Feasible Mathematics

A so-called "effective" algorithm may require arbitrarily large finite amounts of time and space resources, and hence may not be practical in the real world. A "feasible" algorithm is one which only requires a limited amount of space and/or time for execution; the general idea is that a feasible algorithm is one which may be practical on today's or at least tomorrow's computers. There is no definitive analogue of Church's thesis giving a mathematical definition of feasibility; however, the most widely studied mathematical model of feasible computability is polynomial-time computability. Feasible Mathematics includes both the study of feasible computation from a mathematical and logical point of view and the reworking of traditional mathematics from the point of view of feasible computation. The diversity of Feasible Mathematics is illustrated by the. contents of this volume which includes papers on weak fragments of arithmetic, on higher type functionals, on bounded linear logic, on sub recursive definitions of complexity classes, on finite model theory, on models of feasible computation for real numbers, on vector spaces and on recursion theory. The vVorkshop on Feasible Mathematics was sponsored by the Mathematical Sciences Institute and was held at Cornell University, June 26-28, 1989.

Contenu

Parity and the Pigeonhole Principle.- Computing over the Reals (or an Arbitrary Ring) Abstract.- On Model Theory for Intuitionistic Bounded Arithmetic with Applications to Independence Results.- Sequential, Machine Independent Characterizations of the Parallel Complexity Classes AlogTIME, ACk NCk and NC.- Characterizations of the Basic Feasible Functionals of Finite Type.- Functional Interpretations of Feasibly Constructive Arithmetic - Abstract.- Polynomial-time Combinatorial Operators are Polynomials.- Isols and Kneser Graphs.- Stockmeyer Induction.- Probabilities of Sentences about Two Linear Orderings.- Bounded Linear Logic: a Modular Approach to Polynomial Time Computability, Extended Abstract.- On Finite Model Theory (Extended Abstract).- Computational Models for Feasible Real Analysis.- Inverting a One-to-One Real Function is Inherently Sequential.- On Bounded ?11 Polynomial Induction.- Subrecursion and Lambda Representation over Free Algebras (Preliminary Summary).- Complexity-Theoretic Algebra: Vector Space Bases.- When is every Recursive Linear Ordering of Type ? Recursively Isomorphic to a Polynomial Time Linear Ordering over the Natural Numbers in Binary Form?.