On vectorizations of unary generalized quantifiers

Archive for Mathematical Logic 51 (3-4):241-255 (2012)
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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 )



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Definability hierarchies of general quantifiers.Lauri Hella - 1989 - Annals of Pure and Applied Logic 43 (3):235.
Hierarchies of monadic generalized quantifiers.Kerkko Luosto - 2000 - Journal of Symbolic Logic 65 (3):1241-1263.
δ-Logics and generalized quantifiers.J. A. Makowsky - 1976 - Annals of Mathematical Logic 10 (2):155.

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