# On the bounded theories of finite trees

## Abstract

The *theory of finite trees* is the full first-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 *non-elementary complexity* [14].

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*∈ω, with *d* written in binary notation, 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 complexity-tailored Ehrenfeucht-Fraïssé 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.

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