We say that a first order formula Φ distinguishes a structure M over vocabulary L from another structure M ′ over the same vocabulary if Φ is true on M but false on M ′. A formula Φ defines an L-structure M if Φ distinguishes M from any other non-isomorphic L-structure M ′. A formula Φ identifies an n-element L-structure M if Φ distinguishes M from any other non-isomorphic n-element L-structure...

In his pioneering work Mostowski (1957) made the very insightful suggestion that quantifiers are special higher order relations, relations between relations (where sets are unary relations). This move led eventually to the generalised quantifier theory (GQT). In particular it made it possible to study ”classical” quantifiers using familiar tools of the elementary theory of relations. This simpl...

Abstract For every q ∈ N let FOq denote the class of sentences of first-order logic FO of quantifier rank at most q. If a graph property can be defined in FOq, then it can be decided in time O(n). Thus, minimizing q has favorable algorithmic consequences. Many graph properties amount to the existence of a certain set of vertices of size k. Usually this can only be expressed by a sentence of qua...

In property testing a small, random sample of an object is taken and one wishes to distinguish with high probability between the case where it has a desired property and the case where it is far from having the property. Much of the recent work has focused on graphs. In the present paper three generalized models for testing relational structures are introduced and relationships between these va...

We propose a refinement of the usual Ehrenfeucht-Fräıssé game. The new game will help us make finer distinctions than the traditional one. In particular, it can be used to measure not only quantifier rank but also lengths of conjunctions and disjunctions needed for expressing a given property. Our game is similar to the game in [1] and in [5]. The most common measure of complexity of a first or...

Natural language quantifiers (e.g., few, many, not much) and probability expressions (e.g., likely, possible, not probable) are used to communicate information about quantities, frequencies, and probabilities. A great deal of psychological research addressed the role of these expressions in communication and thinking. Some of the problems addressed include how quantifiers and probability expres...

The language of arithmetical theories considered here is L = +, ×, ≤, 0, 1 in which the symbols are interpreted as usual in elementary mathematics. Robinson's arithmetic is denoted by Q; it is a finitely axiomatized basic theory of the function and predicate symbols in L. Peano's arithmetic PA is the first-order theory that extends Q by the induction schema for any L-formula ϕ(x): ϕ(0) & ∀x(ϕ(x...

1. Let us agree on the following notation. Lower-case Latin letters from a to n (inclusively) with indices and without them will be used as variables for nonnegative integers, the remaining lower-case Latin letters will be used as variables for integers. Analogously, lower case Greek letters from to will be used as metavariables for nonnegative integers, and the rest of the Greek letters will b...

Abstract We survey optimization problems that allow natural simple formulations with one existential and universal quantifier. summarize the theoretical background from computational complexity theory, we present a multitude of illustrating examples. discuss connections to robust bilevel optimization, explain reasons why operational research community should be interested in aspects this area.