By Heinz-Dieter Ebbinghaus, Jörg Flum

It is a completely revised and enlarged moment variation (the first version was once released within the "Perspectives in Mathematical good judgment" sequence in 1995) that offers the most result of descriptive complexity conception, that's, the connections among axiomatizability of periods of finite buildings and their complexity with admire to time and area bounds. The logics which are very important during this context comprise fixed-point logics, transitive closure logics, and in addition sure infinitary languages; their version concept is studied in complete aspect. The publication is written in any such method that the respective elements on version idea and descriptive complexity thought might be learn independently.

The publication provides the most result of descriptive complexity idea, that's, the connections among axiomatizability of periods of finite buildings 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 in addition definite infinitary languages; their version conception is studied in complete element. different themes comprise DATALOG languages, quantifiers and oracles, 0-1 legislation, and optimization and approximation difficulties. The ebook is written in any such approach that the respective components on version conception and descriptive complexity idea might be learn independently. This moment variation is a completely revised and enlarged model of the unique text.

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Now, for the second quantifier 3x, the spoiler selects in B the element e, thereby getting a witness ior B \= 3xy < x[d]. Obviously the old value c for x is no longer relevant. Therefore, the play above may be represented more informatively by x-box ^-box first move second move third move A B A B A B a * c * a b c d ? 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.

Pr) be the structure obtained by splitting the cycle (I>/,Pi,... ,P^) into two cycles by removing the edges (a_,a), (6_,6) and adding edges {b-,a), (a-,b) instead; more formally: E[ := (^A{(«-,«),(&-,^)})U{(6_,a),(a_,6)}. ,Pr)=m {Di,E[,Pu • • -^Pr)- Proof. Immediate by Hanf's Theorem, since both structures have the same number of 3"^-balls of any given isomorphism type. 4 For r = {E,Pi,... 4-2. Let I > IQ and (Qi, P i , . . , Pr) be a r-structure, where Qi is the Gaifman graph Q{Vi) ofVi, that is, Qi is a cycle of length / + 1.

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. Then in a structure of cardinality > IQ := {i -h l)(2-3"^ H- 1) there must be two points with disjoint 3"^-balls of the same isomorphism type. 3 Suppose (X>/,Pi,... ,Pr) is a r-structure (r as in the preceding lemma) containing elements a and 6 with disjoint and isomorphic 3"^balls.