Henkin quantifiers and complete problems

Certain applications of logic to computer science, for example those discus: in [4], have the following features: only finite models are of interest, first-order logic is not sufficiently expressive. Accordingly, various extensions first-order logic have been proposed and analyzed for finite structures. " analysis often reveals connections with computational complexity, and ind some of the extensions were proposed just for the purpose of obtaining s connections. (See [4], [7] and the works cited there.) It is natural to consider ~ those extensions of first-order logic which appear in the logical literature whose introduction was motivated by considerations quite distant from comp~ science. One such extension, perhaps the oldest that can be meaningfully applie~ finite models, is the logic with partially ordered quantification introduce¢ Henkin [6]. We show in this paper that formulas obtained from first-order a by application of these so-called Henkin quantifiers axe able to express . complete properties of finite structures, for example 3-colorability of t graphs. This expressive power persists even if rather stringent restrictions imposed on the Henkin quantifiers. However, Henkin quantifiers of extremely restricted sort, which we call narrow Henkin quantifiers, express, applied to first-order formulas, exactly the co-N~ properties of finite structu We use the customary notations and conventions of first-order logic equality, except that we also allow Boolean variables, a~, f l , . . . , which over {true, false}. These can easily be eliminated by coding, but the coding ,~