Results 1 
4 of
4
Locality of Queries Definable in Invariant FirstOrder Logic with Arbitrary Builtin Predicates
"... Abstract. We consider firstorder formulas over relational structures which may use arbitrary numerical predicates. We require that the validity of the formula is independent of the particular interpretation of the numerical predicates and refer to such formulas as Arbinvariant firstorder. Our mai ..."
Abstract

Cited by 4 (2 self)
 Add to MetaCart
(Show Context)
Abstract. We consider firstorder formulas over relational structures which may use arbitrary numerical predicates. We require that the validity of the formula is independent of the particular interpretation of the numerical predicates and refer to such formulas as Arbinvariant firstorder. Our main result shows a Gaifman locality theorem: two tuples of a structure with n elements, having the same neighborhood up to distance (log n) ω(1), cannot be distinguished by Arbinvariant firstorder formulas. When restricting attention to word structures, we can achieve the same quantitative strength for Hanf locality. In both cases we show that our bounds are tight. Our proof exploits the close connection between Arbinvariant firstorder formulas and the complexity class AC 0, and hinges on the tight lower bounds for parity on constantdepth circuits. 1
LOCALITY FROM CIRCUIT LOWER BOUNDS
, 2012
"... We study the locality of an extension of firstorder logic that captures graph queries computable in AC 0, i.e., by families of polynomialsize constantdepth circuits. The extension considers firstorder formulas over relational structures which may use arbitrary numerical predicates in such a way ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
(Show Context)
We study the locality of an extension of firstorder logic that captures graph queries computable in AC 0, i.e., by families of polynomialsize constantdepth circuits. The extension considers firstorder formulas over relational structures which may use arbitrary numerical predicates in such a way that their truth value is independent of the particular interpretation of the numerical predicates. We refer to such formulas as Arbinvariant firstorder. We consider the two standard notions of locality, Gaifman and Hanf locality. Our main result gives a Gaifman locality theorem: An Arbinvariant firstorder formula cannot distinguish between two tuples that have the same neighborhood up to distance (log n) c,wheren represents the number of elements in the structure and c is a constant depending on the formula. When restricting attention to string structures, we achieve the same quantitative strength for Hanf locality. In both cases we show that our bounds are tight. We also present an application of our results to the study of regular languages. Our proof exploits the close connection between firstorder formulas and the complexity class AC 0 and hinges on the tight lower bounds for parity on constantdepth circuits.
On optimal proof systems and logics for PTIME
"... We prove that TAUT has a poptimal proof system if and only if a logic related to least fixedpoint logic captures polynomial time on all finite structures. Furthermore, we show that TAUT has no effective poptimal proof system if NTIME(hO(1)) 6 ⊆ DTIME(hO(log h)) for every time constructible and i ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
(Show Context)
We prove that TAUT has a poptimal proof system if and only if a logic related to least fixedpoint logic captures polynomial time on all finite structures. Furthermore, we show that TAUT has no effective poptimal proof system if NTIME(hO(1)) 6 ⊆ DTIME(hO(log h)) for every time constructible and increasing function h. 1.