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International Conference on Rewriting Techniques and Applications

RTA 2008: Rewriting Techniques and Applications pp 142–156Cite as

Combining Equational Tree Automata over AC and ACI Theories

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 5117))

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 combination of 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-modular property 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) →x rather 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?

Research supported by ONR Grant N00014-02-1-0715.

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Authors and Affiliations

  1. University of Illinois at Urbana-Champaign,  

    Joe Hendrix

  2. National Institute of Advanced Industrial Science and Technology,  

    Hitoshi Ohsaki

Authors
  1. Joe Hendrix
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  2. Hitoshi Ohsaki
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Andrei Voronkov

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Hendrix, J., Ohsaki, H. (2008). Combining Equational Tree Automata over AC and ACI Theories. In: Voronkov, A. (eds) Rewriting Techniques and Applications. RTA 2008. Lecture Notes in Computer Science, vol 5117. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-70590-1_10

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  • DOI: https://doi.org/10.1007/978-3-540-70590-1_10

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-70588-8

  • Online ISBN: 978-3-540-70590-1

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