From polynomial time queries to graph structure theory
Proceedings of the 13th International Conference on Database Theory
Fixed-point definability and polynomial time
CSL'09/EACSL'09 Proceedings of the 23rd CSL international conference and 18th EACSL Annual conference on Computer science logic
Fixed-point definability and polynomial time on chordal graphs and line graphs
Fields of logic and computation
Experimental descriptive complexity
Logic and Program Semantics
Pebble games with algebraic rules
ICALP'12 Proceedings of the 39th international colloquium conference on Automata, Languages, and Programming - Volume Part II
Fixed-point definability and polynomial time on graphs with excluded minors
Journal of the ACM (JACM)
Maximum Matching and Linear Programming in Fixed-Point Logic with Counting
LICS '13 Proceedings of the 2013 28th Annual ACM/IEEE Symposium on Logic in Computer Science
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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 rk_p for determining the rank of a matrix over the finite field GF_p 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 rank operators are surprisingly expressive, even in the absence of fixed-point operators. We show that FO+rk_p can define deterministic and symmetric transitive closure. This allows us to show that, on ordered structures, FO+rk_p captures the complexity class MOD_pL, for all prime values of p.