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.

翻译：本文提出一种隐式（即机器无关）复杂性方法，通过离散常微分方程（ODE）视角研究由多项式规模、常深度电路（包含模常数计数的门）所计算的问题。迄今为止，递归理论的特征化仅针对包含模2和模6计数门的常深度电路（即分别对应FAC0[2]和FAC0[6]类）所计算的函数。本文证明，采用ODE模式而非有界递归可实现更精细的分析，从而为所有FAC0[n]（n∈N）类（即包含模n计数门电路所计算的函数）提供（统一）特征化。受ODE模式的语法形式启发，我们进一步提出一阶有界理论，以捕获这些类别中可证明的全函数。