MPI-INF/SWS Research Reports 1991-2017

# MPI-I-96-2-008

## On the decision complexity of the bounded theories of trees

### Vorobyov, Sergei

#### November 1996, 26 pages.

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##### Status: available - back from printing

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 $c>0$.

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URL to this document: http://domino.mpi-inf.mpg.de/internet/reports.nsf/NumberView/1996-2-008

BibTeX
@TECHREPORT{Vorobyov96-RR-008,
AUTHOR = {Vorobyov, Sergei},
TITLE = {On the decision complexity of the bounded theories of trees},
TYPE = {Research Report},
INSTITUTION = {Max-Planck-Institut f{\"u}r Informatik},