Recursion and proof theoretical characterizations of small circuit classes with modulo counting via discrete differential equations (long version)

Melissa Antonelli; Arnaud Durand; Rui Li

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arXiv · 2605.23805

Recursion and proof theoretical characterizations of small circuit classes with modulo counting via discrete differential equations (long version)

Melissa Antonelli·Arnaud Durand·Rui Li

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Abstract

The paper proposes an implicit (i.e., machine-independent) complexity approach to studying computation by polynomial-size, constant-depth circuits with gates counting modulo a constant through the lens of discrete ordinary differential equations (ODEs). So far, recursion-theoretic characterizations have been provided for functions computed by circuits of constant depth, including gates counting modulo 2 and 6 only (i.e., for the classes FAC0[2] and FAC0[6], resp.). In this paper, it is shown that considering ODE schemas, rather than bounded recursion, allows for a more fine-grained analysis, leading to (uniform) characterizations for all classes FAC0[n] (n \in N), i.e. functions computed by circuits including counting modulo n gates. Inspired by the syntactic form of the ODE schemas, we go further in this direction and present first-order bounded theories for capturing provably total functions in each of these classes.

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Melissa Antonelli, Arnaud Durand, Rui Li. 2026-05-22. Recursion and proof theoretical characterizations of small circuit classes with modulo counting via discrete differential equations (long version). https://arxiv.org/abs/2605.23805

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