Jump to content

Counting quantification

From Wikipedia, the free encyclopedia

A counting quantifier is a mathematical term for a quantifier of the form "there exists at least k elements that satisfy property X". In first-order logic with equality, counting quantifiers can be defined in terms of ordinary quantifiers, so in this context they are a notational shorthand. However, they are interesting in the context of logics such as two-variable logic with counting that restrict the number of variables in formulas. Also, generalized counting quantifiers that say "there exists infinitely many" are not expressible using a finite number of formulas in first-order logic.

Definition in terms of ordinary quantifiers[edit]

Counting quantifiers can be defined recursively in terms of ordinary quantifiers.

Let denote "there exist exactly ". Then

Let denote "there exist at least ". Then

See also[edit]

References[edit]

  • Erich Graedel, Martin Otto, and Eric Rosen. "Two-Variable Logic with Counting is Decidable." In Proceedings of 12th IEEE Symposium on Logic in Computer Science LICS `97, Warschau. 1997. Postscript file OCLC 282402933