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Completions of µalgebras
 In Proceedings of the Twentieth Annual IEEE Symposium on Logic in Computer Science (LICS 2005
, 2005
"... A µalgebra is a model of a first order theory that is an extension of the theory of bounded lattices, that comes with pairs of terms (f, µx.f) where µx.f is axiomatized as the least prefixed point of f, whose axioms are equations or equational implications. Standard µalgebras are complete meaning ..."
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A µalgebra is a model of a first order theory that is an extension of the theory of bounded lattices, that comes with pairs of terms (f, µx.f) where µx.f is axiomatized as the least prefixed point of f, whose axioms are equations or equational implications. Standard µalgebras are complete meaning that their lattice reduct is a complete lattice. We prove that any non trivial quasivariety of µalgebras contains a µalgebra that has no embedding into a complete µalgebra. We focus then on modal µalgebras, i.e. algebraic models of the propositional modal µcalculus. We prove that free modal µalgebras satisfy a condition – reminiscent of Whitman’s condition for free lattices – which allows us to prove that (i) modal operators are adjoints on free modal µalgebras, (ii) least prefixed points of Σ1operations satisfy the constructive relation µx.f = W n≥0 f n (⊥). These properties imply the following statement: the MacNeilleDedekind completion of a free modal µalgebra is a complete modal µalgebra and moreover the canonical embedding preserves all the operations in the class Comp(Σ1, Π1) of the fixed point alternation hierarchy.
A bialgebraic review of regular expressions, deterministic automata and languages
 Techn. Rep. ICISR05003, Inst. for Computing and Information Sciences, Radboud Univ
, 2005
"... To Joseph Goguen on the occasion of his 65th birthday1. Abstract. This papers reviews the classical theory of deterministic automata and regular languages from a categorical perspective. The basis is formed by Rutten’s description of the Brzozowski automaton structure in a coalgebraic framework. We ..."
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To Joseph Goguen on the occasion of his 65th birthday1. Abstract. This papers reviews the classical theory of deterministic automata and regular languages from a categorical perspective. The basis is formed by Rutten’s description of the Brzozowski automaton structure in a coalgebraic framework. We enlarge the framework to a socalled bialgebraic one, by including algebras together with suitable distributive laws connecting the algebraic and coalgebraic structure of regular expressions and languages. This culminates in a reformulated proof via finality of Kozen’s completeness result. It yields a complete axiomatisation of observational equivalence (bisimilarity) on regular expressions. We suggest that this situation is paradigmatic for (theoretical) computer science as the study of “generated behaviour”.
Automatic Equivalence Proofs for Nondeterministic
, 1303
"... A notion of generalized regular expressions for a large class of systems modeled as coalgebras, and an analogue of Kleene’s theorem and Kleene algebra, were recently proposed by a subset of the authors of this paper. Examples of the systems covered include infinite streams, deterministic automata, M ..."
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A notion of generalized regular expressions for a large class of systems modeled as coalgebras, and an analogue of Kleene’s theorem and Kleene algebra, were recently proposed by a subset of the authors of this paper. Examples of the systems covered include infinite streams, deterministic automata, Mealy machines and labelled transition systems. In this paper, we present a novel algorithm to decide whether two expressions are bisimilar or not. The procedure is implemented in the automatic theorem prover CIRC, by reducing coinduction to an entailment relation between an algebraic specification and an appropriate set of equations. We illustrate the generality of the tool with three examples: infinite streams of real numbers, Mealy machines and labelled transition systems. 1.
Departamento de Ciência de Computadores
, 2014
"... This paper gives a new presentation of Kozen’s proof of Kleene algebra completeness featured in his article A completeness theorem for Kleene algebras and the algebra of regular events. A few new variants are introduced, shortening the proof. Specifically, we directly construct an εfree automaton t ..."
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This paper gives a new presentation of Kozen’s proof of Kleene algebra completeness featured in his article A completeness theorem for Kleene algebras and the algebra of regular events. A few new variants are introduced, shortening the proof. Specifically, we directly construct an εfree automaton to prove an equivalent to Kleene’s representation theorem (implementing Glushkov’s instead of Thompson’s construction), and we bypass the use of minimal automata by directly implementing a MyhillNerode equivalence relation on the union of equivalent deterministic automata. 1