Infinitely generated semigroups and polynomial complexity

TL;DR

This paper investigates the monoid of polynomial-time right-ideal morphisms, proving it is not finitely generated, which leads to new separation results in computational complexity theory related to P versus NP.

Contribution

It introduces a machine model for RM_2^P, proves its non-finite generation, and derives implications for time-complexity separations in the P versus NP problem.

Findings

RM_2^P is not finitely generated

Separation results for time-complexity classes

Advances the functional approach to P vs NP

Abstract

This paper continues the functional approach to the P-versus-NP problem, begun in [1]. Here we focus on the monoid RM_2^P of right-ideal morphisms of the free monoid, that have polynomial input balance and polynomial time-complexity. We construct a machine model for the functions in RM_2^P, and evaluation functions. We prove that RM_2^P is not finitely generated, and use this to show separation results for time-complexity.