Higher-order unification is equational unification for βη-conversion. But it is not first-order equational unification, as substitution has to avoid capture. In this paper higher-order unification is reduced to first-order equational unification in a suitable theory: the λσ-calculus of explicit substitutions.

The main concern of this paper is the design of a noetherian and confluent normalization for LK 2 (that is, classical second order predicate logic presented as a sequent calculus). The method we present is powerful: since it allows us to recover as fragments formalisms as seemingly different as Girard's LC and Parigot's , FD ([9, 11, 27, 31]), delineates other viable systems as well, and gives means to extend the Krivine/Leivant paradigm of `programming-with-proofs' ([22, 23]) to classical logic; it is painless: since we reduce strong normalization and confluence to the same properties for linear logic (for non-additive proof nets, to be precise) using appropriate embeddings (so-called decorations); it is unifying: it organizes known solutions in a simple pattern that makes apparent the how and why of their making. A comparison of our method to that of embedding LK into LJ (intuitionistic sequent calculus) brings to the fore the latter's defects for these `deconstructi...

We prove that the class of stable models is incomplete with respect to pure λ-calculus. More precisely, we show that no stable model has the same theory as the strongly stable version of Park's model. This incompleteness proof can be adapted to the continuous case, giving an incompleteness proof for this case which is much simpler than the original proof by Honsell an Ronchi della Rocca. Moreover, we isolate a very simple finite set, F , of equations and inequations, which has neither a stable nor a continuous model, and which is included in Th(P fs ) and in T

We present a new system of intersection types for a composition-free calculus of explicit substitutions with a rule for garbage collection, and show that it characterizes those terms which are strongly normalizing. This system extends previous work on the natural generalization of the classical intersection types system, which characterized head normalization and weak normalization, but was not complete for strong normalization. An important role is played by the notion of available variable in a term, which is a generalization of the classical notion of free variable.

We investigate an automated program synthesis system based on the paradigm of programming by proofs. To automatically extract a -term that computes a recursive function given by a set of equations the system must nd a formal proof of the totality of the given function. Because of the particular logical framework, usually such approaches make it dicult to use techniques such as those in rewriting theory. We overcome this diculty for the automated system that we consider by exploiting product types. As a consequence, this would enable the incorporation of termination techniques used in other areas while still extracting programs.

and second order typed λ-calculi

We prove that any pair of derivations, without structural rules, of F ` G and G ` F , where F , G are first-order formulas `without any qualities', in a constrained classical sequent calculus LK j p , define a computational isomorphism up to an equivalence on derivations based upon reversibility properties of logical rules. This result gives a rationale behind the success of Girard's denotational semantics for classical logic, in which all standard `linear' boolean equations are satisfied.

We prove that any pair of derivations, without structural rules, of F ) G and G ) F , where F , G are rst-order formulas `without any qualities', in a constrained classical sequent calculus LK p , denes a computational isomorphism up to an equivalence on derivations based upon reversibility properties of logical rules. This result gives a rationale behind the success of Girard's denotational semantics for classical logic, in which all standard `linear' boolean equations are satised. 1 Introduction 1.1 A patch of paradise to be broadened In recent work [1] devoted to the proof theory of classical logic, we embarked on the project of overcoming the obstacles that prevent cut from being a decent binary operation on the set of classical sequent derivations. To clarify what we mean by decency, let us have a look at the world of simply typed -calculus, which, seen from a normalization-as-computation point of view, is something close to a patch of paradise. danos@logique...

We give a first order formulation of Church's type theory in which types are mere sets. This formulation is obtained by replacing -calculus by a language of combinators (skolemized comprehension schemes), introducing a distinction between propositions and their contents, relativizing quantifiers and at last replacing typing predicates by membership to some sets. The theory obtained this way has both a type theoretical flavor and a set theoretical one. Like set theory, it is a first order theory, and it uses only one notion of collection. Like type theory, it gives an explicit notation for objects, a primitive notion of function and propositions are objects.

The simply typed -calculus is known to be complete with respect to models of the form Sets . More formally, that means that given any simply typed -theory T, T ` t 1 = t 2 i for all T-models [[ ]] in Sets , for all categories C , we have [[t 1 ]] = [[t 2 ]]. It follows from a result by Awodey that it is enough to look at models in Sets for P a poset. In this thesis, I will describe explicitly how this more powerful completeness result follows from a result in [2]. As models of the form Sets for P a poset resemble the Kripke models familiar from intuitionistic logic, they are relatively easy for non-category theorists to understand. We hope that the simpler semantics result in new applications of the simply typed -calculus. We also describe how this gives a complete semantics of the simply typed -calculus in a certain category of posets.