Shrinking games and local formulas

AbstractGaifman's normal form theorem showed that every first-order sentence of quantifier rank n is equivalent to a Boolean combination of “scattered local sentences”, where the local neighborhoods have radius at most 7n−1. This bound was improved by Lifsches and Shelah to 3×4n−1. We use Ehrenfeucht–Fraı̈ssé type games with a “shrinking horizon” to get a spectrum of normal form theorems of the Gaifman type, depending on the rate of shrinking. This spectrum includes the result of Lifsches and Shelah, with a more easily understood proof and with the bound on the radius improved to 4n−1. We also obtain bounds for a normal form theorem of Schwentick and Barthelmann

Almost everywhere elimination of probability quantifiers

AbstractWe obtain an almost everywhere quantifier elimination for (the noncritical fragment of) the logic with probability quantifiers, introduced by the first author in [10]. This logic has quantifiers like ∃≥3/4y which says that “for at least 3/4 of all y”. These results improve upon the 0-1 law for a fragment of this logic obtained by Knyazev [11]. Our improvements are:1. We deal with the quantifier ∃≥ry, where y is a tuple of variables.2. We remove the closedness restriction, which requires that the variables in y occur in all atomic subformulas of the quantifier scope.3. Instead of the unbiased measure where each model with universe n has the same probability, we work with any measure generated by independent atomic probabilities PR for each predicate symbol R.4. We extend the results to parametric classes of finite models (for example, the classes of bipartite graphs, undirected graphs, and oriented graphs).5. We extend the results to a natural (noncritical) fragment of the infinitary logic with probability quantifiers.6. We allow each PR, as well as each r in the probability quantifier (∃≥ry), to depend on the size of the universe.</jats:p

Rank hierarchies for generalized quantifiers

We show that for each n and m, there is an existential first order sentence that is NOT logically equivalent to a sentence of quantifier rank at most m in infinitary logic augmented with all generalized quantifiers of arity at most n. We use this to show the strictness of the quantifier rank hierarchies for various logics ranging from existential (or universal) fragments of first-order logic to infinitary logics augmented with arbitrary classes of generalized quantifiers of bounded arity.The sentence above is also shown to be equivalent to a first-order sentence with at most n+2 variables (free and bound). This gives the strictness of the quantifier rank hierarchies for various logics with only n+2 variables. The proofs use the bijective Ehrenfeucht-Fraïsse game and a modification of the building blocks of Hella. © The Author, 2010. Published by Oxford University Press. All rights reserved