## Model checking for π-calculus using proof search (2005)

Venue: | CONCUR, volume 3653 of LNCS |

Citations: | 15 - 5 self |

### BibTeX

@INPROCEEDINGS{Tiu05modelchecking,

author = {Alwen Tiu},

title = {Model checking for π-calculus using proof search},

booktitle = {CONCUR, volume 3653 of LNCS},

year = {2005},

pages = {36--50},

publisher = {Springer}

}

### OpenURL

### Abstract

Abstract. Model checking for transition systems specified in π-calculus has been a difficult problem due to the infinite-branching nature of input prefix, name-restriction and scope extrusion. We propose here an approach to model checking for π-calculus by encoding it into a logic which supports reasoning about bindings and fixed points. This logic, called F Oλ ∆ ∇ , is a conservative extension of Church’s Simple Theory of Types with a “generic ” quantifier. By encoding judgments about transitions in pi-calculus into this logic, various conditions on the scoping of names and restrictions on name instantiations are captured naturally by the quantification theory of the logic. Moreover, standard implementation techniques for (higher-order) logic programming are applicable for implementing proof search for this logic, as illustrated in a prototype implementation discussed in this paper. The use of logic variables and eigenvariables in the implementation allows for exploring the state space of processes in a symbolic way. Compositionality of properties of the transitions is a simple consequence of the meta theory of the logic (i.e., cut elimination). We illustrate the benefits of specifying systems in this logic by studying several specifications of modal logics for pi-calculus. These specifications are also executable directly in the prototype implementation of F Oλ ∆ ∇. 1

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Citation Context ...s a premise for every member of that set: if that set is empty, then the premise is proved.s3 Logical specification of one-step transition We consider the late transition system for the π-calculus in =-=[16]-=-, but we shall follow the operational semantics of π-calculus presented in [21]. The syntax of processes is defined as follows P ::= 0 | ¯xy.P | x(y).P | τ.P | (x)P | [x = y]P | P|P | P + P | !P. We u... |

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Citation Context ...which can potentially be used in the implementation of F Oλ ∆∇ . We also plan to study other process calculi and their related notions of equivalences and modal logics, in particular the spi-calculus =-=[1]-=- and its related notions of bisimulation. Acknowledgment. The author would like to thank the anonynimous referees for useful comments and suggestions. This work is based partly on a joint work of the ... |

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Citation Context ...e is proved.s3 Logical specification of one-step transition We consider the late transition system for the π-calculus in [16], but we shall follow the operational semantics of π-calculus presented in =-=[21]-=-. The syntax of processes is defined as follows P ::= 0 | ¯xy.P | x(y).P | τ.P | (x)P | [x = y]P | P|P | P + P | !P. We use the notation P, Q, R, S and T to denote processes. Names are denoted by lowe... |

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Citation Context .... An automated proof search engine for a fragment of F Oλ ∆∇ has been implemented [24]. It was essentially done by plugging together different existing implementation: higherorder pattern unification =-=[12, 18]-=-, stream-based approach to back-tracking, and parser for λ-terms. On top of this prototype implementation several specifications of process calculi and bisimulation have been implemented. 1 In most ca... |

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Citation Context ...antiation into the more general setting of proof theory for quantifiers in formal logic. More specifically, we encode judgments about transitions in π-calculus and several modal logics for π-calculus =-=[17]-=- into a meta logic, and proof search is used to model the operational semantics of these judgments. This meta logic, called F Oλ ∆∇ [15], is an extension of Church’s Simple Theory of Types (but withou... |

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Citation Context ...tisfiability of modal formulae, cannot be expressed directly in these logics. To encode such notions, it is necessary to move to a richer logic. Recent developments in the proof theory of definitions =-=[10, 11]-=- have shown that mustbehaviour can indeed be captured in logics extended with this proof-theoretical notion of definitions. In a logic with definitions, an atomic proposition may be “defined” by anoth... |

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Citation Context ...F Oλ ∆∇ [15], is an extension of Church’s Simple Theory of Types (but without quantification over propositions, so the logic is essentially first-order) with a proof theoretical notion of definitions =-=[22]-=- and a new “generic” quantifier, ∇. The quantifier ∇, roughly summarized, facilitates reasoning about binders (more details will be given later). We summarize our approach as follows. λ-tree syntax. W... |

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Citation Context ...ansitions in π-calculus and several modal logics for π-calculus [17] into a meta logic, and proof search is used to model the operational semantics of these judgments. This meta logic, called F Oλ ∆∇ =-=[15]-=-, is an extension of Church’s Simple Theory of Types (but without quantification over propositions, so the logic is essentially first-order) with a proof theoretical notion of definitions [22] and a n... |

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Citation Context ...he introduction rule for the judgment A is in effect done by replacing A with B during proof search. This notion of definitions is an extension of work by Schroeder-Heister [22], Eriksson [5], Girard =-=[8]-=-, Stärk [23] and McDowell and Miller [10]. These inference rules for definitions allow for modest reasoning about the fixed points of definitions. ∇R ∀R ∃RsDefinition 1. A definition clause is written... |

35 | A Logical Framework for Reasoning about Logical Specifications
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Citation Context ...is implicit, since X is outside the scope of y and therefore cannot be instantiated with y. The adequacy of our encoding is stated in the following lemma and proposition (their proofs can be found in =-=[25]-=-). Lemma 3. The function [.] is a bijection between α-equivalence classes of expressions. Proposition 4. Let P and Q be processes and α an action. Let ¯n be a list of free names containing the free na... |

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Citation Context ...tisfiability of modal formulae, cannot be expressed directly in these logics. To encode such notions, it is necessary to move to a richer logic. Recent developments in the proof theory of definitions =-=[10, 11]-=- have shown that mustbehaviour can indeed be captured in logics extended with this proof-theoretical notion of definitions. In a logic with definitions, an atomic proposition may be “defined” by anoth... |

28 | A finitary version of the calculus of partial inductive definitions
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Citation Context ...tion of A. The introduction rule for the judgment A is in effect done by replacing A with B during proof search. This notion of definitions is an extension of work by Schroeder-Heister [22], Eriksson =-=[5]-=-, Girard [8], Stärk [23] and McDowell and Miller [10]. These inference rules for definitions allow for modest reasoning about the fixed points of definitions. ∇R ∀R ∃RsDefinition 1. A definition claus... |

27 | Tabled Higher-Order Logic Programming
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Citation Context ... programming. Its use in implementing model checkers has been demonstrated in XSB [27] and also in [20]. Implementation of tabled deduction for higher-order logic programming has also been studied in =-=[19]-=-, which can potentially be used in the implementation of F Oλ ∆∇ . We also plan to study other process calculi and their related notions of equivalences and modal logics, in particular the spi-calculu... |

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Citation Context ...can be downloaded from the author’s website: http://www.loria.fr/˜tiu.sspecification of the operational semantics of late π-calculus in Section 3. The materials in these two sections have appeared in =-=[15, 26]-=-; they are included here since the main results of this paper are built on them. Section 4 presents the specification of modal logics along with the adequacy results. Section 5 gives an overview of a ... |

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Citation Context ...el names. This is a problem which has been studied extensively,sof course, due to the importance of π-calculus. A non-exhaustive list of existing works includes the work on history dependent automata =-=[6]-=- model of mobile processes, specific programming logics and decision procedures for model checking mobile processes [3, 4], the spatial logic model checker [2] using Gabbay-Pitts permutation technique... |

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Citation Context ...ntifier, ∇. The quantifier ∇, roughly summarized, facilitates reasoning about binders (more details will be given later). We summarize our approach as follows. λ-tree syntax. We use the λ-tree syntax =-=[14]-=- to encode syntax with bindings. It is a variant of higher-order abstract syntax, where syntax of arbitrary system is encoded as λ-terms and the λ-abstraction is used to encode bindings within express... |

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Citation Context ...tion rule for the judgment A is in effect done by replacing A with B during proof search. This notion of definitions is an extension of work by Schroeder-Heister [22], Eriksson [5], Girard [8], Stärk =-=[23]-=- and McDowell and Miller [10]. These inference rules for definitions allow for modest reasoning about the fixed points of definitions. ∇R ∀R ∃RsDefinition 1. A definition clause is written ∀¯x[p ¯t △ ... |

6 |
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Citation Context ...xhaustive list of existing works includes the work on history dependent automata [6] model of mobile processes, specific programming logics and decision procedures for model checking mobile processes =-=[3, 4]-=-, the spatial logic model checker [2] using Gabbay-Pitts permutation techniques [7], and implementation using logic programming [27]. The approach to model checking π-caculus (or mobile processes in g... |

6 | A logic encoding of the π-calculus: model checking mobile processes using tabled resolution
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Citation Context ...ogics and decision procedures for model checking mobile processes [3, 4], the spatial logic model checker [2] using Gabbay-Pitts permutation techniques [7], and implementation using logic programming =-=[27]-=-. The approach to model checking π-caculus (or mobile processes in general) taken in this paper is based on the proof theory of sequent calculus, by casting the problem of reasoning about scoping and ... |

3 | Level 0/1 Prover: A tutorial
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Citation Context ...ightforwardly, using only the standard tools and techniques used in higher-order logic programming and theorem provers. An automated proof search engine for a fragment of F Oλ ∆∇ has been implemented =-=[24]-=-. It was essentially done by plugging together different existing implementation: higherorder pattern unification [12, 18], stream-based approach to back-tracking, and parser for λ-terms. On top of th... |

2 |
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Citation Context ...s the work on history dependent automata [6] model of mobile processes, specific programming logics and decision procedures for model checking mobile processes [3, 4], the spatial logic model checker =-=[2]-=- using Gabbay-Pitts permutation techniques [7], and implementation using logic programming [27]. The approach to model checking π-caculus (or mobile processes in general) taken in this paper is based ... |