## On the Completeness of the Equations for the Kleene Star in bisimulation

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@MISC{Fokkink_onthe,

author = {Wan Fokkink},

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

year = {}

}

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### Abstract

A classical result from Redko [14] says that there does not exist a complete finite equational axiomatization for the Kleene star modulo trace equivalence. Fokkink and Zantema [9] showed that there does exist a complete finite equational axiomatization for the Kleene star up to strong bisimulation equivalence. Their proof is based on a sophisticated term rewriting analysis. In this paper, we present amuch simpler and shorter completeness proof. Furthermore, we show that the three equations for the Kleene star are all essential for this completeness result.

### Citations

1286 | A structural approach to operational semantics
- Plotkin
- 1981
(Show Context)
Citation Context ... closed process terms in T(BPA (A)), that is, the process terms which do not contain any variables. 2.2 Operational Semantics Table 1 presents an operational semantics for T(BPA (A)) in Plotkin style =-=[19]-=-, where variables are taken to be atomic actions, that is, variable x can execute x and then terminate; the special symbol p represents (successful) termination. ff ff \Gamma! p P ff \Gamma! p P +Q ff... |

655 |
Concurrency and automata on infinite sequences
- Park
- 1981
(Show Context)
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 ... |

376 |
Representation of events in nerve nets and finite automata
- Kleene
- 1951
(Show Context)
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... |

275 |
Regular Algebra and Finite Machines
- Conway
- 1971
(Show Context)
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... |

201 |
Process Algebra
- Baeten, Weijland
- 1990
(Show Context)
Citation Context ...erm rewriting analysis. The completeness proof in [13] is deplorably long and complicated. Therefore, the completeness result itself was presented in the recent handbook chapter of Baeten and Verhoef =-=[6]-=-, but its proof was omitted because it was considered beyond the scope of that paper. Here, a simpler completeness proof is proposed, which is based on induction on the structure of process terms. Thi... |

122 |
A complete inference system for a class of regular behaviours
- Milner
- 1984
(Show Context)
Citation Context ...el 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 the empty process ffl, which includes a conditional axiom which stems from Salomaa [21] in the setting of language theory. He a... |

108 | A congruence theorem for structured operational semantics with predicates and negative premises. Nordic Journal of Computing 2 (2), 274–302. A. Proof of Theorem 9 We show that the problem of deciding whether a universal two-counter machine diverges on inp
- Verhoef
- 1995
(Show Context)
Citation Context ... 0 , then there is a transition S ff \Gamma! S 0 such that R 0 B S 0 , - if R B S and R ff \Gamma! p , then S ff \Gamma! p . The action rules in Table 1 are in the `path' format of Baeten and Verhoef =-=[5]-=-. Hence, bisimulation equivalence is a congruence with respect to all the operators, which means that if P $ P 0 and Q $ Q 0 , then P+Q $ P 0 +Q 0 and PQ $ P 0 Q 0 and P Q $ P 0sQ 0 . See [5] for the ... |

96 |
Decidability of bisimulation equivalence for processes generating context-free languages
- Baeten, Bergstra, et al.
- 1993
(Show Context)
Citation Context ...are normed, which means that they are able to terminate in finitely many transitions. The norm of a process yields the length of the shortest termination trace of this process; this notion stems from =-=[4]-=-. Norm can be defined inductively as follows. jffj = 1 jP +Qj = minfjP j; jQjg jP Qj = jP j + jQj jP Qj = jQj: Note that bisimilar processes have the same norm. Definition 4. P 0 is a derivative of P ... |

86 | Termination of term rewriting by semantic labelling
- Zantema
- 1995
(Show Context)
Citation Context ...where the left-hand side can be obtained from the right-hand side by the elimination of function symbols. Termination is obtained by means of the advanced technique of semantic labelling from Zantema =-=[24]-=-. Hence, each process term is provably equal to a ground normal form, which cannot be reduced by the conditional rewrite system. Finally, a painstaking case analysis learns that if two ground normal f... |

77 |
Two complete axiom systems for the algebra of regular events
- Salomaa
- 1966
(Show Context)
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... |

57 |
A complete axiomatisation for observational congruence of finitestate behaviours
- Milner
- 1989
(Show Context)
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... |

54 | Ntyft/ntyxt rules reduce to ntree rules
- Fokkink, Glabbeek
- 1996
(Show Context)
Citation Context ...Q $ P 0sQ 0 . See [5] for the definition of the path format, and for a proof of this congruence result. Their proof uses the extra assumption that the rules are well-founded; Fokkink and Van Glabbeek =-=[12]-=- showed that this requirement 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 prefi... |

51 |
Process algebra with iteration and nesting
- Bergstra, Bethke, et al.
- 1994
(Show Context)
Citation Context ...amma1 ) for each ns2, and this infinite number of equivalences cannot be expressed in finitely many equations. This reasoning is also valid in BPA modulo trace equivalence. Bergstra, Bethke and Ponse =-=[7]-=- studied BPA modulo bisimulation equivalence, and they suggested a finite equational axiomatization for it. Fokkink and Zantema [13] proved that this axiomatization is complete, by means of a sophisti... |

36 | Concurrency and automata on in nite sequences - Park - 1981 |

33 | Basic process algebra with iteration: Completeness of its equational axioms
- Fokkink, Zantema
(Show Context)
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... |

29 | Axiomatizing prefix iteration with silent steps
- Aceto, Fokkink, et al.
- 1996
(Show Context)
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... |

29 |
On defining relations for the algebra of regular events
- Redko
- 1964
(Show Context)
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... |

27 |
Verification of an alternating bit protocol by means of process algebra
- Bergstra, Klop
- 1986
(Show Context)
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... |

26 | A complete axiomatization for branching bisimulation congruence of finite-state behaviours
- Glabbeek
- 1993
(Show Context)
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... |

26 | Representation of events in nerve nets and nite automata, in: C.E - Kleene - 1956 |

21 | A complete equational axiomatization for prefix iteration
- Fokkink
- 1994
(Show Context)
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 ... |

17 | Bisimulation is not finitely (first order) equationally axiomatisable
- Sewell
- 1994
(Show Context)
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 ... |

16 |
Graphes canoniques de graphes algébriques
- Caucal
- 1990
(Show Context)
Citation Context ... derivative of P if P can evolve into P 0 by zero or more transitions. A derivative P 0 of P is proper if P can evolve into P 0 by one or more transitions. The following lemma, which is due to Caucal =-=[9]-=-, is typical for normed processes. Lemma 5. Let PQ $ RS. By symmetry we may assume jQjsjSj. We can distinguish two cases: - either P $ R and Q $ S, - or there is a proper derivative P 0 of P such that... |

13 | An equational axiomatization for multi-exit iteration
- ACETO, FOKKINK
- 1997
(Show Context)
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... |

5 |
bisimulation is pomset equivalence on a parallel language without explicit internal choice
- Step
- 1993
(Show Context)
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... |

4 | Bisimulation is not nitely ( rst order) equationally axiomatisable - Sewell - 1994 |

3 |
Graphes canoniques et graphes algebriques
- Caucal
- 1990
(Show Context)
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! ... |

3 | Regular algebra and nite machines - Conway - 1971 |

3 | On de ning relations for the algebra of regular events - Redko - 1964 |

2 | Veri cation of an alternating bit protocol by means of process algebra - Bergstra, Klop - 1986 |