Characterizing Polynomial Ramsey Quantifiers

Probabilistic Quantifiers and Games

We consider inclusion relations among a multitude of classical complexity classes and classes with probabilistic components. A key tool is a method for characterizing such classes in terms of the ordinary quantifiers 3 and V together with a quantifier 3+, which means roughly “for most,” applied to polynomial-time predicates. This approach yields a uniform treatment which leads to easier proofs ...

Ramsey Theory Is Needed for Solving Definability Problems of Generalized Quantifiers

In recent years, generalized quantifiers (see [H3]) have received quite a lot of novel interest because of their applications to computer science and linguistics. Their definability theory has made considerable progress during the last decade, which will be the subject of the next section. The proofs of many of these results often use results of Ramsey theory, such as theorems of van der Waerde...

Definability properties and the congruence closure

We introduce a natural class of quantifiers Th containing all monadic type quantifiers, all quantifiers for linear orders, quantifiers for isomorphism, Ramsey type quantifiers, and plenty more, showing that no sublogic of Lo~o,(Th) or countably compact regular sublogic of L~o,(Th), properly extending L,o,o, satisfies the uniform reduction property for quotients. As a consequence, none of these ...

A Dichotomy Result for Ramsey Quantifiers

The quantifier complexity of polynomial-size iterated definitions in first-order logic

We refine the constructions of Ferrante-Rackoff and Solovay on iterated definitions in first-order logic and their expressibility in with polynomial size formulas. These constructions introduce additional quantifiers; however, we show that these extra quantifiers range over only finite sets. We prove optimal upper and lower bounds on the quantifier complexity of polynomial size formulas obtaine...