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Journal of Logic and Computation 1995 5(2):213-226; doi:10.1093/logcom/5.2.213
© 1995 by Oxford University Press
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Original Articles

Generalized Quantifiers and Logical Reducibilities

ANUJ DAWAR

Department of Computer Science, University College of Swansea Swansea SA2 8PP, UK. E-mail: a.dawar{at}swansea.ac.uk

We consider the problem of defining a logic that captures exactly the polynomial time computable properties of finite structures, by extending first-order logic (FO) and fixed-point logic (FP) by means of Lindstrom quantifiers. In particular,we define infinite uniform sequences of quantifiers and show that these correspond to a natural notion of logical reducibility. We show that if there is any logic that captures the complexity class PTTME, in the sense of a recursive enumeration of PTIME properties, then there is one that is an extension of FO by a uniform sequence of quantifiers. This is established through a general result linking the existence of complete problems for a complexity class to the existence of recursive index sets for the class, for a wide variety of complexity classes.

Keywords: Finite model theory; polynomial time computability; complexity theory; logical reducibilities.


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