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Abstract not found

We extend the recent approach of Papadimitrou and Yannakakis that relates the approximation properties of optimization problems to their logical representation. Our work builds on results by Kolaitis and Thakur who sytematically studied the expressibility classes Max \Sigma n and Max \Pi n of maximization problems and showed that they form a short hierarchy of four levels. The two lowest levels, Max \Sigma 0 and Max \Sigma 1 coincide with the classes Max Snp and Max Np of Papadimitriou and Yannakakis; they contain only problems that are approximable in polynomial time up to a constant factor and thus provide a logical criterion for approximability. However, there are computationally very easy maximization problems, such as Maximum Connected Component (MCC) that fail to satisfy this criterion. We modify these classes by allowing the formulae to contain predicates that are definable in least fixpoint logic. In addition, we maximize not only over relations but also over constants. We cal...

We consider infinitary logic with two variable symbols and counting quantifiers, C 2 , and its intersection with Ptime on finite relational structures. In particular we exhibit a Ptime canonization procedure for finite relational structures which provides unique representatives up to equivalence in C 2 . As a consequence we obtain a recursive presentation for the class of all those queries on arbitrary finite relational structures which are both Ptime and definable in C 2 . The proof renders a succinct normal form representation of this non-trivial semantically defined fragment of Ptime. Through specializations of the proof techniques similar results apply with respect to the logic L 2 , infinitary logic with two variable symbols, itself. 1 Introduction and main results Canonization addresses problems of the following kind: Given a notion of equivalence between relational structures of a certain type, is there an algorithm that determines unique representatives for each equi...

Craig's interpolation theorem and Beth's definability theorem are classical results in model theory that fail when only finite structures are considered. Gurevich (Toward Logic Tailored for Computational Complexity, Springer Lecture Notes in Mathematics 1104 (1984) 175--216) has shown that for any logic that captures polynomial time, the analogues of these theorems (on finite structures) are equivalent to certain open statements in complexity theory. By results of Grollmann and Selman (Complexity measures for public-key cryptosystems, SIAM J. Computing 17 (1988), 309--335) these statements are false if and only if there exist certain kinds of one-way functions (for polynomial time). We extend these results and give direct proofs of the correspondance between one-way functions for a complexity class C and the definability and interpolation principles for any logic that captures C. 1 Introduction Intuitively a one-way function is a function which is easy to compute and hard ...

We develop a variant of Least Fixed Point logic based on First Order logic with a relaxed version of guarded quantification. We develop a Game Theoretic Semantics of this logic, and find that under reasonable conditions, guarding quantification does not reduce the expressibility of Least Fixed Point logic. But guarding quantification increases worst-case time complexity.

We consider extensions of fixed-point logic by means of generalized quantifiers in the context of descriptive complexity. By the well-known theorem of Immerman and Vardi, fixed-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 quantifiers 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 quantifiers can cover all of PTime. These limitations are proved on the basis of the semantically motivated notion of simplicity; in particula...