On the decision complexity of the bounded theories of trees
Sergei G. Vorobyov · 1996
The theory of finite trees is the full first-order theory of equality in the Herbrand universe (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 non-elementary complexity [Vor96a]. To overcome the intractability of the theory of finite trees, we introduce in this paper the bounded theory of finite trees. This theory replaces the usual equality =, interpreted as identity, with the infinite family of approximate equalities "down to a fixed given depth" f= d g d2! , 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 off. By using a refinement of Ferrante-Rackoff's complexitytailored Ehrenfeucht-Fraiss'e games, we demonstrate that the bounded theory of finite trees can be decided within linear double exponential space 2 2 cn (n is the length of input) for some constant c > 0.