Run supports and initial algebra supports of weighted automata

TL;DR

This paper investigates conditions under which the support of run semantics matches the support of initial algebra semantics in weighted automata over words and trees, focusing on strong bimonoids and their properties.

Contribution

It characterizes when run and initial algebra supports are equal in weighted automata using properties of strong bimonoids, including zero-sum-free conditions.

Findings

Support equality characterized by strongly zero-sum-free bimonoids

Extension of results to automata over trees with bi-strongly zero-sum-free bimonoids

Analysis of images of semantics functions in this context

Abstract

We consider weighted automata over words and over trees where the weight algebras are strong bimonoids, i.e., semirings which may lack distributivity. It is well known that, for each such weighted automaton, its run semantics and its initial algebra semantics can be different, due to the presence of nondeterminism and the absence of distributivity. Here we investigate the question under which conditions on the strong bimonoid the support of the run semantics equals the support of the initial algebra semantics. We prove a characterization of this equality in terms of strongly zero-sum-free strong bimonoids (for weighted automata over words) and in terms of bi-strongly zero-sum-free strong bimonoids (for weighted automata over trees). We also consider shortly the images of the two semantics functions.