Semantics of Higher-Order Recursion Schemes

Higher-order recursion schemes are recursive equations defining new operations from given ones called "terminals". Every such recursion scheme is proved to have a least interpreted semantics in every Scott's model of \lambda-calculus in which the terminals are interpreted as continuous operations. For the uninterpreted semantics based on infinite \lambda-terms we follow the idea of Fiore, Plotkin and Turi and work in the category of sets in context, which are presheaves on the category of finite sets. Fiore et al showed how to capture the type of variable binding in \lambda-calculus by an endofunctor H\lambda and they explained simultaneous substitution of \lambda-terms by proving that the presheaf of \lambda-terms is an initial H\lambda-monoid. Here we work with the presheaf of rational infinite \lambda-terms and prove that this is an initial iterative H\lambda-monoid. We conclude that every guarded higher-order recursion scheme has a unique uninterpreted solution in this monoid

Syntactic Monoids in a Category

The syntactic monoid of a language is generalized to the level of a symmetric monoidal closed category D. This allows for a uniform treatment of several notions of syntactic algebras known in the literature, including the syntactic monoids of Rabin and Scott (D = sets), the syntactic semirings of Polak (D = semilattices), and the syntactic associative algebras of Reutenauer (D = vector spaces). Assuming that D is an entropic variety of algebras, we prove that the syntactic D-monoid of a language L can be constructed as a quotient of a free D-monoid modulo the syntactic congruence of L, and that it is isomorphic to the transition D-monoid of the minimal automaton for L in D. Furthermore, in case the variety D is locally finite, we characterize the regular languages as precisely the languages with finite syntactic D-monoids

Varieties of Languages in a Category

Eilenberg's variety theorem, a centerpiece of algebraic automata theory, establishes a bijective correspondence between varieties of languages and pseudovarieties of monoids. In the present paper this result is generalized to an abstract pair of algebraic categories: we introduce varieties of languages in a category C, and prove that they correspond to pseudovarieties of monoids in a closed monoidal category D, provided that C and D are dual on the level of finite objects. By suitable choices of these categories our result uniformly covers Eilenberg's theorem and three variants due to Pin, Polak and Reutenauer, respectively, and yields new Eilenberg-type correspondences

Tree constructions of free continuous algebras

AbstractContinuous algebras are algebras endowed with a partial order which is complete with respect to specified joins and such that the operations preserve these specified joins. We prove the existence of free continuous algebras by actually giving a concrete description of them in terms of trees, for any type of algebras and any choice of the “specified” joins

Carmen: Software Component Model Checker

International audienceThe challenge of model checking of isolated software components becomes more and more relevant with the boom of component-oriented technologies [20]. An important issue here is how to verify an open model representing an isolated software component (also referred as the missing environment problem in [17]). In this paper, we propose on-the-fly simulation of the component environment to address the issue. We employ behavior protocols [18] and a system coordinating two model checkers: Java PathFinder [4] and BPChecker [15]. This approach allows us to enclose the model represent- ing the behavior of a given component and consequently to exhaustively verify the model. Our solution was implemented as the Carmen tool [1]. We demonstrate scalability of our approach on real-life examples and show that, in comparison with the COMBAT model checker [17], we bring better performance, and also exhaustive and correct verification