Bounded-depth, polynomial-size circuits for symmetric functions
Theoretical Computer Science
Definability by constant-depth polynomial-size circuits
Information and Control
Journal of Computer and System Sciences - 3rd Annual Conference on Structure in Complexity Theory, June 14–17, 1988
Journal of Computer and System Sciences
Relational expressive power of constraint query languages
Journal of the ACM (JACM)
Languages defined with modular counting quantifiers
Information and Computation
An Ehrenfeucht-Fraïssé Approach to Collapse Results for First-Order Queries over Embedded Databases
STACS '01 Proceedings of the 18th Annual Symposium on Theoretical Aspects of Computer Science
Padding and the Expressive Power of Existential Second-Order Logics
CSL '97 Selected Papers from the11th International Workshop on Computer Science Logic
Applications of Alfred Tarski's Ideas in Database Theory
CSL '01 Proceedings of the 15th International Workshop on Computer Science Logic
Embedded Finite Models, Stability Theory and the Impact of Order
LICS '98 Proceedings of the 13th Annual IEEE Symposium on Logic in Computer Science
A theorem on probabilistic constant depth Computations
STOC '84 Proceedings of the sixteenth annual ACM symposium on Theory of computing
LICS '01 Proceedings of the 16th Annual IEEE Symposium on Logic in Computer Science
Arithmetic, first-order logic, and counting quantifiers
ACM Transactions on Computational Logic (TOCL)
The descriptive complexity approach to LOGCFL
STACS'99 Proceedings of the 16th annual conference on Theoretical aspects of computer science
Constraint Databases
An Ehrenfeucht-Fraïssé game approach to collapse results in database theory
Information and Computation
Extensional Uniformity for Boolean Circuits
CSL '08 Proceedings of the 22nd international workshop on Computer Science Logic
Languages with bounded multiparty communication complexity
STACS'07 Proceedings of the 24th annual conference on Theoretical aspects of computer science
Extensional Uniformity for Boolean Circuits
SIAM Journal on Computing
Characterizing definability of second-order generalized quantifiers
WoLLIC'11 Proceedings of the 18th international conference on Logic, language, 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
On the expressive power of FO[+]
LATA'10 Proceedings of the 4th international conference on Language and Automata Theory and Applications
Non-definability of Languages by Generalized First-order Formulas over (N,+)
LICS '12 Proceedings of the 2012 27th Annual IEEE/ACM Symposium on Logic in Computer Science
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A language L over an alphabet A is said to have a neutral letter if there is a letter e@?A such that inserting or deleting e's from any word in A^* does not change its membership or non-membership in L. The presence of a neutral letter affects the definability of a language in first-order logic. It was conjectured that it renders all numerical predicates apart from the order predicate useless, i.e., that if a language L with a neutral letter is not definable in first-order logic with linear order, then it is not definable in first-order logic with any set N of numerical predicates. Named after the location of its first, flawed, proof this conjecture is called the Crane Beach conjecture (CBC, for short). The CBC is closely related to uniformity conditions in circuit complexity theory and to collapse results in database theory. We investigate the CBC in detail, showing that it fails for N={+,x}, or, possibly stronger, for any set N that allows counting up to the m times iterated logarithm, for any constant m. On the positive side, we prove the conjecture for the case of all monadic numerical predicates, for the addition predicate +, for the fragment BC(@S"1) of first-order logic, for regular languages, and for languages over a binary alphabet. We explain the precise relation between the CBC and so-called natural-generic collapse results in database theory. Furthermore, we introduce a framework that gives better understanding of what exactly may cause a failure of the conjecture.