Closure of Tree Automata Languages under Innermost Rewriting

Authors:
Adrià Gascón;Guillem Godoy;Florent Jacquemard
Affiliations:
Technical University of Catalonia, Jordi Girona 1, Barcelona, Spain11The first two authors were supported by Spanish Min. of Educ. and Science by the FORMALISM project (TIN2007-66523) and by the L ...;Technical University of Catalonia, Jordi Girona 1, Barcelona, Spain11The first two authors were supported by Spanish Min. of Educ. and Science by the FORMALISM project (TIN2007-66523) and by the L ...;INRIA Futurs & LSV, UMR CNRS/ENS Cachan, France
Venue:
Electronic Notes in Theoretical Computer Science (ENTCS)
Year:
2009

Citing 11
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Abstract

Preservation of regularity by a term rewriting system (TRS) states that the set of reachable terms from a tree automata (TA) language (aka regular term set) is also a TA language. It is an important and useful property, and there have been many works on identifying classes of TRS ensuring it; unfortunately, regularity is not preserved for restricted classes of TRS like shallow TRS. Nevertheless, this property has not been studied for important strategies of rewriting like the innermost strategy - which corresponds to the call by value computation of programming languages. We prove that the set of innermost-reachable terms from a TA language by a shallow TRS is not necessarily regular, but it can be recognized by a TA with equality and disequality constraints between brothers. As a consequence we conclude decidability of regularity of the reachable set of terms from a TA language by innermost rewriting and shallow TRS. This result is in contrast with plain (not necessarily innermost) rewriting for which we prove undecidability. We also show that, like for plain rewriting, innermost rewriting with linear and right-shallow TRS preserves regularity.