The generating power of weighted tree automata with initial algebra semantics

TL;DR

The paper analyzes the images of the initial algebra semantics of weighted tree automata over strong bimonoids, focusing on finiteness and universality properties. It constructs a weakly locally finite but not locally finite strong bimonoid and proves a universality result: when the ranked alphabet contains a binary symbol, wta can generate all elements of any finitely generated strong bimonoid via initial algebra semantics. This leads to infinite image phenomena for certain bimonoids (contrasting with the finite-image behavior of weighted string automata) and implies that, for finitely generated semirings, there exist wta whose run semantics cover the entire semiring. The results underscore the heightened generative power of wta on trees relative to strings and open avenues for further finiteness-characterization research in this setting.

Abstract

We consider the images of the initial algebra semantics of weighted tree automata over strong bimonoids (hence also over semirings). These images are subsets of the carrier set of the underlying strong bimonoid. We consider locally finite, weakly locally finite, and bi-locally finite strong bimonoids. We show that there exists a strong bimonoid which is weakly locally finite and not locally finite. We also show that if the ranked alphabet contains a binary symbol, 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 weakly locally finite strong bimonoids which are not locally finite, weighted tree automata can generate infinite images provided that the input ranked alphabet contains at least one binary symbol. This is in sharp contrast to the setting of weighted string automata, where each such image is known to be finite. 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.

Figures (2)

• Figure 1: An overview for finite image property with respect to initial algebra semantics. In the figure lf = locally finite, wlf = weakly locally finite, and bi-lf = bi-locally finite. Moreover, Fin means that, for each wta $\mathcal{A}$ over such an alphabet and such a strong bimonoid $\mathsf{B}$, $\mathrm{im}([\![\mathcal{A}]\!]^{\mathrm{init}})$ is finite.
• Figure 2: The $(\Sigma,\mathsf{B})$-wta $\mathcal{A}$ of the proof of Theorem \ref{['lm:closure-of-finite-set-i-recognizable']}, where the three occurrences of the state $q_0$ have to be identified.

Theorems & Definitions (32)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• proof
• proof
• Example 2.5
• Example 2.6
• Lemma 3.1