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Elements Of Finite Model Theory (Texts in Theoretical Computer Science. An Eatcs Series)
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Foundations and Trends® in Theoretical Computer Science
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SAT'11 Proceedings of the 14th international conference on Theory and application of satisfiability testing
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Electronic Notes in Theoretical Computer Science (ENTCS)
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
Parameterized Complexity of DPLL Search Procedures
ACM Transactions on Computational Logic (TOCL)
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We prove a lower bound of ω(nk/4) on the size of constant-depth circuits solving the k-clique problem on n-vertex graphs (for every constant k). This improves a lower bound of ω(nk/89d2) due to Beame where d is the circuit depth. Our lower bound has the advantage that it does not depend on the constant d in the exponent of n, thus breaking the mold of the traditional size-depth tradeoff. Our k-clique lower bound derives from a stronger result of independent interest. Suppose fn :0,1n/2 → {0,1} is a sequence of functions computed by constant-depth circuits of size O(nt). Let G be an Erdos-Renyi random graph with vertex set {1,...,n} and independent edge probabilities n-α where α ≤ 1/2t-1. Let A be a uniform random k-element subset of {1,...,n} (where k is any constant independent of n) and let KA denote the clique supported on A. We prove that fn(G) = fn(G ∪ KA) asymptotically almost surely. These results resolve a long-standing open question in finite model theory (going back at least to Immerman in 1982). The m-variable fragment of first-order logic, denoted by FOm, consists of the first-order sentences which involve at most m variables. Our results imply that the bounded variable hierarchy FO1 ⊂ FO2 ⊂ ... ⊂ FOm ⊂ ... is strict in terms of expressive power on finite ordered graphs. It was previously unknown that FO3 is less expressive than full first-order logic on finite ordered graphs.