3,554 research outputs found
Satisfaction classes in nonstandard models of first-order arithmetic
A satisfaction class is a set of nonstandard sentences respecting Tarski's
truth definition. We are mainly interested in full satisfaction classes, i.e.,
satisfaction classes which decides all nonstandard sentences. Kotlarski,
Krajewski and Lachlan proved in 1981 that a countable model of PA admits a
satisfaction class if and only if it is recursively saturated. A proof of this
fact is presented in detail in such a way that it is adaptable to a language
with function symbols. The idea that a satisfaction class can only see finitely
deep in a formula is extended to terms. The definition gives rise to new
notions of valuations of nonstandard terms; these are investigated. The notion
of a free satisfaction class is introduced, it is a satisfaction class free of
existential assumptions on nonstandard terms.
It is well known that pathologies arise in some satisfaction classes. Ideas
of how to remove those are presented in the last chapter. This is done mainly
by adding inference rules to M-logic. The consistency of many of these
extensions is left as an open question.Comment: Thesis for the degree of licentiate of philosophy, 74 pages, 4
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Motivic integration and the Grothendieck group of pseudo-finite fields
We survey our recent work on an extension of the theory of motivic
integration, called arithmetic motivic integration. We developed this theory to
understand how p-adic integrals of a very general type depend on p.Comment: 11 page
Meta-Kernelization using Well-Structured Modulators
Kernelization investigates exact preprocessing algorithms with performance
guarantees. The most prevalent type of parameters used in kernelization is the
solution size for optimization problems; however, also structural parameters
have been successfully used to obtain polynomial kernels for a wide range of
problems. Many of these parameters can be defined as the size of a smallest
modulator of the given graph into a fixed graph class (i.e., a set of vertices
whose deletion puts the graph into the graph class). Such parameters admit the
construction of polynomial kernels even when the solution size is large or not
applicable. This work follows up on the research on meta-kernelization
frameworks in terms of structural parameters.
We develop a class of parameters which are based on a more general view on
modulators: instead of size, the parameters employ a combination of rank-width
and split decompositions to measure structure inside the modulator. This allows
us to lift kernelization results from modulator-size to more general
parameters, hence providing smaller kernels. We show (i) how such large but
well-structured modulators can be efficiently approximated, (ii) how they can
be used to obtain polynomial kernels for any graph problem expressible in
Monadic Second Order logic, and (iii) how they allow the extension of previous
results in the area of structural meta-kernelization
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