MPI-I-96-2-008

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 [Vorobyov CADE'96]. 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'' $\{=^d\}_{d\in\omega}$, with $d$ written in binary, and $s=^dt$ 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 complexity-tailored Ehrenfeucht-Fraisse 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 $c>0$.