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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.
A tactic for deciding Kleene algebras
 In 1st Coq Workshop. Tech. Univ
, 2009
"... We present a Coq reflexive tactic for deciding equalities or inequalities in Kleene algebras. This tactic is part of a larger project, whose aim is to provide tools for reasoning about binary relations in Coq: binary relations form a Kleene algebra, where the star operation is the reflexive transiti ..."
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We present a Coq reflexive tactic for deciding equalities or inequalities in Kleene algebras. This tactic is part of a larger project, whose aim is to provide tools for reasoning about binary relations in Coq: binary relations form a Kleene algebra, where the star operation is the reflexive transitive closure. Our tactic relies on an initiality theorem, whose proof goes by replaying finite automata algorithms in an algebraic way, using matrices.
A Coalgebraic Approach to Kleene Algebra with Tests
 In volume 82(1) of ENTCS
, 2003
"... Kleene Algebra with Tests is an extension of Kleene Algebra, the algebra of regular expressions, which can be used to reason about programs. We develop a coalgebraic theory of Kleene Algebra with Tests, along the lines of the coalgebraic theory of regular expressions based on deterministic automata. ..."
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Cited by 9 (0 self)
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Kleene Algebra with Tests is an extension of Kleene Algebra, the algebra of regular expressions, which can be used to reason about programs. We develop a coalgebraic theory of Kleene Algebra with Tests, along the lines of the coalgebraic theory of regular expressions based on deterministic automata. Since the known automatatheoretic presentation of Kleene Algebra with Tests does not lend itself to a coalgebraic theory, we define a new interpretation of Kleene Algebra with Tests expressions and a corresponding automatatheoretic presentation. One outcome of the theory is a coinductive proof principle, that can be used to establish equivalence of our Kleene Algebra with Tests expressions.
Controlflow semantics for assemblylevel dataflow graphs
 8th Intl. Seminar on Relational Methods in Computer Science, RelMiCS 2005, volume 3929 of LNCS
, 2006
"... Abstract. As part of a larger project, we have built a declarative assembly language that enables us to specify multiple code paths to compute particular quantities, giving the instruction scheduler more flexibility in balancing execution resources for superscalar execution. Since the key design poi ..."
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Abstract. As part of a larger project, we have built a declarative assembly language that enables us to specify multiple code paths to compute particular quantities, giving the instruction scheduler more flexibility in balancing execution resources for superscalar execution. Since the key design points for this language are to only describe data flow, have builtin facilities for redundancies, and still have code that looks like assembler, by virtue of consisting mainly of assembly instructions, we are basing the theoretical foundations on dataflow graph theory, and have to accommodate also relational aspects. Using functorial semantics into a Kleene category of “hyperpaths”, we formally capture the dataflowwithchoice aspects of this language and its implementation, providing also the framework for the necessary correctness proofs. 1
Automatic proof generation in Kleene algebra
 In Proc. 10th Int. Conf. Relational Methods in Computer Science (RelMiCS10) and 5th Int. Conf Applications of Kleene Algebra (AKA5), volume 4988 of LNCS
, 2008
"... Abstract. In this paper, we develop the basic theory of disimulations, a type of relation between two automata which witnesses equivalence. We show that many standard constructions in the theory of automata such as determinization, minimization, inaccessible state removal, et al., are instances of d ..."
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Abstract. In this paper, we develop the basic theory of disimulations, a type of relation between two automata which witnesses equivalence. We show that many standard constructions in the theory of automata such as determinization, minimization, inaccessible state removal, et al., are instances of disimilar automata. Then, using disimulations, we define an “algebraic ” proof system for the equational theory of Kleene algebra in which a proof essentially consists of a sequence of matrices encoding automata and disimulations between them. We show that this proof system is complete for the equational theory of Kleene algebra, and that proofs in this system can be constructed by a PSPACE transducer. 1
MyhillNerode relations on automatic systems and the completeness of Kleene algebra
 In STACS 2001 (Dresden), volume 2010 of Lecture
"... Abstract. It is well known that finite square matrices over a Kleene algebra again form a Kleene algebra. This is also true for infinite matrices under suitable restrictions. One can use this fact to solve certain infinite systems of inequalities over a Kleene algebra. Automatic systems are a specia ..."
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Abstract. It is well known that finite square matrices over a Kleene algebra again form a Kleene algebra. This is also true for infinite matrices under suitable restrictions. One can use this fact to solve certain infinite systems of inequalities over a Kleene algebra. Automatic systems are a special class of infinite systems that can be viewed as infinitestate automata. Automatic systems can be collapsed using Myhill–Nerode relations in much the same way that finite automata can. The Brzozowski derivative on an algebra of polynomials over a Kleene algebra gives rise to a triangular automatic system that can be solved using these methods. This provides an alternative method for proving the completeness of Kleene algebra. 1
CL: A Logic for Reasoning about Legal Contracts – Semantics
, 2008
"... The work reported here is concerned with the definition of a logic (which we call CL) for reasoning about legal contracts. The report presents the syntax of the logic and the associated semantics. There are two semantics presented: one is defined with respect to linear structures (i.e. traces of act ..."
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The work reported here is concerned with the definition of a logic (which we call CL) for reasoning about legal contracts. The report presents the syntax of the logic and the associated semantics. There are two semantics presented: one is defined with respect to linear structures (i.e. traces of actions) and is intended for runtime monitoring of executions of contracts; the second semantics is given over branching structures (i.e. Kripkelike structures) and is intended for reasoning about contracts in a static manner (i.e. modelchecking and theorem proving). In the first part of the report we present the theoretical results underlying the branching semantics. It presents an algebra of actions and restates some of previous results presented in another report, as well as new results useful for the definition of the branching semantics and for the proofs. The rest of the report is concerned with the definition of the two semantics. Moreover, several
On Hoare logic, Kleene algebra, and types
 Computer Science Department, Cornell University
, 1999
"... We show that propositional Hoare logic is subsumed by the type calculus of typed Kleene algebra augmented with subtypes and typecasting. Assertions are interpreted as typecast operators. Thus Hoarestyle reasoning with partial correctness assertions reduces to typechecking in this system. 1 ..."
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We show that propositional Hoare logic is subsumed by the type calculus of typed Kleene algebra augmented with subtypes and typecasting. Assertions are interpreted as typecast operators. Thus Hoarestyle reasoning with partial correctness assertions reduces to typechecking in this system. 1
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.
An Algebraic Calculus of Database Preferences
 Mathematics of Program Construction
, 2012
"... Abstract. Preference algebra, an extension of the algebra of database relations, is a wellstudied field in the area of personalized databases. It allows modelling user wishes by preference terms; they represent strict partial orders telling which database objects the user prefers over other ones. T ..."
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Abstract. Preference algebra, an extension of the algebra of database relations, is a wellstudied field in the area of personalized databases. It allows modelling user wishes by preference terms; they represent strict partial orders telling which database objects the user prefers over other ones. There are a number of constructors that allow combining simple preferences into quite complex, nested ones. A preference term is then used as a database query, and the results are the maximal objects according to the order it denotes. Depending on the size of the database, this can be computationally expensive. For optimisation, preference queries and the corresponding terms are transformed using a number of algebraic laws. So far, the correctness proofs for such laws have been performed by hand and in a pointwise fashion. We enrich the standard theory of relational databases to an algebraic framework that allows completely pointfree reasoning about complex preferences. This blackbox view is amenable to a treatment in firstorder logic and hence to fully automated proofs using offtheshelf verification tools. We exemplify the use of the calculus with some nontrivial laws, notably concerning socalled preference prefilters which perform a preselection to speed up the computation of the maximal objects proper.