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Finite equational bases in process algebra: Results and open questions
 Processes, Terms and Cycles: Steps on the Road to Infinity, LNCS 3838
, 2005
"... Abstract. 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. ..."
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Abstract. 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 stateoftheart in axiomatizations for richer process algebras with constructs like sequential and parallel composition. 1
A finite basis for failure semantics
 In Proceedings 32nd Colloquium on Automata, Languages and Programming (ICALP’05), Lisbon, LNCS 3580
, 2005
"... Abstract. We present a finite ωcomplete axiomatization for the process algebra BCCSP modulo failure semantics, in case of a finite alphabet. This solves an open question by Groote [12]. 1 ..."
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Abstract. We present a finite ωcomplete axiomatization for the process algebra BCCSP modulo failure semantics, in case of a finite alphabet. This solves an open question by Groote [12]. 1
On finite alphabets and infinite bases: From ready pairs to possible worlds
 In Proceedings 7th Conference on Foundations of Software Science and Computation Structures (FOSSACS’04), Barcelona, LNCS 2987
, 2004
"... Abstract. We prove that if a finite alphabet of actions contains at least two elements, then the equational theory for the process algebra BCCSP modulo any semantics no coarser than readiness equivalence and no finer than possible worlds equivalence does not have a finite basis. This semantic range ..."
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Abstract. We prove that if a finite alphabet of actions contains at least two elements, then the equational theory for the process algebra BCCSP modulo any semantics no coarser than readiness equivalence and no finer than possible worlds equivalence does not have a finite basis. This semantic range includes ready trace equivalence. 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 7 (3 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.
On finite alphabets and infinite bases III: Simulation
 Proc. CONCUR’06, LNCS 4137
, 2006
"... Abstract. This paper studies the (in)equational theory of simulation preorder and equivalence over the process algebra BCCSP. We prove that in the presence of a finite alphabet with at least two actions, the (in)equational theory of BCCSP modulo simulation preorder or equivalence does not have a fin ..."
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Abstract. This paper studies the (in)equational theory of simulation preorder and equivalence over the process algebra BCCSP. We prove that in the presence of a finite alphabet with at least two actions, the (in)equational theory of BCCSP modulo simulation preorder or equivalence does not have a finite basis. In contrast, in the presence of an alphabet that is infinite or a singleton, the equational theory for simulation equivalence does have a finite basis. 1
ON HENK BARENDREGT’S FAVORITE OPEN PROBLEM
"... Abstract. H is the λtheory extending βconversion by identifying all closed unsolvables. A longstanding open problem of H. Barendregt states that the range property holds in H. Here we discuss what we know about the problem. We also make some remarks on the λtheoryHω (the closure ofHunder the ωr ..."
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Abstract. H is the λtheory extending βconversion by identifying all closed unsolvables. A longstanding open problem of H. Barendregt states that the range property holds in H. Here we discuss what we know about the problem. We also make some remarks on the λtheoryHω (the closure ofHunder the ωrule and βconversion). 1.
Under consideration for publication in J. Functional Programming 1 The Lambda Calculus is Algebraic
"... This paper serves as a selfcontained, tutorial introduction to combinatory models of the untyped lambda calculus. We focus particularly on the interpretation of free variables. We argue that free variables should not be interpreted as elements in a model, as is usually done, but as indeterminates. ..."
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This paper serves as a selfcontained, tutorial introduction to combinatory models of the untyped lambda calculus. We focus particularly on the interpretation of free variables. We argue that free variables should not be interpreted as elements in a model, as is usually done, but as indeterminates. We claim that the resulting interpretation is more natural and leads to a closer correspondence between models and theories. In particular, it solves the problem of the notorious ξrule, which asserts that equations should be preserved under binders, and which fails to be sound for the usual interpretation.
THE OMEGA RULE IS Π1 1COMPLETE IN THE λβCALCULUS
, 2008
"... Vol. 5 (2:6) 2009, pp. 1–21 www.lmcsonline.org ..."
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THE OMEGA RULE IS Π1 1COMPLETE IN THE λβCALCULUS
, 903
"... Abstract. In a functional calculus, the so called ωrule states that if two terms P and Q applied to any closed term N return the same value (i.e. PN = QN), then they are equal (i.e. P = Q holds). As it is well known, in the λβcalculus the ωrule does not hold, even when the ηrule (weak extensiona ..."
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Abstract. In a functional calculus, the so called ωrule states that if two terms P and Q applied to any closed term N return the same value (i.e. PN = QN), then they are equal (i.e. P = Q holds). As it is well known, in the λβcalculus the ωrule does not hold, even when the ηrule (weak extensionality) is added to the calculus. A longstanding problem of H. Barendregt (1975) concerns the determination of the logical power of the ωrule when added to the λβcalculus. In this paper we solve the problem, by showing that the resulting theory is Π 1 1Complete.