An Exponential Lower Bound for the Size of Monotone Real Circuits

AbstractWe prove a lower bound, exponential in the eighth root of the input length, on the size of monotone arithmetic circuits that solve an NP problem related to clique detection. The result is more general than the famous lower bound of Razborov and Andreev, because the gates of the circuit are allowed to compute arbitrary monotone binary real-valued functions (including AND and OR). Our proof is relatively simple and direct and uses the method of counting bottlenecks. The generalization was proved independently by Pudlák using a different method, who also showed that the result can be used to obtain an exponential lower bound on the size of unrestricted cutting plane proofs in the propositional calculus

Lower Bounds for Monotone Counting Circuits

A {+,x}-circuit counts a given multivariate polynomial f, if its values on 0-1 inputs are the same as those of f; on other inputs the circuit may output arbitrary values. Such a circuit counts the number of monomials of f evaluated to 1 by a given 0-1 input vector (with multiplicities given by their coefficients). A circuit decides $f$ if it has the same 0-1 roots as f. We first show that some multilinear polynomials can be exponentially easier to count than to compute them, and can be exponentially easier to decide than to count them. Then we give general lower bounds on the size of counting circuits.Comment: 20 page

An Exponential Lower Bound for the Size of Monotone Real Circuits

AbstractWe prove a lower bound, exponential in the eighth root of the input length, on the size of monotone arithmetic circuits that solve an NP problem related to clique detection. The result is more general than the famous lower bound of Razborov and Andreev, because the gates of the circuit are allowed to compute arbitrary monotone binary real-valued functions (including AND and OR). Our proof is relatively simple and direct and uses the method of counting bottlenecks. The generalization was proved independently by Pudlák using a different method, who also showed that the result can be used to obtain an exponential lower bound on the size of unrestricted cutting plane proofs in the propositional calculus

An Exponential Lower Bound for the Size of Monotone Real Circuits

We prove a lower bound, exponential in the eighth root of the input length, on the size of monotone arithmetic circuits that solve an NP problem related to clique detection. The result is more general than the famous lower bound of Razborov and Andreev, because the gates of the circuit are allowed to compute arbitrary monotone binary real-valued functions (including AND and OR). Our proof is relatively simple and direct, and uses the method of counting bottlenecks. The generalization was proved independently by Pudl'ak using a different method, who also showed that the result can be used to obtain an exponential lower bound on the size of unrestricted cutting plane proofs in the propositional calculus. 1 Introduction The Razborov/Andreev [Raz85] [AB87] [And85] exponential lower bound on the size of monotone Boolean circuits which detect cliques represented a breakthrough in the theory of monotone circuit complexity. The proof introduced the method of approximation, which has been use..