Logic

Finite Model Theory by Heinz-Dieter Ebbinghaus, Jörg Flum

By Heinz-Dieter Ebbinghaus, Jörg Flum

This can be a completely revised and enlarged moment variation (the first variation was once released within the "Perspectives in Mathematical common sense" sequence in 1995) that provides the most result of descriptive complexity idea, that's, the connections among axiomatizability of sessions of finite buildings and their complexity with appreciate to time and house bounds. The logics which are very important during this context comprise fixed-point logics, transitive closure logics, and likewise convinced infinitary languages; their version conception is studied in complete element. The publication is written in the sort of manner that the respective components on version conception and descriptive complexity concept might be learn independently.

The e-book offers the most result of descriptive complexity concept, that's, the connections among axiomatizability of periods of finite constructions and their complexity with appreciate to time and area bounds. The logics which are vital during this context contain fixed-point logics, transitive closure logics, and likewise definite infinitary languages; their version thought is studied in complete aspect. different subject matters contain DATALOG languages, quantifiers and oracles, 0-1 legislation, and optimization and approximation difficulties. The ebook is written in this type of means that the respective elements on version idea and descriptive complexity idea can be learn independently. This moment version is a completely revised and enlarged model of the unique text.

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Example text

G. Di - { 0 , . . , / } , El := {(i, i + l)\i 0. Then there is anlo > 1 such that for any I > IQ and any r-structure of the form (P/, P i , . . , P^) there are a,b £ Di with disjoint and isomorphic 3"^-balls. Proof For the structures under consideration any 3^-ball contains exactly 2 - 3 ^ -\- 1 elements (note that Pi,... ,Pr are unary and therefore do not influence the distances induced by the underlying digraphs).

B e d where the a;-boxes and the y-boxes always contain the actual value for x and y, respectively, and * stands for an empty box. 50 3. More on Games This example motivates how to adapt the Ehrenfeucht-Fraisse method to our situation: As always, fix a vocabulary r. By convention, * will not belong to the universe of any structure. For a £ {AU {*})^,a = a i . a^, let supp(a) := {i | a^ G A} be the support of a, and if a G A, let a j denote ai... ai-iaai+i . . ag. For a G (AU{*})* and b G (5U{*})* we say that a i-)- 6 is an s-partial isomorphism from A to B, if supp(a) = supp(6) and a' \-^b is a partial isomorphism from A to B, where a' and b are the subsequences of a and b with indices in the support.

2 Suppose r = {E, P i , . . , P^} where P i , . . , P^ are unary, and let m > 0. Then there is anlo > 1 such that for any I > IQ and any r-structure of the form (P/, P i , . . , P^) there are a,b £ Di with disjoint and isomorphic 3"^-balls. Proof For the structures under consideration any 3^-ball contains exactly 2 - 3 ^ -\- 1 elements (note that Pi,... ,Pr are unary and therefore do not influence the distances induced by the underlying digraphs). Let i be the number of possible isomorphism types of 3^-balls.

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