12 research outputs found
A combinatorial proof of the extension property for partial isometries
We present a short and self-contained proof of the extension property for
partial isometries of the class of all finite metric spaces.Comment: 7 pages, 1 figure. Minor revision. Accepted to Commentationes
Mathematicae Universitatis Carolina
EPPA for two-graphs and antipodal metric spaces
We prove that the class of finite two-graphs has the extension property for
partial automorphisms (EPPA, or Hrushovski property), thereby answering a
question of Macpherson. In other words, we show that the class of graphs has
the extension property for switching automorphisms. We present a short,
self-contained, purely combinatorial proof which also proves EPPA for the class
of integer valued antipodal metric spaces of diameter 3, answering a question
of Aranda et al.
The class of two-graphs is an important new example which behaves differently
from all the other known classes with EPPA: Two-graphs do not have the
amalgamation property with automorphisms (APA), their Ramsey expansion has to
add a graph, it is not known if they have coherent EPPA and even EPPA itself
cannot be proved using the Herwig--Lascar theorem.Comment: 14 pages, 3 figure
Querying the Guarded Fragment
Evaluating a Boolean conjunctive query Q against a guarded first-order theory
F is equivalent to checking whether "F and not Q" is unsatisfiable. This
problem is relevant to the areas of database theory and description logic.
Since Q may not be guarded, well known results about the decidability,
complexity, and finite-model property of the guarded fragment do not obviously
carry over to conjunctive query answering over guarded theories, and had been
left open in general. By investigating finite guarded bisimilar covers of
hypergraphs and relational structures, and by substantially generalising
Rosati's finite chase, we prove for guarded theories F and (unions of)
conjunctive queries Q that (i) Q is true in each model of F iff Q is true in
each finite model of F and (ii) determining whether F implies Q is
2EXPTIME-complete. We further show the following results: (iii) the existence
of polynomial-size conformal covers of arbitrary hypergraphs; (iv) a new proof
of the finite model property of the clique-guarded fragment; (v) the small
model property of the guarded fragment with optimal bounds; (vi) a
polynomial-time solution to the canonisation problem modulo guarded
bisimulation, which yields (vii) a capturing result for guarded bisimulation
invariant PTIME.Comment: This is an improved and extended version of the paper of the same
title presented at LICS 201