Reinhard Kahle

The proof-theoretic analysis of transfinitely iterated fixed point theories

This article provides the proof-theoretic analysis of the transfinitely iterated fixed point theories (alpha) and (<alpha); the exact proof-theoretic ordinals of these systems are presented.

On Applicative Theories

Systems of explicit mathematics were introduced in Feferman [7, 9] in order to give a logical account to Bishop-style constructive mathematics, and they soon turned out to be very important for the proof-theoretic analysis of subsystems of second order arithmetic and set theory. Moreover, systems of explicit mathematics provide a logical framework for functional programming languages.

Universes in explicit mathematics

Abstract This paper deals with universes in explicit mathematics. After introducing some basic definitions, the limit axiom and possible ordering principles for universes are discussed. Later, we turn to least universes, strictness and name induction. Special emphasis is put on theories for explicit mathematics with universes which are proof-theoretically equivalent to Fefermans T 0 .

Proof Theory in Computer Science

A typed lambda calculus with recursion in all finite types is defined such that the first order terms exactly characterize the parallel complexity class NC. This is achieved by use of the appropriate forms of recursion (concatenation recursion and logarithmic recursion), a ramified type structure and imposing of a linearity constraint.

Recursion Schemata for NCk

We give a recursion-theoretic characterization of the complexity classes NCkfor ki¾? 1. In the spirit of implicit computational complexity, it uses no explicit bounds in the recursion and also no separation of variables is needed. It is based on three recursion schemes, one corresponds to time (time iteration), one to space allocation (explicit structural recursion) and one to internal computations (mutual in place recursion). This is, to our knowledge, the first exact characterization of NCkby function algebra over infinite domains in implicit complexity.

Introduction: Proof-theoretic Semantics

According to the model-theoretic view, which still prevails in logic, semantics is primarily denotational. Meanings are denotations of linguistic entities. The denotations of individual expressions are objects, those of predicate signs are sets, and those of sentences are truth values. The meaning of an atomic sentence is determined by the meanings of the individual and predicate expressions this sentence is composed of, and the meaning of a complex sentence is determined by the meanings of its constituents. A consequence is logically valid if it transmits truth from its premisses to its conclusion, with respect to all interpretations. Proof systems are shown to be correct by demonstrating that the consequences they generate are logically valid. This basic conception also underlies most alternative logics such as intensional or partial logics. In these logics, the notion of a model is more involved than in the classical case, but the view of proofs as entities which are semantically dependent on denotational meanings remains unchanged. Proof-theoretic semantics proceeds the other way round, assigning proofs or deductions an autonomous semantic role from the very onset, rather than explaining this role in terms of truth transmission. In proof-theoretic semantics, proofs are not merely treated as syntactic objects as in Hilbert’s formalist philosophy of mathematics, but as entities in terms of which meaning and logical consequence can be explained. The programme of proof-theoretic semantics can be traced back to Gentzen (1934). Seminal papers by Tait, Martin-Löf, Girard and Prawitz were published in 1967 and 1971. An explicit formulation of a semantic validity notion for generalized deductions with respect to arbitrary justifications was given by Prawitz (1973). Much of the philosophical groundwork for proof-theoretic semantics was laid by Dummett from the 1970s on, culminating in Dummett (1991). MartinLöf ’s type theory, whose philosophical foundation is proof-theoretic semantics, became a full-fledged theory in the 1970s aswell (seeMartinLöf 1975, 1982). The term ‘‘proof-theoretic semantics’’ was proposed by the second editor in a lecture in Stockholm in 1987.

Formalizing non-termination of recursive programs

Abstract In applicative theories the recursion theorem provides a term rec which solves recursive equations. However, it is not provable that a solution obtained by rec is minimal. In the present paper we introduce an applicative theory in which it is possible to define a least fixed point operator. Still, our theory has a standard recursion theoretic interpretation.

Frege structures for partial applicative theories

Due to strictness problems, usually the syntactical definition of Frege structures is conceived as a truth theory for total applicative theories. To investigate Frege structures in a partial framework we can follow two ways. First, simply by ignoring undefinedness in the truth definition. Second, by introducing of a certain notion of pointer. Both approaches are compatible with the traditional formalizations of Frege structures and preserve the main results, namely abstraction and the proof-theoretic strength.

Truth in Applicative Theories

We give a survey on truth theories for applicative theories. It comprises Frege structures, universes for Frege structures, and a theory of supervaluation. We present the proof-theoretic results for these theories and show their syntactical expressive power. In particular, we present as a novelty a syntactical interpretation of ID1 in a applicative truth theory based on supervaluation.

Towards an implicit characterization of NC k

We define a hierarchy of term systems Tk by means of restrictions of the recursion schema. We essentially use a pointer technique together with tiering. We prove Tk⊆NCk⊆Tk+1, for k ≥2. Special attention is put on the description of T2 and T3 and on the proof of T2⊆NC2⊆T3. Such a hierarchy yields a characterization of NC.