Speedup for Natural Problems and Noncomputability

TL;DR

This paper explores the relationship between noncomputability and superpolynomial speedups in algorithms and proof systems, highlighting implications for P vs NP and the nature of optimal algorithms.

Contribution

It establishes an equivalence between non-halting recognition, superpolynomial speedups, and proof length speedups, linking noncomputability to computational and proof complexity.

Findings

Equivalence between non-recognition of non-halting Turing machines and superpolynomial speedups.

Superpolynomial speedups imply P != NP.

Connection between 'no algorithm' and 'no best algorithm' properties in complexity theory.

Abstract

A resource-bounded version of the statement "no algorithm recognizes all non-halting Turing machines" is equivalent to an infinitely often (i.o.) superpolynomial speedup for the time required to accept any coNP-complete language and also equivalent to a superpolynomial speedup in proof length in propositional proof systems for tautologies, each of which implies P!=NP. This suggests a correspondence between the properties 'has no algorithm at all' and 'has no best algorithm' which seems relevant to open problems in computational and proof complexity.