Polynomial time operations in explicit mathematics

In this paper we study (self-)applicative theories of operations and binary words in the context of polynomial time computability. We propose a first order theory PTO which allows full self-application and whose provably total functions on = {0, 1}* are exactly the polynomial time computable functions. Our treatment of PTO is proof-theoretic and very much in the spirit of reductive proof theor

Some theories with positive induction of ordinal strength φω0

This paper deals with: (i) the theory which results from by restricting induction on the natural numbers to formulas which are positive in the fixed point constants, (ii) the theory BON(μ) plus various forms of positive induction, and (iii) a subtheory of Peano arithmetic with ordinals in which induction on the natural numbers is restricted to formulas which are Σ in the ordinals. We show that these systems have proof-theoretic strength φω

An extended predicative definition of the Mahlo universe

This article, which will be reviewed by Zentralblatt Math, contains the first predicative definition of the Mahlo universe, by extending the concept of predicativity. This is a break through result, since it introduces a methodology which allows to justify proof theoretically much stronger theories than were known before predicatively.Before this article predicativity was limited to inductive recursive definition, and it was widely believed that it is impossible to go beyond that notion in a predicative way. With this article for the first time this barrier has been passed using a novel approach

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

A Semantic Approach to Illative Combinatory Logic

This work introduces the theory of illative combinatory algebras, which is closely related to systems of illative combinatory logic. We thus provide a semantic interpretation for a formal framework in which both logic and computation may be expressed in a unified manner. Systems of illative combinatory logic consist of combinatory logic extended with constants and rules of inference intended to capture logical notions. Our theory does not correspond strictly to any traditional system, but draws inspiration from many. It differs from them in that it couples the notion of truth with the notion of equality between terms, which enables the use of logical formulas in conditional expressions. We give a consistency proof for first-order illative combinatory algebras. A complete embedding of classical predicate logic into our theory is also provided. The translation is very direct and natural