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arXiv · 2207.07450

Z-polyregular functions

Thomas Colcombet·Gaëtan Douéneau-Tabot·Aliaume Lopez
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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.

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Thomas Colcombet, Gaëtan Douéneau-Tabot, Aliaume Lopez. 2022-07-15. Z-polyregular functions. https://arxiv.org/abs/2207.07450

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