Single-Path Restarting Tree Automata

Abstract

Restarting tree automata are an extension of top-down tree automata that incorporate transformations of trees through the execution of certain size-reducing rewrite operations. An input tree is repeatedly rewritten until a simple tree is obtained that is then accepted without further rewrites. Accordingly, these automata can be seen as term-rewriting systems with an incorporated regular control realizing parallel rewrites on independent branches. Here we introduce and study two restricted types of restarting tree automata by restricting the options for the regular control. The first variant we consider is the single-path restarting tree automaton, which is obtained from the general model by restricting it to the ability to pass down information along a single path only. In this way it is enforced that rewrites are executed in a strictly sequential way. Interestingly, single-path restarting tree automata reduce the tree languages they recognize to a proper subclass of the class of regular tree languages. Nevertheless, many of the results on the general model of restarting automata carry over to this variant. The second variant we study is the ground-rewrite restarting tree automaton. It is required to perform its size-reducing rewrite steps only on ground terms of bounded height. Accordingly, these automata can be interpreted as ground term-rewriting systems with additional regular control. Although they are much less expressive than the general model, it turns out that due to an inherent synchronization mechanism they can still accept certain non-regular tree languages. Finally, we consider the combination of both restrictions.

The results presented here are mostly taken from Heiko Stamer’s doctoral dissertation [8], where the proofs can be found in full detail.