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Abstract:
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 $c>0$.
