On the constant-depth complexity of k-clique
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Reductions for monotone Boolean circuits
Theoretical Computer Science
Linear-size log-depth negation-limited inverter for k-tonic binary sequences
Theoretical Computer Science
Ehrenfeucht-Fraïssé Games on Random Structures
WoLLIC '09 Proceedings of the 16th International Workshop on Logic, Language, Information and Computation
Limiting Negations in Formulas
ICALP '09 Proceedings of the 36th International Colloquium on Automata, Languages and Programming: Part I
Note: Limiting negations in non-deterministic circuits
Theoretical Computer Science
Linear-size log-depth negation-limited inverter for k-tonic binary sequences
TAMC'07 Proceedings of the 4th international conference on Theory and applications of models of computation
Limiting negations in bounded treewidth and upward planar circuits
MFCS'10 Proceedings of the 35th international conference on Mathematical foundations of computer science
On the negation-limited circuit complexity of sorting and inverting k-tonic sequences
COCOON'06 Proceedings of the 12th annual international conference on Computing and Combinatorics
Reductions for monotone boolean circuits
MFCS'06 Proceedings of the 31st international conference on Mathematical Foundations of Computer Science
Negation-Limited complexity of parity and inverters
ISAAC'06 Proceedings of the 17th international conference on Algorithms and Computation
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In this paper, we investigate the lower bound on the number of gates in a Boolean circuit that computes the clique function with a limited number of negation gates. To derive strong lower bounds on the size of such a circuit we develop a new approach by combining three approaches: the restriction applied to constant depth circuits due to Håstad, the approximation method applied to monotone circuits due to Razborov, and the boundary covering developed in the present paper. We prove that if a circuit $C$ with at most $\floor{(1/6) \log \log m}$ negation gates detects cliques of size $(\log m)^{3(\log m)^{1/2}}$ in a graph with $m$ vertices, then $C$ contains at least $2^{(1/5)(\log m)^{(\log m)^{1/2}}}$ gates. No nontrivial lower bounds on the size of such circuits were previously known, even if we restrict the number of negation gates to be a constant. Moreover, it follows from a result of Fischer [{\it Lect. Notes Comput. Sci.,} 33 (1974), pp. 71--82] that if one can improve the number of negation gates from $\floor{(1/6)\log\log m}$ to $\floor{2\log m}$ in the statement, then we have P $\neq$ NP. We also show that the problem of lower bounding the negation-limited circuit complexity can be reduced to the one of lower bounding the maximum of the monotone circuit complexity of the functions in a certain class of monotone functions.