this paper deals exclusively with intuitionistic logic (in fact, only the implicative fragment), we require succedents to be a single consequent formula. Natural deduction systems, which we choose to call N-systems, are symbolic logics generally given via introduction and elimination rules for the logical connectives which operate on the right, i.e., they manipulate the succedent formula. Examples are Gentzen's NJ and NK (Gentzen 1935). Logical deduction systems are given via left-introduction and right-introduction rules for the logical connectives. Although others have called these systems "sequent calculi", we call them L-systems to avoid confusion with other systems given in sequent style. Examples are Gentzen's LK and LJ (Gentzen 1935). In this paper we are primarily interested in L-systems. The advantage of N-systems is that they seem closer to actual reasoning, while L-systems on the other hand seem to have an easier proof theory. L-systems are often extended with a "cut" rule as part of showing that for a given L-system and N-system, the derivations of each system can be encoded in the other. For example, NK proves the same as LK + cut (Gentzen 1935). Proof Normalization. A system is consistent when it is impossible to prove false, i.e., derive absurdity from zero assumptions. A system is analytic (has the analycity property) when there is an e#ective method to decompose any conclusion sequent into simpler premise sequents from which the conclusion can be obtained by some rule in the system such that the conclusion is derivable i# the premises are derivable (Maenpaa 1993). To achieve the goals of consistency and analycity, it has been customary to consider

Abstract. In this paper we propose a Weak Lambda Calculus called λPw having explicit operators for Pattern Matching and Substitution. This formalism is able to specify functions defined by cases via pattern matching constructors as done by most modern functional programming languages such as OCAML. We show the main property enjoyed by λPw, namely subject reduction, confluence and strong normalization. 1

We present a simple term calculus with an explicit control of erasure and duplication of substitutions, enjoying a sound and complete correspondence with the intuitionistic fragment of Linear Logic’s proof-nets. We show the operational behaviour of the calculus and some of its fundamental properties such as confluence, preservation of strong normalisation, strong normalisation of simply-typed terms, step by step simulation of β-reduction and full composition.

We present CLL, a concurrent programming language that symmetrically integrates functional and concurrent logic programming. First, a core functional language is obtained from a proof-term assignment to a variant of intuitionistic linear logic, called FOMLL, via the Curry-Howard isomorphism. Next, we introduce a Chemical Abstract Machine (CHAM) whose molecules are typed terms of this functional language. Rewrite rules for this CHAM are derived by augmenting proof-search rules for FOMLL with proof-terms. We show that this CHAM is a powerful concurrent language and that the linear connectives ⊗, ∃, ⊕, ⊸ and & correspond to process-calculi connectives for parallel composition, name restriction, internal choice, input prefixing and external choice respectively. We also demonstrate that communication and synchronization between CHAM terms can be performed through proof-search on the types of terms. Finally, we embed this CHAM as a construct in our functional language to allow interleaving functional and concurrent computation in CLL.

In this paper, we show how pattern matching can be seen to arise from a proof term assignment for the focused sequent calculus. This use of the Curry-Howard correspondence allows us to give a novel coverage checking algorithm, and makes it possible to give a rigorous correctness proof for the classical pattern compilation strategy of building decision trees via matrices of patterns. Categories and Subject Descriptors F.4.1 [Mathematical Logic]:

Abstract. Superdeduction is a method specially designed to ease the use of first-order theories in predicate logic. The theory is used to enrich the deduction system with new deduction rules in a systematic, correct and complete way. A proof-term language and a cut-elimination reduction already exist for superdeduction, both based on Christian Urban’s work on classical sequent calculus. However Christian Urban’s calculus is not directly related to the Curry-Howard correspondence contrarily to the λµ˜µ-calculus which relates straightaway to the λ-calculus. This short paper is my first step towards a further exploration of the computational content of superdeduction proofs, for I extend the λµ˜µ-calculus in order to obtain a proofterm langage together with a cut-elimination reduction for superdeduction. I also prove strong normalisation for this extension of the λµ˜µ-calculus.