Minimal equational representations of recognizable tree languages

Abstract.

 A tree language is congruential if it is the union of finitely many classes of a finitely generated congruence on the term algebra. It is well known that congruential tree languages are the same as recognizable tree languages. An equational representation is an ordered pair (E, P) , where E is either a ground term equation system or a ground term rewriting system, and P is a finite set of ground terms. We say that (E, P) represents the congruential tree language L which is the union of those ?^* _E -classes containing an element of P, i.e., for which L=⋃{[p]_? ^* _E ∣p∈P}. We define two sorts of minimality for equational representations. We introduce the cardinality vector (∣E∣, ∣P∣) of an equational representation (E, P). Let ? _l and ? _a denote the lexicographic and antilexicographic orders on the set of ordered pairs of nonnegative integers, respectively. Let L be a congruential tree language. An equational representation (E, P) of L with ? _l -minimal (? _a -minimal) cardinality vector is called ? _l -minimal (? _a -minimal). We compute, for an L given by a deterministic bottom-up tree automaton, both a ? _l -minimal and a ? _a -minimal equational representation of L.