These decision problems remain open also in the case of a single constraint graph, though I would argue that in this case, at least, it should be decidable. Other natural model theoretic issues, lacking a clear motivation in graph theory, have not been addressed to date.
Bazzoni and J. In the talk, we focus on a key set-theoretic lemma which derives from earlier work of Shelah, Mekler and Fuchs. The necessity to choose elements from nonempty sets arises in computing but the epsilon operator doesn't fit the bill.
Besides, the extension of first-order logic with the epsilon operator is complex: its expressivity equals that of the existential monadic second-order logic. We introduce a different choice operator, d , that fits computing better. While e is a fixed choice operator, d is an independent choice operator. Different evaluations of d X for the same X may produce different results. Furthermore, there is no correlation between different evaluations. Somewhat surprisingly, the extension of first-order logic with delta does not increase the expressive power of first-order logic.
See details in Journal of Symbolic Logic 3 65 Coles, G. LaForte, and R. Downey, and will appear in [ 1 ].
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It was shown by C. Jockusch and R. We study what more can be said about such an A under the additional hypothesis that the pseudojump operator J is nontrivial , i. A sample theorem is that A can be chosen to be d. Our most surprising result is that upper cone avoidance fails, i. References  R.
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Coles, R. Downey, C. Jockusch and G. LaForte, Completing pseudojump operators. Pure Appl. Logic, to appear. Shore, Pseudo jump operators I: The r. The idea is to find nice invariants for a class of mathematical objects, up to isomorphism or other equivalence, or else to say, in some concrete way, that there are no nice invariants. There are results on classification in different branches of logic.
I will describe some recent work in computable structure theory, which was inspired by work in descriptive set theory. In recursion theory one asks how unsolvable is a problem, while in computational complexity theory one asks how difficult, in terms of time, storage, and other measures, it is to solve a problem.
In this lecture I will describe the influences that traditional recursion theory have had on computational complexity theory. Some of the topics I will cover are resource bounded reducibilities, hierarchies, complexity classes, and structural properties of sets. I will describe some of the highlights in computational complexity in the past 30 years.
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We will discuss the proof of this result, along with an approach for bounding the quantifier complexity of trivial, uncountably categorical theories as a function of the rank of the theory. Some of this is joint work with Alf Dolich and Alex Raichev. One consequence they drew is that an omega-stable, omega-categorical theory is not finitely axiomatizable.
What part of the full structure theory is sufficient for the non-finite axiomatizability? The key fact is that an omega-stable, omega-categorical theory is one-based. Alex has written an excellent thesis in the area of computable model theory. The latter is a subject that nicely combines model-theoretic ideas with delicate recursion-theoretic constructions. In his thesis, Alex begins by reviewing the essential model-theoretic facts, especially the Baldwin-Lachlan result about uncountably categorical theories. This he follows with a brief discussion of recursion theory, including mention of the priority method.
First-order Model Theory
The deepest part of the thesis concerns the study of the recursive spectrum of an uncountably categorical theory, i. This is a deep and very active area of contemporary research in computable model theory which Alex discusses in considerable detail. The exposition is very good and therefore the thesis makes a nice introduction to the subject for a wide community of logicians. Read more Read less. No customer reviews.