Logics with Rank Operators

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Logics with Rank Operators
—We introduce extensions of first-order logic (FO) and fixed-point logic (FP) with operators that compute the rank of a definable matrix. These operators are generalizations of the counting operations in FP+C (i.e. fixed-point logic with counting) that allow us to count the dimension of a definable vector space, rather than just count the cardinality of a definable set. The logics we define have data complexity contained in polynomial time and all known examples of polynomial time queries that are not definable in FP+C are definable in FP+rk, the extension of FP with rank operators. For each prime number p and each positive integer n, we have rank operators rkp for determining the rank of a matrix over the finite field GFp defined by a formula over n-tuples. We compare the expressive power of the logics obtained by varying the values p and n can take. In particular, we show that increasing the arity of the operators yields an infinite hierarchy of expressive power. The r...
Anuj Dawar, Martin Grohe, Bjarki Holm, Bastian Lau
Added 24 May 2010
Updated 24 May 2010
Type Conference
Year 2009
Where LICS
Authors Anuj Dawar, Martin Grohe, Bjarki Holm, Bastian Laubner
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