A Nivat Theorem for Weighted Alternating Automata over Commutative Semirings

TL;DR

This paper extends Nivat's theorem to weighted alternating automata over commutative semirings, providing a characterization via weighted finite tree automata and tree homomorphisms, and explores logical and decidability aspects.

Contribution

It introduces a Nivat-like characterization for weighted alternating automata over commutative semirings, linking them to weighted finite tree automata and tree homomorphisms.

Findings

Weighted alternating automata can be characterized as concatenations of weighted finite tree automata and specific tree homomorphisms.

The class of recognized series is closed under inverse homomorphisms but not under homomorphisms.

The ZERONESS problem for weighted alternating automata over rational numbers is decidable.

Abstract

In this paper, we give a Nivat-like characterization for weighted alternating automata over commutative semirings (WAFA). To this purpose we prove that weighted alternating can be characterized as the concatenation of weighted finite tree automata (WFTA) and a specific class of tree homomorphism. We show that the class of series recognized by weighted alternating automata is closed under inverses of homomorphisms, but not under homomorphisms. We give a logical characterization of weighted alternating automata, which uses weighted MSO logic for trees. Finally we investigate the strong connection between weighted alternating automata and polynomial automata. Using the corresponding result for polynomial automata, we are able to prove that the ZERONESS problem for weighted alternating automata with the rational numbers as weights is decidable.