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An Efficient Coq Tactic for Deciding Kleene Algebras
, 2009
"... We present a reflexive tactic for deciding the equational theory of Kleene algebras in the Coq proof assistant. This tactic relies on a careful implementation of efficient finite automata algorithms, so that it solves casual equations almost instantaneously. The corresponding decision procedure was ..."
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We present a reflexive tactic for deciding the equational theory of Kleene algebras in the Coq proof assistant. This tactic relies on a careful implementation of efficient finite automata algorithms, so that it solves casual equations almost instantaneously. The corresponding decision procedure was proved correct and complete; correctness is established w.r.t. any model (including binary relations), by formalising Kozen’s initiality theorem.
Untyping Typed Algebraic Structures and Colouring Proof Nets of Cyclic Linear Logic
 COMPUTER SCIENCE LOGIC, CZECH REPUBLIC
, 2010
"... We prove “untyping” theorems: in some typed theories (semirings, Kleene algebras, residuated lattices, involutive residuated lattices), typed equations can be derived from the underlying untyped equations. As a consequence, the corresponding untyped decision procedures can be extended for free to th ..."
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We prove “untyping” theorems: in some typed theories (semirings, Kleene algebras, residuated lattices, involutive residuated lattices), typed equations can be derived from the underlying untyped equations. As a consequence, the corresponding untyped decision procedures can be extended for free to the typed settings. Some of these theorems are obtained via a detour through fragments of cyclic linear logic, and give rise to a substantial optimisation of standard proof search algorithms.
Coinductive Proof Techniques for Language Equivalence
"... Abstract. Language equivalence can be checked coinductively by establishing a bisimulation on suitable deterministic automata. We improve and extend this technique with bisimulationupto, which is an enhancement of the bisimulation proof method. First, we focus on the regular operations of union, c ..."
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Abstract. Language equivalence can be checked coinductively by establishing a bisimulation on suitable deterministic automata. We improve and extend this technique with bisimulationupto, which is an enhancement of the bisimulation proof method. First, we focus on the regular operations of union, concatenation and Kleene star, and illustrate our method with new proofs of classical results such as Arden’s rule. Then we extend our enhanced proof method to incorporate language complement and intersection. Finally we define a general format of behavioural differential equations, in which one can define operations on languages for which bisimulationupto is a sound proof technique. 1
http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva207378 Program Analysis and Verification based on Kleene Algebra in Isabelle/HOL
"... N.B. When citing this work, cite the original published paper. Permanent link to this version: ..."
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N.B. When citing this work, cite the original published paper. Permanent link to this version:
Untyping Typed Algebraic Structures
"... Algebraic structures sometimes need to be typed. For example, matrices over a ring form a ring, but the product is a only a partial operation: dimensions have to agree. Therefore, an easy way to look at matrices algebraically is to consider “typed rings”. We prove some “untyping ” theorems: in some ..."
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Algebraic structures sometimes need to be typed. For example, matrices over a ring form a ring, but the product is a only a partial operation: dimensions have to agree. Therefore, an easy way to look at matrices algebraically is to consider “typed rings”. We prove some “untyping ” theorems: in some algebras (semirings, Kleene algebras, residuated monoids), types can be reconstructed from valid untyped equalities. As a consequence, the corresponding untyped decision procedures can be extended to the typed setting.
Proving Language Inclusion and Equivalence by Coinduction
, 2014
"... 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)simulationupto. We give general conditions on language operations for which bisimulationupto i ..."
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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)simulationupto. We give general conditions on language operations for which bisimulationupto 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).
ProjectTeam Sardes System Architecture for Reflective Distributed Computing Environments Grenoble RhôneAlpes
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Partial Derivative Automata Formalized in Coq
"... Abstract. In this paper we present a computer assisted proof of the correctness of a partial derivative automata construction from a regular expression within the Coq proof assistant. This proof is part of a formalization of Kleene algebra and regular languages in Coq towards their usage in program ..."
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Abstract. In this paper we present a computer assisted proof of the correctness of a partial derivative automata construction from a regular expression within the Coq proof assistant. This proof is part of a formalization of Kleene algebra and regular languages in Coq towards their usage in program certification. 1
Nominal Kleene Coalgebra
, 2015
"... We develop the coalgebraic theory of nominal Kleene algebra, including an alternative languagetheoretic semantics, a nominal extension of the Brzozowski derivative, and a bisimulationbased decision procedure for the equational theory. ..."
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We develop the coalgebraic theory of nominal Kleene algebra, including an alternative languagetheoretic semantics, a nominal extension of the Brzozowski derivative, and a bisimulationbased decision procedure for the equational theory.