Vectorization of a class of structures is a natural notion in finite model theory. Roughly speaking, vectorizations allow tuples to be treated similarly to elements of structures. The importance of vectorizations is highlighted by the fact that if the complexity class PTIME corresponds to a logic with reasonable syntax, then it corresponds to a logic generated via vectorizations by a single generalized quantifier (Dawar in J Log Comput 5(2):213–226, 1995). It is somewhat surprising, then, that there have been few systematic (...) studies of the expressive power of vectorizations of various quantifiers. In the present paper, we consider the simplest case: the cardinality quantifiers C S. We show that, in general, the expressive power of the vectorized quantifier logic $${{\rm FO}(\{{\mathsf C}_S^{(n)}\, | \, n \in \mathbb{Z}_+\})}$$ is much greater than the expressive power of the non-vectorized logic FO(C S ). (shrink)

A combinatorial criterium is given when a monadic quantifier is expressible by means of universe-independent monadic quantifiers of width n. It is proved that the corresponding hierarchy does not collapse. As an application, it is shown that the second resumption (or vectorization) of the Hartig quantifier is not definable by monadic quantifiers. The techniques rely on Ramsey theory.

The concept of a generalized quantifier of a given similarity type was defined in [12]. Our main result says that on finite structures different similarity types give rise to different classes of generalized quantifiers. More exactly, for every similarity type t there is a generalized quantifier of type t which is not definable in the extension of first order logic by all generalized quantifiers of type smaller than t. This was proved for unary similarity types by Per Lindström [17] with (...) a counting argument. We extend his method to arbitrary similarity types. (shrink)

The concept of a generalized quantifier of a given similarity type was defined in [12]. Our main result says that on finite structures different similarity types give rise to different classes of generalized quantifiers. More exactly, for every similarity typetthere is a generalized quantifier of typetwhich is not definable in the extension of first order logic by all generalized quantifiers of type smaller thant. This was proved for unary similarity types by Per Lindström [17] with a counting argument. We extend (...) his method to arbitrary similarity types. (shrink)

We prove that if ℵα is uncountable and regular, then the Beth-closure of Lωω(Qα) is not a sublogic of L∞ω(Qn), where Qn is the class of all n-ary generalized quantifiers. In particular, B(Lωω(Qα)) is not a sublogic of any finitely generated logic; i.e., there does not exist a finite set Q of Lindstrom quantifiers such that B(Lωω(Qα)) ≤ Lωω(Q).

We prove that if $\aleph_\alpha$ is uncountable and regular, then the Beth-closure of $\mathscr{L}_{\omega\omega}(Q_\alpha)$ is not a sublogic of $\mathscr{L}_{\infty\omega}(\mathbf{Q}_n)$, where $\mathbf{Q}_n$ is the class of all $n$-ary generalized quantifiers. In particular, $B(\mathscr{L}_{\omega\omega}(Q_\alpha))$ is not a sublogic of any finitely generated logic; i.e., there does not exist a finite set $\mathbf{Q}$ of Lindstrom quantifiers such that $B(\mathscr{L}_{\omega\omega}(Q_\alpha)) \leq \mathscr{L}_{\omega\omega}(\mathbf{Q})$.

We introduce a new framework for classifying logics on finite structures and studying their expressive power. This framework is based on the concept of almost everywhere equivalence of logics, that is to say, two logics having the same expressive power on a class of asymptotic measure 1. More precisely, if L, L ′ are two logics and μ is an asymptotic measure on finite structures, then $\scr{L}\equiv _{\text{a.e.}}\scr{L}^{\prime}(\mu)$ means that there is a class C of finite structures with μ (C)=1 (...) and such that L and L ′ define the same queries on C. We carry out a systematic investigation of $\equiv _{\text{a.e.}}$ with respect to the uniform measure and analyze the $\equiv _{\text{a.e.}}$ -equivalence classes of several logics that have been studied extensively in finite model theory. Moreover, we explore connections with descriptive complexity theory and examine the status of certain classical results of model theory in the context of this new framework. (shrink)

We carry out a systematic investigation of the definability of linear order on classes of finite rigid structures. We obtain upper and lower bounds for the expressibility of linear order in various logics that have been studied extensively in finite model theory, such as least fixpoint logic LFP, partial fixpoint logic PFP, infinitary logic Lω∞ω with a finite number of variables, as well as the closures of these logics under implicit definitions. Moreover, we show that the upper and lower bounds (...) established here cannot be made substantially tighter, unless outstanding conjectures in complexity theory are resolved at the same time. (shrink)