281 research outputs found
On the Scope of the Universal-Algebraic Approach to Constraint Satisfaction
The universal-algebraic approach has proved a powerful tool in the study of
the complexity of CSPs. This approach has previously been applied to the study
of CSPs with finite or (infinite) omega-categorical templates, and relies on
two facts. The first is that in finite or omega-categorical structures A, a
relation is primitive positive definable if and only if it is preserved by the
polymorphisms of A. The second is that every finite or omega-categorical
structure is homomorphically equivalent to a core structure. In this paper, we
present generalizations of these facts to infinite structures that are not
necessarily omega-categorical. (This abstract has been severely curtailed by
the space constraints of arXiv -- please read the full abstract in the
article.) Finally, we present applications of our general results to the
description and analysis of the complexity of CSPs. In particular, we give
general hardness criteria based on the absence of polymorphisms that depend on
more than one argument, and we present a polymorphism-based description of
those CSPs that are first-order definable (and therefore can be solved in
polynomial time).Comment: Extended abstract appeared at 25th Symposium on Logic in Computer
Science (LICS 2010). This version will appear in the LMCS special issue
associated with LICS 201
Uniformly defining valuation rings in Henselian valued fields with finite or pseudo-finite residue fields
We give a definition, in the ring language, of Z_p inside Q_p and of F_p[[t]]
inside F_p((t)), which works uniformly for all and all finite field
extensions of these fields, and in many other Henselian valued fields as well.
The formula can be taken existential-universal in the ring language, and in
fact existential in a modification of the language of Macintyre. Furthermore,
we show the negative result that in the language of rings there does not exist
a uniform definition by an existential formula and neither by a universal
formula for the valuation rings of all the finite extensions of a given
Henselian valued field. We also show that there is no existential formula of
the ring language defining Z_p inside Q_p uniformly for all p. For any fixed
finite extension of Q_p, we give an existential formula and a universal formula
in the ring language which define the valuation ring
On Kirchberg's Embedding Problem
Kirchberg's Embedding Problem (KEP) asks whether every separable C
algebra embeds into an ultrapower of the Cuntz algebra . In this
paper, we use model theory to show that this conjecture is equivalent to a
local approximate nuclearity condition that we call the existence of good
nuclear witnesses. In order to prove this result, we study general properties
of existentially closed C algebras. Along the way, we establish a
connection between existentially closed C algebras, the weak expectation
property of Lance, and the local lifting property of Kirchberg. The paper
concludes with a discussion of the model theory of . Several
results in this last section are proven using some technical results concerning
tubular embeddings, a notion first introduced by Jung for studying embeddings
of tracial von Neumann algebras into the ultrapower of the hyperfinite II
factor.Comment: 42 pages; final version to appear in the Journal of Functional
Analysi
A survey of local-global methods for Hilbert's Tenth Problem
Hilbert's Tenth Problem (H10) for a ring R asks for an algorithm to decide
correctly, for each , whether the
diophantine equation has a solution in R. The celebrated
`Davis-Putnam-Robinson-Matiyasevich theorem' shows that {\bf H10} for
is unsolvable, i.e.~there is no such algorithm. Since then,
Hilbert's Tenth Problem has been studied in a wide range of rings and fields.
Most importantly, for {number fields and in particular for }, H10
is still an unsolved problem. Recent work of Eisentr\"ager, Poonen,
Koenigsmann, Park, Dittmann, Daans, and others, has dramatically pushed forward
what is known in this area, and has made essential use of local-global
principles for quadratic forms, and for central simple algebras. We give a
concise survey and introduction to this particular rich area of interaction
between logic and number theory, without assuming a detailed background of
either subject. We also sketch two further directions of future research, one
inspired by model theory and one by arithmetic geometry
Fields and Fusions: Hrushovski constructions and their definable groups
An overview is given of the various expansions of fields and fusions of
strongly minimal sets obtained by means of Hrushovski's amalgamation method, as
well as a characterization of the groups definable in these structures
A Fragment of Dependence Logic Capturing Polynomial Time
In this paper we study the expressive power of Horn-formulae in dependence
logic and show that they can express NP-complete problems. Therefore we define
an even smaller fragment D-Horn* and show that over finite successor structures
it captures the complexity class P of all sets decidable in polynomial time.
Furthermore we study the question which of our results can ge generalized to
the case of open formulae of D-Horn* and so-called downwards monotone
polynomial time properties of teams
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