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MPI-INF/SWS Research Reports 1991-2017

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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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Hide details for BibTeXBibTeX
@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},
  ADDRESS = {Im Stadtwald, D-66123 Saarbr{\"u}cken, Germany},
  NUMBER = {MPI-I-96-2-008},
  MONTH = {November},
  YEAR = {1996},
  ISSN = {0946-011X},
}