Combining Equational Tree Automata over AC and ACI Theories

  • Authors:

  • Joe Hendrix;Hitoshi Ohsaki

  • Affiliations:

  • University of Illinois at Urbana-Champaign,;National Institute of Advanced Industrial Science and Technology,

  • Venue:
  • RTA '08 Proceedings of the 19th international conference on Rewriting Techniques and Applications
  • Year:
  • 2008

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Abstract

In this paper, we study combining equational tree automata in two different senses: (1) whether decidability results about equational tree automata over disjoint theories ${\mathcal{E}}_1$ and ${\mathcal{E}}_2$ imply similar decidability results in the combined theory${\mathcal{E}}_1 \cup {\mathcal{E}}_2$; (2) checking emptiness of a language obtained from the Boolean combinationof regular equational tree languages. We present a negative result for the first problem. Specifically, we show that the intersection-emptiness problem for tree automata over a theory containing at least one AC symbol, one ACI symbol, and 4 constants is undecidable despite being decidable if either the AC or ACI symbol is removed. Our result shows that decidability of intersection-emptiness is a non-modularproperty even for the union of disjoint theories. Our second contribution is to show a decidability result which implies the decidability of two open problems: (1) If idempotence is treated as a rule f(x,x) 驴xrather than an equation f(x,x) = x, is it decidable whether an AC tree automata accepts an idempotent normal form? (2) If ${\mathcal{E}}$ contains a single ACI symbol and arbitrary free symbols, is emptiness decidable for a Boolean combination of regular ${\mathcal{E}}$-tree languages?