A herbrandized functional interpretation of classical first-order logic

We introduce a new typed combinatory calculus with a type constructor that, to each type σ, associates the star type σ^∗ of the nonempty finite subsets of elements of type σ. We prove that this calculus enjoys the properties of strong normalization and confluence. With the aid of this star combinatory calculus, we define a functional interpretation of first-order predicate logic and prove a corresponding soundness theorem. It is seen that each theorem of classical first-order logic is connected with certain formulas which are tautological in character. As a corollary, we reprove Herbrand’s theorem on the extraction of terms from classically provable existential statements.info:eu-repo/semantics/publishedVersio

Asymptotically almost all \lambda-terms are strongly normalizing

We present quantitative analysis of various (syntactic and behavioral) properties of random \lambda-terms. Our main results are that asymptotically all the terms are strongly normalizing and that any fixed closed term almost never appears in a random term. Surprisingly, in combinatory logic (the translation of the \lambda-calculus into combinators), the result is exactly opposite. We show that almost all terms are not strongly normalizing. This is due to the fact that any fixed combinator almost always appears in a random combinator

A proof of strong normalisation using domain theory

Ulrich Berger presented a powerful proof of strong normalisation using domains, in particular it simplifies significantly Tait's proof of strong normalisation of Spector's bar recursion. The main contribution of this paper is to show that, using ideas from intersection types and Martin-Lof's domain interpretation of type theory one can in turn simplify further U. Berger's argument. We build a domain model for an untyped programming language where U. Berger has an interpretation only for typed terms or alternatively has an interpretation for untyped terms but need an extra condition to deduce strong normalisation. As a main application, we show that Martin-L\"{o}f dependent type theory extended with a program for Spector double negation shift.Comment: 16 page

Infinitary Combinatory Reduction Systems: Normalising Reduction Strategies

We study normalising reduction strategies for infinitary Combinatory Reduction Systems (iCRSs). We prove that all fair, outermost-fair, and needed-fair strategies are normalising for orthogonal, fully-extended iCRSs. These facts properly generalise a number of results on normalising strategies in first-order infinitary rewriting and provide the first examples of normalising strategies for infinitary lambda calculus

Derivation Lengths Classification of G\"odel's T Extending Howard's Assignment

Let T be Goedel's system of primitive recursive functionals of finite type in the lambda formulation. We define by constructive means using recursion on nested multisets a multivalued function I from the set of terms of T into the set of natural numbers such that if a term a reduces to a term b and if a natural number I(a) is assigned to a then a natural number I(b) can be assigned to b such that I(a) is greater than I(b). The construction of I is based on Howard's 1970 ordinal assignment for T and Weiermann's 1996 treatment of T in the combinatory logic version. As a corollary we obtain an optimal derivation length classification for the lambda formulation of T and its fragments. Compared with Weiermann's 1996 exposition this article yields solutions to several non-trivial problems arising from dealing with lambda terms instead of combinatory logic terms. It is expected that the methods developed here can be applied to other higher order rewrite systems resulting in new powerful termination orderings since T is a paradigm for such systems

A Finite Model Property for Intersection Types

We show that the relational theory of intersection types known as BCD has the finite model property; that is, BCD is complete for its finite models. Our proof uses rewriting techniques which have as an immediate by-product the polynomial time decidability of the preorder <= (although this also follows from the so called beta soundness of BCD).Comment: In Proceedings ITRS 2014, arXiv:1503.0437

Partial Applicative Theories and Explicit Substitutions

Systems based on theories with partial self-application are relevant to the formalization of constructive mathematics and as a logical basis for functional programming languages. In the literature they are either presented in the form of partial combinatory logic or the partial A calculus, and sometimes these two approaches are erroneously considered to be equivalent. In this paper we address some defects of the partial λ calculus as a constructive framework for partial functions. In particular, the partial λ calculus is not embeddable into partial combinatory logic and it lacks the standard recursion-theoretic model. The main reason is a concept of substitution, which is not consistent with a strongly intensional point of view. We design a weakening of the partial λ calculus, which can be embedded into partial combinatory logic. As a consequence, the natural numbers with partial recursive function application are a model of our system. The novel point will be the use of explicit substitutions, which have previously been studied in the literature in connection with the implementation of functional programming language

Computation in director string calculus

In this thesis we introduce a modified version of Director String Calculus (MDSC) which preserves the applicative structure of the original lambda terms and captures the strong reduction as opposed to weak reduction of the original Director String Calculus (DSC). Furthermore, our reduction system provides an environment which supports the nonatomic nature of substitution operation and hence can lend itself to parallel and optimal reduction. We shall compare our reduction method with other reduction methods, and discuss some of the advantages and disadvantages of our method