which go beyond pure equational Horn logic: either the rules are impure by involving nonequational sideconditions, or they are schematically infinitary like the congruence rule which is not Horn. It is an open question whether these complications cannot be avoided in the prooftheoretically and computationally clean and powerful setting of secondorder equational Horn logic. This paper presents a positive and a negative result regarding axiomatisability of observational congruence in equational Horn logic. Firstly, we show how Milner’s impure rule system can be reworked into a pure Horn axiomatisation that is complete for guarded processes. Secondly, we prove that for unguarded processes, both Milner’s and Bloom / Ésik’s axiomatisations are incomplete without the congruence rule, and neither system has a complete extension in rank 1 equational axioms. It remains open whether there are higherrank equational axioms or Horn rules which would render Milner’s or Bloom / Ésik’s axiomatisations complete for unguarded processes.
An axiomatic study of infinite basis
"... Abstract. We present a new technique for studying the waxiomatizability of the semantics in the ltbtspectrum. Although Fokkink et al. have recently solved most of the problems still open, our main goal is to shed light on them, and to simply and unify their proofs. Besides, we will focus on preord ..."
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Abstract. We present a new technique for studying the waxiomatizability of the semantics in the ltbtspectrum. Although Fokkink et al. have recently solved most of the problems still open, our main goal is to shed light on them, and to simply and unify their proofs. Besides, we will focus on preorders (and make a fundamental use of the axiom for ready simulation semantics) instead of equivalences, since they give rise to much simpler proofs. 1