Functional Closure Properties of Finite $\mathbb{N}$-weighted Automata

TL;DR

This work analyzes which functional transformations preserve the class $\mathsf{\#FA}$ of functions realized by finite $\mathbb{N}$-weighted automata. It proves a sharp dichotomy: univariate functional closures coincide exactly with ultimately PORC functions, and multivariate closures are precisely finite sums of finite products of univariate ultimately PORC functions. For multivariate polynomials, a concrete criterion based on dominating terms in the binomial basis articulates when a polynomial defines a closure property, and this extends to monotone graph varieties via a promise-closure framework tied to the vanishing ideal $I(S)$. The results provide unconditional, oracle-free classifications and connect combinatorial interpretation with algebraic geometry through graph varieties and polynomial representations, offering a robust toolkit for deriving combinatorial identities from automata-theoretic constructions.

Abstract

We determine all functional closure properties of finite $\mathbb{N}$-weighted automata, even all multivariate ones, and in particular all multivariate polynomials. We also determine all univariate closure properties in the promise setting, and all multivariate closure properties under certain assumptions on the promise, in particular we determine all multivariate closure properties where the output vector lies on a monotone algebraic graph variety.

Theorems & Definitions (14)

• Definition 1
• Definition 2: Simple NFA
• Lemma 3: Folklore
• Lemma 3: Addition
• Lemma 3: Multiplication
• Lemma 3: Subtraction of constants
• Lemma 3: Clamping
• Lemma 3: Comparison with constants
• Lemma 3: Division by constants
• Lemma 3: Modular arithmetic