Generalized Quantifiers for Simple Properties

We consider extensions of xed-point logic by means of generalized quantiiers in the context of descriptive complexity. By the well-known theorem of Immerman and Vardi, xed-point logic captures PTime over linearly ordered structures. It fails, however , to express even most fundamental structural properties, like simple cardinality properties, in the absence of order. In the present investigation we concentrate on extensions by generalized quantiiers which serve to adjoin simple or basic structural properties. An abstract notion of simplicity is put forward which isolates those structural properties, that can be characterized in terms of a concise structural invariant. The key examples are provided by all monadic and cardinality properties in a very general sense. The main theorem establishes that no extension by any family of such simple quantiiers can cover all of PTime. These limitations are proved on the basis of the semantically motivated notion of simplicity; in particular there is no implicit bound on the arities of the generalized quantiiers involved. Quite to the contrary , the natural applications concern innnite families of quantiiers adjoining certain structural properties across all arities in a uniform way.