Proof Theory

A note on full intuitionistic linear logic

Applicative theories form the basis of Feferman's systems of explicit mathematics, which have been introduced in the 1970s In an applicative universe, all individuals may be thought of as operations, which can freely be applied to each other: self-application is meaningful, but not necessarily total It has turned out that theories with self-application provide a natural setting for studying notions of abstract computability, especially from a proof-theoretic perspective This paper is concerned with the study of (unramified) bounded applicative theories which have a strong relationship to classes of computational complexity We propose new applicative systems whose provably total functions coincide with the functions computable in polynomial time, polynomial space, polynomial time and linear space, as well as linear space Our theories can be regarded as applicative analogues of traditional systems of bounded arithmetic We are also interested in higher-type features of our systems; in particular, it is shown that Cook and Urquhart's system PVω is directly contained in a natural applicative theory of polynomial strength

Theories with self-application and computational complexity

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.

On Applicative Theories

In this paper we characterize varieties with the amalgamation property and the congruence extension property by means of a condition of proof-theoretic nature resembling a version of Craig’s interpolation property due to Maehara (1961).

On a Form of Equational Interpolation Property

Combinatorial theorists have for some time been showing that certain partial orderings are well-partial-orderings (w.p.o.’s). De Jongh and Parikh showed that w.p.o.’s are just those well-founded partial orderings which can be extended to a well-ordering of maximal order type; we call the ordinal thus obtained the maximal order type of the w.p.o. In this paper we calculate, in terms of a system of notations due to Schutte [24], the maximal order types of the w.p.o.’s investigated in Higman [11], and give upper bounds for the maximal order types of the w.p.o.’s investigated in Kruskal [13] and Nash-Williams [16]. As a by-product and an application of de Jongh and Parikh’s work, we give new and easier proofs of Higman’s, Kruskal’s and Nash–Williams’ theorems that the partial orderings considered are indeed w.p.o.’s. We also apply our results to the theory of ordinal notations.

Proof Theory

Citations

A note on full intuitionistic linear logic

Theories with self-application and computational complexity

On Applicative Theories

On a Form of Equational Interpolation Property

Well-Partial Orderings and their Maximal Order Types

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