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, establishing their characterization via tree automata, logical frameworks, and connections to polynomial automata, with decidability results for the ZERONESS problem.

Contribution

It provides a Nivat-like characterization for weighted alternating automata and explores their closure properties, logical descriptions, and relation to polynomial automata.

Findings

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

The class of series recognized by weighted alternating automata is closed under inverse homomorphisms.

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

Abstract

This paper connects the classes of weighted alternating finite automata (WAFA), weighted finite tree automata (WFTA), and polynomial automata (PA). First, we investigate the use of trees in the run semantics for weighted alternating automata and prove that the behavior of a weighted alternating automaton can be characterized as the composition of the behavior of a weighted finite tree automaton and a specific tree homomorphism, if weights are taken from a commutative semiring. Based on this, we give a Nivat-like characterization for weighted alternating automata. Moreover, we show that the class of series recognized by weighted alternating automata is closed under inverses of homomorphisms, but not under homomorphisms. Additionally, 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. We prove: A weighted language is recognized by a weighted alternating automaton iff its reversal in recognized by a polynomial automaton. Using the corresponding result for polynomial automata, we are able to prove that the ZERONESS problem for weighted alternating automata with weights taken from the rational numbers decidable.