A full formalisation of π-calculus theory in the calculus of constructions

Hirschkoff, Daniel

doi:10.1007/bfb0028392

Cited by 50 publications

(36 citation statements)

References 8 publications

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“…However, these properties do not provide an intuitive mathematical framework. In [24], Daniel Hirschkoff formalised a subset of the π-calculus excluding sum, match and mismatch in Coq using de Bruijn indices. The theories formalised was that early bisimulation is a congruence as well as the structural congruence results.…”

Section: Results and Conclusionmentioning

confidence: 99%

Formalising the pi-calculus using nominal logic

Bengtson¹,

Parrow²

2009

Logical Methods in Computer Science

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Abstract. We formalise the pi-calculus using the nominal datatype package, based on ideas from the nominal logic by Pitts et al., and demonstrate an implementation in Isabelle/HOL. The purpose is to derive powerful induction rules for the semantics in order to conduct machine checkable proofs, closely following the intuitive arguments found in manual proofs. In this way we have covered many of the standard theorems of bisimulation equivalence and congruence, both late and early, and both strong and weak in a uniform manner. We thus provide one of the most extensive formalisations of a process calculus ever done inside a theorem prover.A significant gain in our formulation is that agents are identified up to alpha-equivalence, thereby greatly reducing the arguments about bound names. This is a normal strategy for manual proofs about the pi-calculus, but that kind of hand waving has previously been difficult to incorporate smoothly in an interactive theorem prover. We show how the nominal logic formalism and its support in Isabelle accomplishes this and thus significantly reduces the tedium of conducting completely formal proofs. This improves on previous work using weak higher order abstract syntax since we do not need extra assumptions to filter out exotic terms and can keep all arguments within a familiar first-order logic.

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Section: Results and Conclusionmentioning

confidence: 99%

Formalising the pi-calculus using nominal logic

Bengtson¹,

Parrow²

2009

Logical Methods in Computer Science

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“…Aspects of nominal reasoning have been incorporated into the proof assistant Isabelle [Urban and Tasson 2005] and there has been some recent work in formalizing the meta theory of the π-calculus in this framework [Bengtson and Parrow 2007]. Hirschkoff [Hirschkoff 1997] also used Coq but employed deBruijn numbers [Bruijn 1972] instead of explicit names. In the papers that address bisimulation, formalizing names and their scopes, occurrences, freshness, and substitution is considerable work.…”

Section: Related and Future Workmentioning

confidence: 99%

Proof search specifications of bisimulation and modal logics for the π-calculus

Tiu

Miller

2010

ACM Trans. Comput. Logic

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We specify the operational semantics and bisimulation relations for the finite π-calculus within a logic that contains the ∇ quantifier for encoding generic judgments and definitions for encoding fixed points. Since we restrict to the finite case, the ability of the logic to unfold fixed points allows this logic to be complete for both the inductive nature of operational semantics and the coinductive nature of bisimulation. The ∇ quantifier helps with the delicate issues surrounding the scope of variables within π-calculus expressions and their executions (proofs). We shall illustrate several merits of the logical specifications permitted by this logic: they are natural and declarative; they contain no side-conditions concerning names of variables while maintaining a completely formal treatment of such variables; differences between late and open bisimulation relations arise from familar logic distinctions; the interplay between the three quantifiers (∀, ∃, and ∇) and their scopes can explain the differences between early and late bisimulation and between various modal operators based on bound input and output actions; and proof search involving the application of inference rules, unification, and backtracking can provide complete proof systems for one-step transitions, bisimulation, and satisfaction in modal logic. We also illustrate how one can encode the π-calculus with replications, in an extended logic with induction and co-induction.

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“…We can say, therefore, that locks subsume different proof attitudes, such as "proof-irrelevant" approaches, where one is interested only in knowing that some evidence does exist, as well as approaches relying on powerful terminating metalanguages. Indeed, locks allow for a straightforward accommodation of many different proof cultures within a single Logical Framework, which can otherwise be embedded only very deeply (Boulton et al 1992;Hirschkoff 1997) or axiomatically (Honsell et al 2001).…”

Section: Introductionmentioning

confidence: 99%

Plugging-in proof development environments usingLocksin`LF`

Honsell¹,

Liquori²,

Maksimovic

et al. 2018

Math. Struct. Comp. Sci.

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We present two extensions of the LF Constructive Type Theory featuring monadic locks. A lock is a monadic type construct that captures the effect of an external call to an oracle. Such calls are the basic tool for plugging-in, i.e. gluing together, different Type Theories and proof development environments. The oracle can be invoked either to check that a constraint holds or to provide a suitable witness. The systems are presented in the canonical style developed by the "CMU School". The first system, CLLFP , is the canonical version of the system LLFP , presented earlier by the authors. The second system, CLLF P? , features the possibility of invoking the oracle to obtain also a witness satisfying a given constraint. We discuss encodings of Fitch-Prawitz Set theory, call-by-value λ-calculi, systems of Light Linear Logic, and partial functions.

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A full formalisation of π-calculus theory in the calculus of constructions

Cited by 50 publications

References 8 publications

Formalising the pi-calculus using nominal logic

Formalising the pi-calculus using nominal logic

Proof search specifications of bisimulation and modal logics for the π-calculus

Plugging-in proof development environments usingLocksin`LF`

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A full formalisation of π-calculus theory in the calculus of constructions

Cited by 50 publications

References 8 publications

Formalising the pi-calculus using nominal logic

Formalising the pi-calculus using nominal logic

Proof search specifications of bisimulation and modal logics for the π-calculus

Plugging-in proof development environments usingLocksinLF

Contact Info

Product

Resources

About

Plugging-in proof development environments usingLocksin`LF`