## Bisimilarity of Open Terms (2000)

Citations: | 21 - 0 self |

### BibTeX

@MISC{Rensink00bisimilarityof,

author = {Arend Rensink},

title = { Bisimilarity of Open Terms },

year = {2000}

}

### Years of Citing Articles

### OpenURL

### Abstract

Traditionally, in process calculi, relations over open terms, i.e., terms with free process variables, are defined as extensions of closed-term relations: two open terms are related if and only if all their closed instantiations are related. Working in the context of bisimulation, in this paper we study a different approach; we define semantic models for open terms, so-called conditional transition systems, and define bisimulation directly on those models. It turns out that this can be done in at least two different ways, one giving rise to De Simone's formal hypothesis bisimilarity and the other to a variation which we call hypothesis-preserving bisimilarity (denoted t fh and t hp, respectively). For open terms, we have (strict) inclusions t fh /t hp / t ci (the latter denoting the standard ``closed instance' ' extension); for closed terms, the three coincide. Each of these relations is a congruence in the usual sense. We also give an alternative characterisation of t hp in terms of nonconditional transitions, as substitution-closed bisimilarity (denoted t sb). Finally, we study the issue of recursion congruence: we prove that each of the above relations is a congruence with respect to the recursion operator; however, for t ci this result holds under more restrictive conditions than for tfh and thp.]

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Citation Context ... to repair the perceived limitation of standard SOS mentioned above. For that purpose, we adopt a technique from type systems (cf. [27,6]), which ultimately goes back to sequents in formal logic (cf. =-=[11]-=-): we make assumptions or hypotheses about the free variables in a term, and state the existence of transitions under such hypotheses. That is, our basic predicates will not be simple transitions anym... |

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Citation Context ... sense, the behaviour described by one transition system can be said to implement the behaviour described by another. An overview of such implementation relations has been drawn up by Van Glabbeek in =-=[12,14]-=-. The criteria for implementation relations generally include at least reflexivity (so every system implements itself) and transitivity (so implementation steps can be combined), meaning that they are... |

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Citation Context ...rise to sets of operators. 2.2 Conditional transitions We now set out to repair the perceived limitation of standard SOS mentioned above. For that purpose, we adopt a technique from type systems (cf. =-=[27,6]-=-), which ultimately goes back to sequents in formal logic (cf. [11]): we make assumptions or hypotheses about the free variables in a term, and state the existence of transitions under such hypotheses... |

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Citation Context ...ty, �� fh and �� hp , are always preserved by recursion. Section 6 contains a comparison of this paper's results with, among others, the work on SOS formats by De Simone [7], Bloom, Istrail an=-=d Meyer [4]-=- and Groote and Vaandrager [18], on contexts by Larsen and Xinxin [22], on bisimilarity in functional programming by Howe and others [20,16,32] and in higher-order calculi by Sangiorgi [33]. Due to la... |

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Citation Context ...f bisimilarity over conditional transition systems. Apart from closed-instance bisimilarity (�� ci ), which is obtained through the standard approach outlined above, we apply an idea due to De Sim=-=one [7] to defi-=-ne two weaker notions: formal hypothesis and hypothesis preservingsbisimilarity (respectively �� fh and �� hp ). (In the more restricted setting of [7], these notions coincide.) We also give a... |

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Citation Context ...the completeness of the resulting (conditional) transition systems, and secondly, the congruence of the various bisimulation relations with respect to the resulting binders. A starting point could be =-=[9]-=-, which considers binder rules, in a more general setting than ours, in order to study conservativity of signature extensions. ffl It might be useful to apply the idea of conditional transitions to th... |

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Citation Context ...index lists the assumptions made in the past, and is augmented each time a match is made. (A similar setup exists for several other versions of bisimularity, such as symbolic [19], history-preserving =-=[15]-=- and locality [28].) Definition 4.4 Let T be a conditional transition system. ffl A hypotheses-preserving (hp-) bisimulation relation over T is a family of symmetrical relations (R \Gamma ) \Gamma2E `... |

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Citation Context ...plications) with meta-variables rather than conditional transitions with term variables. In fact, this was the original formulation in [7], and has also been studied by Aceto, Bloom and Vaandrager in =-=[1]-=- (who go into the issue of equational proof systems for this relation). As discussed in Section 2, this obviates the possibility to reason about open terms (in the classical sense) and binders like th... |

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Citation Context ...substitution, and moreover, y is not allowed to be free in u, since such free occurrences would be bound in the resulting term. These problems are well documented in the context of the -calculus; see =-=[3]-=-.) Two typical substitution-based rules are: t v u v[t=x] v v[u=x] t v u t[v=x] v u[v=x] The rule on the left expresses that the relation v is preserved by insertion, which is quite close to the prope... |

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