Logical Hierarchies in PTIME

Information And Computation 129 (1):1--19 (1996)
We consider the problem of finding a characterization for polynomial time computable queries on finite structures in terms of logical definability. It is well known that fixpoint logic provides such a characterization in the presence of a built-in linear order, but without linear order even very simple polynomial time queries involving counting are not expressible in fixpoint logic. Our approach to the problem is based on generalized quantifiers. A generalized quantifier isn-ary if it binds any number of formulas, but at mostnvariables in each formula. We prove that, for each natural numbern, there is a query on finite structures which is expressible in fixpoint logic, but not in the extension of first-order logic by any set ofn-ary quantifiers. It follows that the expressive power of fixpoint logic cannot be captured by adding finitely many quantifiers to first-order logic. Furthermore, we prove that, for each natural numbern, there is a polynomial time computable query which is not definable in any extension of fixpoint logic byn-ary quantifiers. In particular, this rules out the possibility of characterizing PTIME in terms of definability in fixpoint logic extended by a finite set of generalized quantifiers
Keywords complexity   computational   descriptive   generalized   quantifiers
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Martin Grohe (1996). Arity Hierarchies. Annals of Pure and Applied Logic 82 (2):103-163.

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