Kleene Algebra with Tests and Coq Tools for while Programs

Pous, Damien

doi:10.1007/978-3-642-39634-2_15

Cited by 30 publications

(26 citation statements)

References 29 publications

(72 reference statements)

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“…Finally, the work that is more related to ours is the one of Pous [62]. This work extends the previous work of the author in the automation of Kleene algebra in Coq which was already discussed previously.…”

Section: Related Worksupporting

confidence: 72%

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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Section: Related Worksupporting

confidence: 72%

Deciding Kleene algebra terms equivalence in Coq

Moreira

Pereira

Sousa

2015

Journal of Logical and Algebraic Methods in Programming

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“…For efficiency reasons, they represent state sets as bounded natural numbers. In [30], Pous describes a complete redesign of [4] based on partial derivatives. As corollary of the completeness proof for an axiomatization of KA, he obtains a proof of Kleene's theorem.…”

Section: Related Workmentioning

confidence: 99%

Regular Language Representations in the Constructive Type Theory of Coq

Doczkal¹,

Smolka²

2018

J Autom Reasoning

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We explore the theory of regular language representations in the constructive type theory of Coq. We cover various forms of automata (deterministic, nondeterministic, oneway, two-way), regular expressions, and the logic WS1S. We give translations between all representations, show decidability results, and provide operations for various closure properties. Our results include a constructive decidability proof for the logic WS1S, a constructive refinement of the Myhill-Nerode characterization of regularity, and translations from twoway automata to one-way automata with verified upper bounds for the increase in size. All results are verified with an accompanying Coq development of about 3000 lines.

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“…However, given the large number of syntactic constructs for PDL and the fact that we only need about a dozen of these basic facts, the gains would be marginal at best. One option to merge the two developments might have been the introduction of a boolean flag, along the lines of the formalization underlying [20], signaling the presence or absence of converse in definitions and lemma statements. This could provide for significant sharing, at the cost of some technical overhead and slightly less natural definitions and lemma statements.…”

Section: Remarks On the Formalizationmentioning

confidence: 99%

Completeness and decidability of converse PDL in the constructive type theory of Coq

Doczkal

Bard

2018

Proceedings of the 7th ACM SIGPLAN International Conference on Certified Programs and Proofs - CPP 2018

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The completeness proofs for Propositional Dynamic Logic (PDL) in the literature are non-constructive and usually presented in an informal manner. We obtain a formal and constructive completeness proof for Converse PDL by recasting a completeness proof by Kozen and Parikh into our constructive setting. We base our proof on a Pratt-style decision method for satisfiability constructing finite models for satisfiable formulas and pruning refutations for unsatisfiable formulas. Completeness of Segerberg's axiomatization of PDL is then obtained by translating pruning refutations to derivations in the Hilbert system. We first treat PDL without converse and then extend the proofs to Converse PDL. All results are formalized in Coq/Ssreflect. IntroductionPropositional Dynamic Logic (PDL) [12,15] is a modal logic developed for reasoning about programs with applications for instance in knowledge representation [3]. The modalities of PDL are given by regular programs (i.e., regular expressions with tests) describing binary relations on states. The formula [α]φ is satisfied by some state if all α-reachable states satisfy φ. The language of programs includes a reflexive transitive closure operation (α * ) causing PDL to be non-compact. Converse PDL (CPDL), also defined in [12], extends PDL with a converse operation on programs. Both PDL and CPDL have the small-model property [12] and are EXPTIME-complete [12,21]. Axiomatizations of PDL and CPDL were first given by Segerberg [22] and shown complete independently by Gabbay [13] and Parikh [19].Our main result is a machine-checked constructive proof that for every PDL or CPDL formula φ one can either construct a proof of ¬φ from the respective axioms or a model of *

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Kleene Algebra with Tests and Coq Tools for while Programs

Cited by 30 publications

References 29 publications

Deciding Kleene algebra terms equivalence in Coq

Deciding Kleene algebra terms equivalence in Coq

Regular Language Representations in the Constructive Type Theory of Coq

Completeness and decidability of converse PDL in the constructive type theory of Coq

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