Descriptive Complexity of Finite Structures: Saving the Quantifier Rank

Details Descriptive Complexity of Finite Structures: Saving the Quantifier Rank Descriptive Complexity of Finite Structures: Saving the Quantifier Rank

2003-05-16

arxiv.org/abs/math/0305244v2

Mathematics

Logic

39 pages

Scientific paper

Given a relational structure M on n elements, let D(M) be the minimum quantifier rank of a first order formula identifying M up to isomorphism in the class of n-element structures. The obvious upper bound is D(M)\le n. We show that if the relations in M have arity at most k, then D(M)<(1-\frac{1}{2k})n+k^2-k+2. The coefficient at n, which equals 1-\frac{1}{2k}, is probably not best possible but this is the first known bound having it strictly below 1 (for fixed k). If one is content to have the worse coefficient 1-\frac{1}{2k^2+2}, then one can choose an identifying formula of a very special form: a prenex formula with at most one quantifier alternation. A few other results in this vein are presented.

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