Infinitely generated semigroups and polynomial complexity

Abstract

This paper continues the functional approach to the [Formula: see text]-vs.-[Formula: see text] problem, begun in [J. C. Birget, Semigroups and one-way functions, Internat. J. Algebra Comput. 25 (1–2) (2015) 3–36]. Here, we focus on the monoid [Formula: see text] 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 [Formula: see text], and evaluation functions. We prove that, [Formula: see text] is not finitely generated, and use this to show separation results for time-complexity.