## On the Completeness of the Equations for the Kleene Star in Bisimulation (1996)

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Venue: | In Proceedings AMAST'96, LNCS 1101 |

Citations: | 8 - 3 self |

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@INPROCEEDINGS{Fokkink96onthe,

author = {Wan Fokkink},

title = {On the Completeness of the Equations for the Kleene Star in Bisimulation},

booktitle = {In Proceedings AMAST'96, LNCS 1101},

year = {1996},

pages = {180--194},

publisher = {Springer}

}

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### Abstract

. A classical result from Redko [20] says that there does not exist a complete finite equational axiomatization for the Kleene star modulo trace equivalence. Fokkink and Zantema [13] showed, by means of a term rewriting analysis, that there does exist a complete finite equational axiomatization for the Kleene star up to strong bisimulation equivalence. This paper presents a simpler and shorter completeness proof. Furthermore, the result is extended to open terms, i.e., to !-completeness. Finally, it is shown that the three equations for the Kleene star are all essential for completeness. 1 Introduction Kleene [15] defined a binary operator x y in the context of finite automata, which denotes the iterate of x on y. Intuitively, the expression x y can choose to execute either x, after which it evolves into x y again, or y, after which it terminates. An advantage of the Kleene star is that on the one hand it can express recursion, while on the other hand it can be captured in eq...

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Citation Context ... Q ff \Gamma! P 0 (P Q) Q ff \Gamma! p P Q ff \Gamma! p Q ff \Gamma! Q 0 P Q ff \Gamma! Q 0 Table 1. Action rules for T(BPA (A)) Process terms are considered modulo bisimulation equivalence from Park =-=[18]-=-. Intuitively, process terms are bisimilar if they have the same branching structure. Definition 1. Two processes P and Q are bisimilar, denoted by P $ Q, if there is a symmetric binary relation B on ... |

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Citation Context ...Furthermore, the result is extended to open terms, i.e., to !-completeness. Finally, it is shown that the three equations for the Kleene star are all essential for completeness. 1 Introduction Kleene =-=[15]-=- defined a binary operator x y in the context of finite automata, which denotes the iterate of x on y. Intuitively, the expression x y can choose to execute either x, after which it evolves into x y a... |

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Citation Context ...d meta-principles such as the Recursive Specification Principle from Bergstra and Klop [8]. Kleene formulated several equations for his operator, notably x y = x(x y) + y. Redko [20] (see also Conway =-=[10]-=-) proved that there does not exist a complete finite equational axiomatization for the Kleene star in language theory. We observe that Redko's proof can be transposed to the binary Kleene star in Basi... |

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Citation Context ...l axiomatization does not exist. Milner [16] formulated an axiomatization for BPA together with the deadlock ffi and the empty process ffl, which includes a conditional axiom which stems from Salomaa =-=[21]-=- in the setting of language theory. He asked whether his axiomatization is complete with respect to bisimulation. The proof that is presented here stems from an, up to now unsuccessful, attempt to sol... |

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Citation Context ... handbook, or at an advanced process algebra course. Also, the proof method employed here is a general strategy, which can be applied to other iteration constructs just as well, see [3, 2]. Following =-=[17, 14, 3]-=-, the preliminaries and the completeness proof focus on open terms, so that we obtain not only completeness, but also !-completeness of the axioms. This last result is new. Finally, it is shown that t... |

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Citation Context ...k@phil.ruu.nl Abstract. A classical result from Redko [20] says that there does not exist a complete finite equational axiomatization for the Kleene star modulo trace equivalence. Fokkink and Zantema =-=[13]-=- showed, by means of a term rewriting analysis, that there does exist a complete finite equational axiomatization for the Kleene star up to strong bisimulation equivalence. This paper presents a simpl... |

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Citation Context ...the Kleene Star in Bisimulation Wan Fokkink Utrecht University, Department of Philosophy Heidelberglaan 8, 3584 CS Utrecht, The Netherlands fokkink@phil.ruu.nl Abstract. A classical result from Redko =-=[20]-=- says that there does not exist a complete finite equational axiomatization for the Kleene star modulo trace equivalence. Fokkink and Zantema [13] showed, by means of a term rewriting analysis, that t... |

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Citation Context ... it can express recursion, while on the other hand it can be captured in equational laws. Hence, one does not need meta-principles such as the Recursive Specification Principle from Bergstra and Klop =-=[8]-=-. Kleene formulated several equations for his operator, notably x y = x(x y) + y. Redko [20] (see also Conway [10]) proved that there does not exist a complete finite equational axiomatization for the... |

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Citation Context ... handbook, or at an advanced process algebra course. Also, the proof method employed here is a general strategy, which can be applied to other iteration constructs just as well, see [3, 2]. Following =-=[17, 14, 3]-=-, the preliminaries and the completeness proof focus on open terms, so that we obtain not only completeness, but also !-completeness of the axioms. This last result is new. Finally, it is shown that t... |

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Citation Context ...ent can be dropped. Note that we give operational semantics to open terms, following [17, 14] for process algebra with abstraction, and [3] for process algebra with the prefix iteration operator from =-=[11]-=-, which is a restricted version of the Kleene star. This approach deviates from the standard approach, which prescribes to give operational semantics to closed terms only, and to give meaning to open ... |

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Citation Context ...f either one of the three equations for the Kleene star is removed from the axiomatization. The proof strategy is to find a model for the axioms minus one of the equations for the Kleene star. Sewell =-=[22]-=- proved that if the deadlock ffi is added to BPA , then a complete finite equational axiomatization does not exist. Milner [16] formulated an axiomatization for BPA together with the deadlock ffi and ... |

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Citation Context ... presentation in a handbook, or at an advanced process algebra course. Also, the proof method employed here is a general strategy, which can be applied to other iteration constructs just as well, see =-=[3, 2]-=-. Following [17, 14, 3], the preliminaries and the completeness proof focus on open terms, so that we obtain not only completeness, but also !-completeness of the axioms. This last result is new. Fina... |

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Citation Context ...e 2 contains an axiom system for T(BPA (A)). It consists of the standard axioms A1-5 together with three axioms BKS1-3 for the binary Kleene star. The most advanced axiom BKS3 originates from Troeger =-=[23]-=-. In the sequel, P = Q will mean that this equality can be derived from the axioms. This axiomatization is sound for T(BPA (A)) with respect to bisimulation equivalence, i.e., if P = Q then P $ Q. Sin... |

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Citation Context ...ish two cases: - either p $ r and q $ s, - or there is a substate p 0 of p such that p $ rp 0 and p 0 q $ s. Proof. We prove this lemma from the following facts A and B. Fact B originates from Caucal =-=[6]-=-. A. If pq $ rs and jqjss, then either q $ s, or there is a substate p 0 of p such that p 0 q $ s. Proof. We apply induction on jpj. First, let jpj = 1. Then p a \Gamma! p for some a, so pq a \Gamma! ... |

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