Reduction Systems Definition 2.2. An Abstract Reduction System (short: ARS) consists of a set A and a sequence ! i of binary relations on A, labelled by some set I. We often drop the label if I is a singleton. We write A j= P if the ARS A = (A; ! i ; : : : ); i 2 I has the property P . Further we write A j= P Q iff A j= P and A j= Q. An ARS A = (A; !) has the diamond property , A j= \Sigma, iff /;! ` !;/. It has the Church-Rosser property (is confluent), A j= CR, iff (A; !!) j= \Sigma. Given an ARS A = (A; !), we write CR(t) as shorthand for (fu j t !! ug; !) j= CR. 4 Stefan Kahrs Under most circumstances, confluence is a useful property of ARSs, mainly because: if (A; !) j= CR, and if two elements x; y 2 A are equivalent w.r.t. the smallest equivalence containing !, then there is a z 2 A such that x!! z //y. Roughly: the ARS decides the equivalence. An ARS A = (A; ! a ; ! b ) commutes directly , A j= CD, iff / a ; ! b ` ! b ; / a . To prove confluence of an ARS, it is sometimes...

In a previous paper [4], we introduced a non-deterministic λ-calculus (λ-LK) whose type system corresponds exactly to Gentzen's cut-free LK [9]. This calculus, however, cannot be provided with a computational interpretation. Some of the constructs act as oracles and, for this reason, it is not possible to define an effective notion of reduction. In the present paper, we address this problem. We consider a weak version of the implicative fragment of λ-LK, and we define for it a relation of reduction that models, at the level of the terms, the appropriate proof-theoretic notion of proof reduction. This reduction relation satisøes several properties of interest, among others, the property of strong normalization. We prove this last result by using a reducibility argument à la Tait.

. It was observed by Curry that when (untyped) -terms can be assigned types, for example, simple types, these terms have nice properties (for example, they are strongly normalizing) . Coppo, Dezani, and Veneri, introduced type systems using conjunctive types, and showed that several important classes of (untyped) terms can be characterized according to the shape of the types that can be assigned to these terms. For example, the strongly normalizable terms, the normalizable terms, and the terms having head-normal forms, can be characterized in some systems D and D \Omega\Gamma The proofs use variants of the method of reducibility. In this paper, we present a uniform approach for proving several meta-theorems relating properties of -terms and their typability in the systems D and D Our proofs use a new and more modular version of the reducibility method. As an application of our metatheorems, we show how the characterizations obtained by Coppo, Dezani, Veneri, and Pottinger, can be easi...

. The main purpose of this paper is to take apart the reducibility method in order to understand how its pieces fit together, and in particular, to recast the conditions on candidates of reducibility as sheaf conditions. There has been a feeling among experts on this subject that it should be possible to present the reducibility method using more semantic means, and that a deeper understanding would then be gained. This paper gives mathematical substance to this feeling, by presenting a generalization of the reducibility method based on a semantic notion of realizability which uses the notion of a cover algebra (as in abstract sheaf theory). A key technical ingredient is the introduction a new class of semantic structures equipped with preorders, called pre-applicative structures. These structures need not be extensional. In this framework, a general realizability theorem can be shown. Kleene's recursive realizability and a variant of Kreisel's modified realizability both fit into this...

Types are important in functional programming. First order logic can be viewed as, using the Curry-Howard isomorphism, a very strict type theory. Proven rst order logic propositions are theorems. Hereby theorems are types and, by the Curry-Howard isomorphism, a ilinearized proofj of a theorem can be viewed as a program having the theorem as its type. A proof of a rst order logic theorem can be created automatically using some strategy for proof search. A generated proof can be huge 1 . Such a generated proof of the theorem can be reduced to be a minimal proof using techniques adopted from the theory of -calculus. Using this technique a program can be specied as a rst order logic theorem, and the automatically derived program hereof can be made eOEcient. Hereby proof search tools implementing the proof reduction can be used to help writing eOEcient programs. To show how this will be done in an environment that can be used for proof search, the proof reduction is implemented in the Is...

The main purpose of this paper is to take apart the reducibility method in order to understand how its pieces t together, and in particular, to recast the conditions on candidates of reducibility as sheaf conditions. There has been a feeling among experts on this subject that it should be possible to present the reducibility method using more semantic means, and that a deeper understanding would then be gained. This paper gives mathematical substance to this feeling, by presenting a generalization of the reducibility method based on a semantic notion of realizability which uses the notion of a cover algebra (as in abstract sheaf theory). A key technical ingredient is the introduction a new class of semantic structures equipped with preorders, called pre-applicative structures. These structures need not be extensional. In this framework, a general realizability theorem can be shown. Kleene's recursive realizability and a variant of Kreisel's modi ed realizability both t into this framework. We are then able to prove a meta-theorem which shows that if a property of realizers satis es some simple conditions, then it holds for the semantic interpretations of all terms. Applying this theorem to the special case of the term model, yields a general theorem for proving properties of typed-terms, in particular, strong normalization and con uence. This approach clari es the reducibility method by showing that the closure conditions on candidates of reducibility can be viewed as sheaf conditions. The above approach is applied to the simply-typed-calculus (with types!,,+,and?), and to the second-order (polymorphic)-calculus (with types! and 82), for which it yields a new theorem.