Algebraic methods in the theory of lower bounds for Boolean circuit complexity
STOC '87 Proceedings of the nineteenth annual ACM symposium on Theory of computing
Finite monoids and the fine structure of NC1
Journal of the ACM (JACM)
Bounded-width polynomial-size branching programs recognize exactly those languages in NC1
Journal of Computer and System Sciences - 18th Annual ACM Symposium on Theory of Computing (STOC), May 28-30, 1986
Non-uniform automata over groups
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Journal of Computer and System Sciences
Finite automata, formal logic, and circuit complexity
Finite automata, formal logic, and circuit complexity
Regular languages defined with generalized quantifiers
Information and Computation
Superlinear lower bounds for bounded-width branching programs
Journal of Computer and System Sciences
Counting modulo quantifiers on finite structures
Information and Computation - Special issue: LICS 1996—Part 1
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Journal of the ACM (JACM)
LICS '01 Proceedings of the 16th Annual IEEE Symposium on Logic in Computer Science
Some Results on Majority Quantifiers over Words
CCC '04 Proceedings of the 19th IEEE Annual Conference on Computational Complexity
Elements Of Finite Model Theory (Texts in Theoretical Computer Science. An Eatcs Series)
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ACM Transactions on Computational Logic (TOCL)
An Ehrenfeucht-Fraïssé game approach to collapse results in database theory
Information and Computation
An algebraic point of view on the crane beach property
CSL'06 Proceedings of the 20th international conference on Computer Science Logic
Linear circuits, two-variable logic and weakly blocked monoids
MFCS'07 Proceedings of the 32nd international conference on Mathematical Foundations of Computer Science
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We consider an extension of first-order logic by modular quantifiers of a fixed modulus q. Drawing on collapse results from finite model theory and techniques of finite semigroup theory, we show that if the only available numerical predicate is addition, then sentences in this logic cannot define the set of bit strings in which the number of 1's is divisible by a prime p that does not divide q. More generally, we completely characterize the regular languages definable in this logic. The corresponding statement, with addition replaced by arbitrary numerical predicates, is equivalent to the conjectured separation of the circuit complexity class ACC from NC1. Thus our theorem can be viewed as proving a highly uniform version of the conjecture.