Separating the polynomial-time hierarchy by oracles
Proc. 26th annual symposium on Foundations of computer science
Relational queries computable in polynomial time
Information and Control
Languages that capture complexity classes
SIAM Journal on Computing
Constant depth circuits, Fourier transform, and learnability
Journal of the ACM (JACM)
Locality of order-invariant first-order formulas
ACM Transactions on Computational Logic (TOCL)
Invariant Definability and P/poly
Proceedings of the 12th International Workshop on Computer Science Logic
The complexity of relational query languages (Extended Abstract)
STOC '82 Proceedings of the fourteenth annual ACM symposium on Theory of computing
Elements Of Finite Model Theory (Texts in Theoretical Computer Science. An Eatcs Series)
Elements Of Finite Model Theory (Texts in Theoretical Computer Science. An Eatcs Series)
LICS '07 Proceedings of the 22nd Annual IEEE Symposium on Logic in Computer Science
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
Addition-Invariant FO and Regularity
LICS '10 Proceedings of the 2010 25th Annual IEEE Symposium on Logic in Computer Science
Deciding First-Order Properties for Sparse Graphs
FOCS '10 Proceedings of the 2010 IEEE 51st Annual Symposium on Foundations of Computer Science
A toolkit for proving limitations of the expressive power of logics
MFCS'12 Proceedings of the 37th international conference on Mathematical Foundations of Computer Science
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We consider first-order 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 Arb-invariant first-order. 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 Arb-invariant first-order 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 Arb-invariant first-order formulas and the complexity class AC0, and hinges on the tight lower bounds for parity on constant-depth circuits.