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  "path": "/abs/2605.23805v1",
  "publishedAt": "2026-05-25T00:00:00.000Z",
  "site": "https://arxiv.org",
  "tags": [
    "Melissa Antonelli",
    "Arnaud Durand",
    "Rui Li"
  ],
  "textContent": "**Authors:** Melissa Antonelli, Arnaud Durand, Rui Li\n\nThe 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.",
  "title": "Recursion and proof theoretical characterizations of small circuit classes with modulo counting via discrete differential equations (long version)"
}