Conceitos essenciais
The functional closure properties of finite N-weighted automata, including all multivariate ones and all multivariate polynomials, are completely characterized.
Resumo
The paper studies the functional closure properties of finite N-weighted automata, which are nondeterministic finite automata that output the number of accepting computation paths on an input word, instead of just whether an accepting path exists.
Key highlights:
The authors determine all functional closure properties of finite N-weighted automata, including all multivariate ones and all multivariate polynomials.
They also determine all univariate closure properties in the promise setting, and all multivariate closure properties under certain assumptions on the promise, in particular where the output vector lies on a monotone algebraic graph variety.
The functional closure properties are precisely the ultimately PORC (Polynomial On Residue Classes) functions, which are functions that can be written as a finite sum of finite products of univariate ultimately PORC functions.
For multivariate polynomials, a polynomial is a functional closure property if and only if all dominating terms in the binomial basis have positive coefficients.
The results provide a complete classification of the functional closure properties of finite N-weighted automata, without relying on any oracle separations, unlike the classification for the larger class #P.