Semantics of linear logic and higher-order model-checking. (Sémantique de la logique linéaire et "model-checking" d'ordre supérieur)

This thesis studies problems of higher-order model-checking from a semantic and logical perspective. Higher-order model-checking is concerned with the verification of properties expressed in monadic second-order logic, specified over infinite trees generated by a class of rewriting systems called higher-order recursion schemes. These systems are equivalent to simply-typed λ-terms with recursion, and can therefore be studied using semantic methods. The more specific purpose of this thesis is to connect higher-order model-checking to a series of advanced ideas in contemporary semantics, such as linear logic and its relational semantics, indexed linear logic, distributive laws between comonads, parametric comonads and tensorial logic. As we will see, all these ingredients meet and combine surprisingly well with higher-order model-checking. The starting point of our approach is the study of the intersection type system of Kobayashi and Ong. This intersection type system enables one to type a higherorder recursion scheme with states of a given automaton, associated with a formula of monadic second-order logic. The recursion scheme is typable with the initial state of the automaton if and only if the infinite tree it represents satisfies the formula of interest. In spite of this soundness-and-completeness result, the original type system by Kobayashi and Ong was not designed with the connection between intersection types and models of linear logic observed by Bucciarelli, Ehrhard, de Carvalho and Terui in mind. Our work has thus been to connect these two fields. Our analysis leads us to the definition of an alternative intersection type system, which enjoys a similar soundness-and-completeness theorem with respect to higherorder model-checking. In contrast to the original type system by Kobayashi and Ong, our modal formulation is the proof-theoretic counterpart of a finitary semantics of linear logic, obtained by composing the traditional exponential modality with a coloring comonad. We equip the semantics of linear logic with an inductive-coinductive fixpoint operator. We obtain in this way a model of the λ-calculus with recursion in which the interpretation of a higher-order recursion scheme is the set of states from which the infinite tree it represents is accepted. The finiteness of the semantics enables us to reestablish several results of decidability for higher-order model-checking problems, among which the selection problem recently formulated and proved by Carayol and Serre. This finitary semantics are inspired from the extensional collapse theorem of Ehrhard, who shows that the relational semantics of linear logic collapses extensionally to the finitary semantics provided by Scott lattices. For that reason, we start in a preliminary approach to define the coloring comonad and the inductivecoinductive fixpoint operator in the quantitative semantics provided by an infinitary (and non-continuous) version of the relational model of linear logic.