## On finite alphabets and infinite bases: From ready pairs to possible worlds (2004)

Venue: | In Proceedings 7th Conference on Foundations of Software Science and Computation Structures (FOSSACS’04), Barcelona, LNCS 2987 |

Citations: | 9 - 7 self |

### BibTeX

@INPROCEEDINGS{Fokkink04onfinite,

author = {Wan Fokkink and Sumit Nain},

title = {On finite alphabets and infinite bases: From ready pairs to possible worlds},

booktitle = {In Proceedings 7th Conference on Foundations of Software Science and Computation Structures (FOSSACS’04), Barcelona, LNCS 2987},

year = {2004},

pages = {182--194},

publisher = {Springer}

}

### Years of Citing Articles

### OpenURL

### Abstract

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

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Citation Context ...n whether a finite algebra is finitely based is undecidable. Other notable examples of ω-incomplete axiomatizations in the literature are the λKβη-calculus (see [25]) and the equational theory of CCS =-=[17]-=-. Therefore laws such as commutativity of parallelism, which are valid in the initial model but which cannot be derived, are often added to the latter equational theory. For such extended equational t... |

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Citation Context ... if for each equation t ≈ u with E ⊢ σ(t) ≈ σ(u) for all closed substitutions σ, we have E ⊢ t ≈ u. The core axioms A1-4 [17] for BCCSP(A) below are sound and complete modulo bisimulation equivalence =-=[23]-=-, which is the finest semantics in van Glabbeek’s linear time - branching time spectrum (see Figure 1). A1 x + y ≈ y + x A2 (x + y) + z ≈ x + (y + z) A3 x + x ≈ x A4 x + 0 ≈ x In the remainder of this... |

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Citation Context ...ectrum. Readiness semantics [22, 27] distinguishes a process by its finite traces, where each finite trace is decorated with the set of initial actions at its ultimate state. In ready trace semantics =-=[4, 26]-=-, each finite trace is decorated with the set of initial actions at all its states. Possible worlds semantics [28] distinguishes a process by the deterministic processes that can be “ready simulated” ... |

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Citation Context ...ectrum. Readiness semantics [22, 27] distinguishes a process by its finite traces, where each finite trace is decorated with the set of initial actions at its ultimate state. In ready trace semantics =-=[4, 26]-=-, each finite trace is decorated with the set of initial actions at all its states. Possible worlds semantics [28] distinguishes a process by the deterministic processes that can be “ready simulated” ... |

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Citation Context ...esults regarding finite ω-complete axiomatizations for BCCSP occur in the literature. Moller [19] proved that the finite axiomatization for BCCSP modulo bisimulation equivalence is ω-complete. Groote =-=[9]-=- presented a similar result for completed trace equivalence, for trace equivalence (in case of an alphabet with more than one element), and for readiness and failures equivalence (in case of an infini... |

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Citation Context ... branching time spectrum of behavioral equivalences for finitely branching, concrete, sequential processes. In this paper we focus on three equivalence relations in this spectrum. Readiness semantics =-=[22, 27]-=- distinguishes a process by its finite traces, where each finite trace is decorated with the set of initial actions at its ultimate state. In ready trace semantics [4, 26], each finite trace is decora... |

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Citation Context ...eted trace equivalence, for trace equivalence (in case of an alphabet with more than one element), and for readiness and failures equivalence (in case of an infinite alphabet). Blom, Fokkink and Nain =-=[5]-=- proved that in case of an infinite alphabet, BCCSP modulo ready trace equivalence does not have a finite (sound and complete) axiomatization. Aceto, Fokkink and Ingólfsdóttir [3] proved a similar neg... |

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Citation Context ...ial model but which cannot be derived, are often added to the latter equational theory. For such extended equational theories, ω-completeness results were presented in the setting of CCS [19] and ACP =-=[6]-=-. Another negative result, for basic process algebra with the binary Kleene star, was reported in [2]: semantics no coarser than completed trace equivalence and no finer than ready simulation equivale... |

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Citation Context ...ic topic of study in universal algebra (see, e.g., [16]), dating back to Lyndon [14]. Murskiĭ [21] proved that “almost all” finite algebras (namely all quasi-primal ones) are finitely based, while in =-=[20]-=- he presented an example of a three-element algebra that has no finite basis. Henkin [12] showed that the algebra of naturals with addition and multiplication is finitely based, while Gurevi˘c [10] sh... |

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Citation Context ...skiĭ [21] proved that “almost all” finite algebras (namely all quasi-primal ones) are finitely based, while in [20] he presented an example of a three-element algebra that has no finite basis. Henkin =-=[12]-=- showed that the algebra of naturals with addition and multiplication is finitely based, while Gurevi˘c [10] showed that after adding exponentiation the algebra is no longer finitely based. McKenzie [... |

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Citation Context ...ations can be derived from E. In applications dealing with theorem proving, ω-completeness of the underlying equational theory often facilitates the production of equational derivations; see [13]. In =-=[11]-=- it was argued that ω-completeness is desirable for the partial evaluation of programs. In universal algebra, ω-completeness is referred to as a basis for the equational theory. The existence of finit... |

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Citation Context ...ve, by showing that the general question whether a finite algebra is finitely based is undecidable. Other notable examples of ω-incomplete axiomatizations in the literature are the λKβη-calculus (see =-=[25]-=-) and the equational theory of CCS [17]. Therefore laws such as commutativity of parallelism, which are valid in the initial model but which cannot be derived, are often added to the latter equational... |

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Citation Context ...om, Fokkink and Nain [5] proved that in case of an infinite alphabet, BCCSP modulo ready trace equivalence does not have a finite (sound and complete) axiomatization. Aceto, Fokkink and Ingólfsdóttir =-=[3]-=- proved a similar negative result for 2-nested simulation equivalence, independent of the cardinality of the alphabet. 1 Groote [9] explicitly left open the question of ω-complete axiomatizations for ... |

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Citation Context ...is referred to as a basis for the equational theory. The existence of finite bases for algebras is a classic topic of study in universal algebra (see, e.g., [16]), dating back to Lyndon [14]. Murskiĭ =-=[21]-=- proved that “almost all” finite algebras (namely all quasi-primal ones) are finitely based, while in [20] he presented an example of a three-element algebra that has no finite basis. Henkin [12] show... |

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Citation Context ...ed with the set of initial actions at its ultimate state. In ready trace semantics [4, 26], each finite trace is decorated with the set of initial actions at all its states. Possible worlds semantics =-=[28]-=- distinguishes a process by the deterministic processes that can be “ready simulated” by the original process. In a ready simulation, the sets of initial actions at a simulated and its simulating stat... |

6 |
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Citation Context ...in [20] he presented an example of a three-element algebra that has no finite basis. Henkin [12] showed that the algebra of naturals with addition and multiplication is finitely based, while Gurevi˘c =-=[10]-=- showed that after adding exponentiation the algebra is no longer finitely based. McKenzie [15] settledsTarski’s Finite Basis Problem in the negative, by showing that the general question whether a fi... |