Papers by Luigi Santocanale
We show how to axiomatize by equations the least prefixed point of an order preserving function a... more We show how to axiomatize by equations the least prefixed point of an order preserving function and discuss the domain of application of the proposed method. Thus, we generalize the well known equational axiomatization of Propositional Dynamic Logic to a complete equational axiomatization of the Boolean Modal μ-Calculus. We show on the other hand that the existence of a term which does not preserve the order is an essential condition for the least prefixed point to be definable by equations.
Theoretical Computer Science, 2003
Abstract: We propose a method to axiomatize by equations the least prexedpoint of an order preser... more Abstract: We propose a method to axiomatize by equations the least prexedpoint of an order preserving function. We discuss its domain of applicationand show that the Boolean Modal -Calculus has a complete equationalaxiomatization. The method relies on the existence of a \closedstructure" and its relationship to the equational axiomatization of ActionLogic is made explicit. The implication operation of a closed
Abstract: for a givenplayer. The two dierent meanings of parity games, the algebraic one and the ... more Abstract: for a givenplayer. The two dierent meanings of parity games, the algebraic one and the combinatorialone, are then shown to coincide. By means of this result we support the claim that the algebraof parity games is the one of -bicomplete categories and that the combinatorics of -bicompletecategories is the one of parity games.1 -Bicomplete CategoriesThe obvious way to dene
Theoretical Informatics and Applications, 2002
ABSTRACT For an arbitrary category, we consider the least class of functors containing the projec... more ABSTRACT For an arbitrary category, we consider the least class of functors containing the projections and closed under finite products, finite coproducts, parameterized initial algebras and parameterized final coalgebras, i.e. the class of functors that are definable by μ-terms. We call the category μ-bicomplete if every μ-term defines a functor. We provide concrete examples of such categories and explicitly characterize this class of functors for the category of sets and functions. This goal is achieved through parity games: we associate to each game an algebraic expression and turn the game into a term of a categorical theory. We show that μ-terms and parity games are equivalent, meaning that they define the same property of being μ-bicomplete. Finally, the interpretation of a parity game in the category of sets is shown to be the set of deterministic winning strategies for a chosen player.
Annals of Pure and Applied Logic, 2007
Given a set Γ of modal formulas of the form γ(x, p), where x occurs positively in γ, the language... more Given a set Γ of modal formulas of the form γ(x, p), where x occurs positively in γ, the language $\mathcal{L}_\sharp({\it \Gamma})$ is obtained by adding to the language of polymodal logic K connectives $\sharp_\gamma$ , γε Γ. Each term $\sharp_\gamma$ is meant to be interpreted as the parametrized least fixed point of the functional interpretation of the term γ(x). Given such a Γ, we construct an axiom system ${\bf K}_\sharp(\Gamma)$ which is sound and complete w.r.t. the concrete interpretation of the language $\mathcal{L}_\sharp({\it \Gamma})$ on Kripke frames. If Γ is finite, then ${\bf K}_\sharp(\Gamma)$ is a finite set of axioms and inference rules.
Given a set Γ of modal formulas of the form γ(x, p), where x occurs positively in γ, the language... more Given a set Γ of modal formulas of the form γ(x, p), where x occurs positively in γ, the language L (Γ ) is obtained by adding to the language of polymodal logic K connectives γ , γ ∈ Γ . Each term γ is meant to be interpreted as the parametrized least fixed point of the functional interpretation of the term γ(x). Given such a Γ , we construct an axiom system K (Γ ) which is sound and complete w.r.t. the concrete interpretation of the language L (Γ ) on Kripke frames. If Γ is finite, then K (Γ ) is a finite set of axioms and inference rules.
Annals of Pure and Applied Logic, 2010
This paper exhibits a general and uniform method to prove completeness for certain modal fixpoint... more This paper exhibits a general and uniform method to prove completeness for certain modal fixpoint logics.
Computing Research Repository, 2008
This paper exhibits a general and uniform method to prove completeness for certain modal fixpoint... more This paper exhibits a general and uniform method to prove completeness for certain modal fixpoint logics.
Uploads
Papers by Luigi Santocanale