Z-polyregular functions

TL;DR

This paper defines Z-polyregular functions from finite words to integers, characterizes them through various formal frameworks, and investigates their logical definability and growth properties.

Contribution

It introduces Z-polyregular functions, provides multiple characterizations, and establishes decidability results for their properties.

Findings

Asymptotic growth rate is computable and linked to logical variable count.

First-order definability of Z-polyregular functions is decidable.

Introduces residual transducers and aperiodicity for semantic characterization.

Abstract

This paper introduces a robust class of functions from finite words to integers that we call Z-polyregular functions. We show that it admits natural characterizations in terms of logics, Z-rational expressions, Z-rational series and transducers. We then study two subclass membership problems. First, we show that the asymptotic growth rate of a function is computable, and corresponds to the minimal number of variables required to represent it using logical formulas. Second, we show that first-order definability of Z-polyregular functions is decidable. To show the latter, we introduce an original notion of residual transducer, and provide a semantic characterization based on aperiodicity.