537 research outputs found
On First-Order Definable Colorings
We address the problem of characterizing -coloring problems that are
first-order definable on a fixed class of relational structures. In this
context, we give several characterizations of a homomorphism dualities arising
in a class of structure
SMT Solving for Functional Programming over Infinite Structures
We develop a simple functional programming language aimed at manipulating
infinite, but first-order definable structures, such as the countably infinite
clique graph or the set of all intervals with rational endpoints. Internally,
such sets are represented by logical formulas that define them, and an external
satisfiability modulo theories (SMT) solver is regularly run by the interpreter
to check their basic properties.
The language is implemented as a Haskell module.Comment: In Proceedings MSFP 2016, arXiv:1604.0038
All reducts of the random graph are model-complete
We study locally closed transformation monoids which contain the automorphism
group of the random graph. We show that such a transformation monoid is locally
generated by the permutations in the monoid, or contains a constant operation,
or contains an operation that maps the random graph injectively to an induced
subgraph which is a clique or an independent set. As a corollary, our
techniques yield a new proof of Simon Thomas' classification of the five closed
supergroups of the automorphism group of the random graph; our proof uses
different Ramsey-theoretic tools than the one given by Thomas, and is perhaps
more straightforward. Since the monoids under consideration are endomorphism
monoids of relational structures definable in the random graph, we are able to
draw several model-theoretic corollaries: One consequence of our result is that
all structures with a first-order definition in the random graph are
model-complete. Moreover, we obtain a classification of these structures up to
existential interdefinability.Comment: Technical report not intended for publication in a journal. Subsumed
by the more recent article 1003.4030. Length 14 pages
The wonderland of reflections
A fundamental fact for the algebraic theory of constraint satisfaction
problems (CSPs) over a fixed template is that pp-interpretations between at
most countable \omega-categorical relational structures have two algebraic
counterparts for their polymorphism clones: a semantic one via the standard
algebraic operators H, S, P, and a syntactic one via clone homomorphisms
(capturing identities). We provide a similar characterization which
incorporates all relational constructions relevant for CSPs, that is,
homomorphic equivalence and adding singletons to cores in addition to
pp-interpretations. For the semantic part we introduce a new construction,
called reflection, and for the syntactic part we find an appropriate weakening
of clone homomorphisms, called h1 clone homomorphisms (capturing identities of
height 1).
As a consequence, the complexity of the CSP of an at most countable
-categorical structure depends only on the identities of height 1
satisfied in its polymorphism clone as well as the the natural uniformity
thereon. This allows us in turn to formulate a new elegant dichotomy conjecture
for the CSPs of reducts of finitely bounded homogeneous structures.
Finally, we reveal a close connection between h1 clone homomorphisms and the
notion of compatibility with projections used in the study of the lattice of
interpretability types of varieties.Comment: 24 page
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