Polynomial-Time Axioms of Choice and Polynomial-Time Cardinality

TL;DR

This paper explores polynomial-time versions of the Axiom of Choice, establishing their equivalence to key complexity hypotheses and developing a theory of polynomial-time cardinality to connect set theory with computational complexity.

Contribution

It introduces polynomial-time formulations of AC, proves their equivalence to complexity hypotheses, and develops a polynomial-time cardinality theory extending previous unary-based work.

Findings

Many classical AC formulations are equivalent to known complexity hypotheses.

A new theory of polynomial-time cardinality is developed for larger alphabets.

Connections between AC, cardinality, and complexity questions are established.

Abstract

There is no single canonical polynomial-time version of the Axiom of Choice (AC); several statements of AC that are equivalent in Zermelo-Fraenkel (ZF) set theory are already inequivalent from a constructive point of view, and are similarly inequivalent from a complexity-theoretic point of view. In this paper we show that many classical formulations of AC, when restricted to polynomial time in natural ways, are equivalent to standard complexity-theoretic hypotheses, including several that were of interest to Selman. This provides a unified view of these hypotheses, and we hope provides additional motivation for studying some of the lesser-known hypotheses that appear here. Additionally, because several classical forms of AC are formulated in terms of cardinals, we develop a theory of polynomial-time cardinality. Nerode & Remmel (Contemp. Math. 106, 1990 and Springer Lec. Notes Math. 1432, 1990) developed a related theory, but restricted to unary sets. Downey (Math. Reviews MR1071525) suggested that such a theory over larger alphabets could have interesting connections to more standard complexity questions, and we illustrate some of those connections here. The connections between AC, cardinality, and complexity questions also allow us to highlight some of Selman's work. We hope this paper is more of a beginning than an end, introducing new concepts and raising many new questions, ripe for further research.