On the Bounded Theories of Finite Trees

The theory of nite trees is the full rst-order theory of equality in the Herbrand universum (the set of ground terms) over a functional signature containing non-unary function symbols and constants. Albeit decidable, this theory turns out to be of the non-elementary complexity in the sense of Kalmar, i.e., it cannot be decided within time or space bounded by any nite xed tower of exponentials 2 2 2 n , where n is the length of input 16]. As a partial remedy to overcome the intractability of the theory of nite trees, we introduce the bounded theory of nite trees. This theory replaces the usual equality =, interpreted as identity, with the innnite family of approximate equalities \down to a xed given depth" f= d gd2!, with d written in binary, and s = d t meaning that the ground terms s and t coincide if all their branches longer than d are cut oo. By using a reenement of Ferrante-Rackoo's complexity-tailored Ehren-feucht-Fra ss e games, we demonstrate that the bounded theory of nite trees can be decided within linear double exponential space 2 2 cn (n is the length of input) for some constant c > 0, i.e., belongs to the complexity class SPACE(2 2 cn).