Deciding Regular Expressions (In-)Equivalence in Coq

Moreira, Nelma; Pereira, David; Sousa, Simão Melo de

doi:10.1007/978-3-642-33314-9_7

Cited by 12 publications

(13 citation statements)

References 20 publications

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“…Recently, various algorithms based on Brzozowski derivatives and bisimulations have been implemented in Isabelle [19] and formalized in type theory, yielding an implementation in Coq [8] (while [8] does not mention bisimulations explicitly, their method is based on constructing a bisimulation). Moreover there is another Coq implementation of regular expression equivalence based on partial derivatives [24]. An efficient algorithm for deciding equivalence in Kleene algebra, based on automata but not on derivatives and bisimulations, was recently implemented in Coq as well [4].…”

Section: Discussion and Related Workmentioning

confidence: 99%

Proving language inclusion and equivalence by coinduction

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Bonsangue

Rütten

2016

Information and Computation

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Language equivalence and inclusion can be checked coinductively by establishing a (bi)simulation on suitable deterministic automata. In this paper we present an enhancement of this technique called (bi)simulation-up-to. We give general conditions on language operations for which bisimulation-up-to is sound. These results are illustrated by a large number of examples, giving new proofs of classical results such as Arden's rule, and involving the regular operations of union, concatenation and Kleene star as well as language equations with complement and intersection, and shuffle (closure).

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Section: Discussion and Related Workmentioning

confidence: 99%

Proving language inclusion and equivalence by coinduction

Rot

Bonsangue

Rütten

2016

Information and Computation

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“…Some of these simplifications can be spectacular: for instance, they need one page to justify the passage between their expressions (24) and (27), while a simple call to hkat does the job; similarly for the page they need between their steps (38) and (43).…”

Section: Flowchart Schemesmentioning

confidence: 99%

Kleene Algebra with Tests and Coq Tools for while Programs

Pous

2013

Lecture Notes in Computer Science

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Abstract. We present a Coq library about Kleene algebra with tests, including a proof of their completeness over the appropriate notion of languages, a decision procedure for their equational theory, and tools for exploiting hypotheses of a certain kind in such a theory. Kleene algebra with tests make it possible to represent if-then-else statements and while loops in most imperative programming languages. They were actually introduced by Kozen as an alternative to propositional Hoare logic. We show how to exploit the corresponding Coq tools in the context of program verification by proving equivalences of while programs, correctness of some standard compiler optimisations, Hoare rules for partial correctness, and a particularly challenging equivalence of flowchart schemes.

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“…In Section 3 we review some of the concepts of formal languages that we need to formalize in order to implement the decision procedure; in Section 4 we describe the formalization of the decision procedure, its proofs of correctness and completeness, and comment on the procedure's computational efficiency; in Section 5 we describe the generalization of the decision procedure to decide KAT terms equivalence, and show how this procedure is useful in program verification; finally, in Section 6 we present our conclusions about the work presented in this paper, and point to future research directions. The work presented here is an extended version of the work previously presented in [30,31], and the corresponding development in Coq is available at [32].…”

Section: Paper Organizationmentioning

confidence: 99%

Deciding Kleene algebra terms equivalence in Coq

Moreira

Pereira

Sousa

2015

Journal of Logical and Algebraic Methods in Programming

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a b s t r a c t Keywords:Proof assistants Regular expressions Kleene algebra with tests Program verification This paper presents a mechanically verified implementation of an algorithm for deciding the equivalence of Kleene algebra terms within the Coq proof assistant. The algorithm decides equivalence of two given regular expressions through an iterated process of testing the equivalence of their partial derivatives and does not require the construction of the corresponding automata. Recent theoretical and experimental research provides evidence that this method is, on average, more efficient than the classical methods based on automata. We present some performance tests, comparisons with similar approaches, and also introduce a generalization of the algorithm to decide the equivalence of terms of Kleene algebra with tests. The motivation for the work presented in this paper is that of using the libraries developed as trusted frameworks for carrying out certified program verification.

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Deciding Regular Expressions (In-)Equivalence in Coq

Cited by 12 publications

References 20 publications

Proving language inclusion and equivalence by coinduction

Proving language inclusion and equivalence by coinduction

Kleene Algebra with Tests and Coq Tools for while Programs

Deciding Kleene algebra terms equivalence in Coq

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