The Value Generating Power of Weighted Tree Automata with Initial Algebra Semantics

2026-08-25

AI Summary

研究解决了在特定条件下加权树自动机生成值的数量问题，通过初始代数语义方法探讨了强双幺半群中值的生成能力。

Abstract

We consider the generating power of the initial algebra semantics of weighted tree automata over strong bimonoids (hence also over semirings) and the question under which conditions the weighted tree automata can produce only finitely many values. We show that there exists a right-distributive strong bimonoid which is bi-locally finite but not locally finite. We also show that if the ranked alphabet contains a symbol with rank at least two, then for any finitely generated strong bimonoid, weighted tree automata can generate, via their initial algebra semantics, all elements of the strong bimonoid. As a consequence of these results, for bi-locally finite right-distributive strong bimonoids which are not locally finite, weighted tree automata can generate infinitely many values, provided that the input ranked alphabet contains a symbol with rank at least two. This is in sharp contrast to the setting of weighted string automata, which can generate only finitely many values. As a further consequence, for any finitely generated semiring, there exists a weighted tree automaton which generates, via its run semantics, all elements of the semiring.

Problem

• weighted tree automata
• initial algebra semantics
• strong bimonoids
• finitely many values
• semirings

Innovation

• weighted tree automata
• initial algebra semantics
• strong bimonoids
• locally finite
• finitely generated