## Psi-Calculi in Isabelle

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Venue: | In Proc of the 22nd Conference on Theorem Proving in Higher Order Logics (TPHOLs), volume 5674 of LNCS |

Citations: | 13 - 4 self |

### BibTeX

@INPROCEEDINGS{Bengtson_psi-calculiin,

author = {Jesper Bengtson and Joachim Parrow},

title = {Psi-Calculi in Isabelle},

booktitle = {In Proc of the 22nd Conference on Theorem Proving in Higher Order Logics (TPHOLs), volume 5674 of LNCS},

year = {},

pages = {99--114}

}

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### Abstract

Abstract. Psi-calculi are extensions of the pi-calculus, accommodating arbitrary nominal datatypes to represent not only data but also communication channels, assertions and conditions, giving it an expressive power beyond the applied pi-calculus and the concurrent constraint picalculus. We have formalised psi-calculi in the interactive theorem prover Isabelle using its nominal datatype package. One distinctive feature is that the framework needs to treat binding sequences, as opposed to single binders, in an efficient way. While different methods for formalising single binder calculi have been proposed over the last decades, representations for such binding sequences are not very well explored. The main effort in the formalisation is to keep the machine checked proofs as close to their pen-and-paper counterparts as possible. We discuss two approaches to reasoning about binding sequences along with their strengths and weaknesses. We also cover custom induction rules to remove the bulk of manual alpha-conversions. 1

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Citation Context ...er than ingenuity and complicated new constructions they should be amenable to proof mechanisation. Our contribution in this paper is to implement a family of application oriented calculi in Isabelle =-=[12]-=-. The calculi we consider are the so called psi-calculi [5], obtained by extending the basic untyped pi-calculus with the following parameters: (1) a set of data terms, which can function as both2 Je... |

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Citation Context ...channels. The main difficulty in representing calculi such as the lambda-, pi- or psicalculi is to find an efficient treatment of binders. Informal proofs often use the Barendregt variable convention =-=[4]-=-, that everything bound is unique. This convention provides a tractable abstraction when doing proofs involving binders, but it has recently been proven to be unsound in the general case [16]. Theorem... |

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Citation Context ...ry many cases to check and the temptation to resort to formulations such as “by analogy with . . . ” or “is easily seen. . . ” can be overwhelming. For examples in point, both the applied pi-calculus =-=[1]-=- and the concurrent constraint pi-calculus [8] have recently been discovered to have flaws or incompletenesses in the sense that the claimed compositionality results do not hold [5]. Since such proofs... |

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Citation Context ...s, but it has recently been proven to be unsound in the general case [16]. Theorem provers have commonly used approaches based on de Bruijn indices [9], higher order abstract syntax, or nominal logic =-=[13]-=-. We use the nominal datatype package in Isabelle [15], and its strategy for dealing with single binders. Recent work by Aydemir et. al. introduce the locally nameless framework [2] which might be an ... |

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Citation Context ...table abstraction when doing proofs involving binders, but it has recently been proven to be unsound in the general case [16]. Theorem provers have commonly used approaches based on de Bruijn indices =-=[9]-=-, higher order abstract syntax, or nominal logic [13]. We use the nominal datatype package in Isabelle [15], and its strategy for dealing with single binders. Recent work by Aydemir et. al. introduce ... |

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Citation Context ...e general case [16]. Theorem provers have commonly used approaches based on de Bruijn indices [9], higher order abstract syntax, or nominal logic [13]. We use the nominal datatype package in Isabelle =-=[15]-=-, and its strategy for dealing with single binders. Recent work by Aydemir et. al. introduce the locally nameless framework [2] which might be an improvement since the infrastructure is small and eleg... |

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Citation Context ... or nominal logic [13]. We use the nominal datatype package in Isabelle [15], and its strategy for dealing with single binders. Recent work by Aydemir et. al. introduce the locally nameless framework =-=[2]-=- which might be an improvement since the infrastructure is small and elegant. One of our main contributions in the present paper is to extend the strategy to finite sequences of binders. Though it is ... |

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Citation Context ...lculus. We also require an entailment relation, a notion of channel equality and composition of assertions. Isabelle has good support for reasoning about parametric systems through the use of locales =-=[3]-=-. 4.1 Substitution properties We require a substitution function on agents. Since terms, assertions and conditions of psi-calculi are parameters, a locale is created to ensure that a set of substituti... |

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Citation Context ...onvention [4], that everything bound is unique. This convention provides a tractable abstraction when doing proofs involving binders, but it has recently been proven to be unsound in the general case =-=[16]-=-. Theorem provers have commonly used approaches based on de Bruijn indices [9], higher order abstract syntax, or nominal logic [13]. We use the nominal datatype package in Isabelle [15], and its strat... |

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Citation Context ...pplied pi-calculus [1] and the concurrent constraint pi-calculus [8] have recently been discovered to have flaws or incompletenesses in the sense that the claimed compositionality results do not hold =-=[5]-=-. Since such proofs often require stamina and attention to detail rather than ingenuity and complicated new constructions they should be amenable to proof mechanisation. Our contribution in this paper... |

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Citation Context .... We are not aware of any other work on implementing calculi of this calibre in a proof assistant such as Isabelle. The closest related work are implementations of the basic pi-calculus, by ourselves =-=[6]-=- and also by others [10, 11, 14]. Neither are we aware of any other general technique for multiple binders, other than the yet unpublished work by Berghofer and Urban which we describe in Section 3. T... |

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Citation Context ...ny other work on implementing calculi of this calibre in a proof assistant such as Isabelle. The closest related work are implementations of the basic pi-calculus, by ourselves [6] and also by others =-=[10, 11, 14]-=-. Neither are we aware of any other general technique for multiple binders, other than the yet unpublished work by Berghofer and Urban which we describe in Section 3. The rest of the paper is structur... |

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(Show Context)
Citation Context ...esort to formulations such as “by analogy with . . . ” or “is easily seen. . . ” can be overwhelming. For examples in point, both the applied pi-calculus [1] and the concurrent constraint pi-calculus =-=[8]-=- have recently been discovered to have flaws or incompletenesses in the sense that the claimed compositionality results do not hold [5]. Since such proofs often require stamina and attention to detail... |

15 | calculus in (co)inductive type theory
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Citation Context ...ny other work on implementing calculi of this calibre in a proof assistant such as Isabelle. The closest related work are implementations of the basic pi-calculus, by ourselves [5] and also by others =-=[9, 10, 13]-=-. Neither are we aware of any other general technique for multiple binders, other than the yet unpublished work by Berghofer and Urban which we describe in Section 3. The rest of the paper is structur... |

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Citation Context ...ny other work on implementing calculi of this calibre in a proof assistant such as Isabelle. The closest related work are implementations of the basic pi-calculus, by ourselves [6] and also by others =-=[10, 11, 14]-=-. Neither are we aware of any other general technique for multiple binders, other than the yet unpublished work by Berghofer and Urban which we describe in Section 3. The rest of the paper is structur... |

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Citation Context ... are used for case analysis of a transition of a particular structure. The nominal package has native support for creating inversion rules for inductively defined predicates using only single binders =-=[6]-=-, but it turns out that the techniques used there are not directly applicable with binding sequences. We have created the inversion rules we actually need and also started to implement a general one. ... |

8 |
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Citation Context ...ny other work on implementing calculi of this calibre in a proof assistant such as Isabelle. The closest related work are implementations of the basic pi-calculus, by ourselves [5] and also by others =-=[9, 10, 13]-=-. Neither are we aware of any other general technique for multiple binders, other than the yet unpublished work by Berghofer and Urban which we describe in Section 3. The rest of the paper is structur... |

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