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 good judgment" sequence in 1995) that provides the most result of descriptive complexity thought, that's, the connections among axiomatizability of periods 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 sure infinitary languages; their version concept is studied in complete element. The e-book is written in this sort of means that the respective components on version conception and descriptive complexity thought should be learn independently.

The e-book provides the most result of descriptive complexity idea, that's, the connections among axiomatizability of periods of finite constructions and their complexity with admire to time and house bounds. The logics which are vital during this context contain fixed-point logics, transitive closure logics, and likewise convinced infinitary languages; their version idea is studied in complete element. different themes contain DATALOG languages, quantifiers and oracles, 0-1 legislation, and optimization and approximation difficulties. The booklet is written in this kind of approach that the respective elements on version conception and descriptive complexity conception could be learn independently. This moment variation is a completely revised and enlarged model of the unique text.

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**Example text**

Now d{e',a) > d{a',a) — d{a',e') > I — r = 2r — r > ri. Therefore, a ^ S {ri,e'), and hence S {ri,e') = S^{ri,e'). 6 As already mentioned in the introduction to this chapter, the algebraic characterization of m-equivalence is due to Fraisse [43], its gametheoretic version to Ehrenfeucht [33]. The j-isomorphism types were introduced by Hintikka [85] in a different context. 5 to Ajtai and Gurevich [7]. Further references for the results in this chapter are [5, 78, 39]. The "local" character as it becomes apparent for first-order logic in the theorems of Hanf and Gaifman is studied in [83, 115, 116, 69].

3Pr x with unary P i , . . 5 Gaifman's Theorem Fix a relational r. Let ^ be a r-structure. A subset M of ^ is l-scattered, if the distance (in the Gaifman graph G{A)) between any two elements of M exceeds /. ^ Gaifman's Theorem states that every first-order sentence is logically equivalent to a boolean combination of such sentences. It thus is a further formulation of ^ Note that, due to 2r-scatteredness, the balls S{r,a) for a G M are pairwise disjoint. 5 Gaifman's Theorem 31 the fact already present in Hanf's Theorem that first-order sentences only capture local properties of structures.

The class CONN of connected finite graphs is not axiomatizable in firstorder logic. 12, CONN is not axiomatizable, since for each m we have ^ 2 - e CONN, ^ 2 - 0 ^ 2 - ^ CONN, ^ 2 - = m ^ 2 - U ^ 2 - . 24 2. The Ehrenfeucht-Praisse Method — The global relation TC (cf. 9), the relation of transitive closure on the class GRAPH of finite graphs, is not first-order definable. In fact, suppose ^^(x, ^) is a first-order formula defining TC on GRAPH. Then CONN would be the class of finite models of \/x\/y {-^x = y -^ ip{x,y)) (and the graph axioms).