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.

We prove tight lower bounds, of up to n ffl , for the monotone depth of functions in monotone-P. As a result we achieve the separation of the following classes. 1. monotone-NC 6= monotone-P. 2. For every i 1, monotone-NC i 6= monotone-NC i+1 . 3. More generally: For any integer function D(n), up to n ffl (for some ffl ? 0), we give an explicit example of a monotone Boolean function, that can be computed by polynomial size monotone Boolean circuits of depth D(n), but that cannot be computed by any (fan-in 2) monotone Boolean circuits of depth less than Const \Delta D(n) (for some constant Const). Only a separation of monotone-NC 1 from monotone-NC 2 was previously known. Our argument is more general: we define a new class of communication complexity search problems, referred to below as DART games, and we prove a tight lower bound for the communication complexity of every member of this class. As a result we get lower bounds for the monotone depth of many functions. In...

Both the bottleneck counting argument [7, 8] and Razborov's approximation method [1, 4, 12] have been used to prove exponential lower bounds for monotone circuits. We show that under the monotone circuit model for every proof by the approximation method, there is a bottleneck counting proof and vice versa. We also illustrate the elegance of the bottleneck counting technique with a simple self-explained example: the proof of a (previously known) lower bound for the 3-CLIQUEn problem by the bottleneck counting argument. Keywords: Computational complexity; Circuit complexity; Monotone circuit complexity 1 Introduction Razborov's proof of an exponential lower bound on the size of monotone Boolean circuits to detect cliques in a graph [12] [1], represented a breakthrough in the theory of monotone circuit complexity. The proof introduced the method of approximation. The method is roughly as follows. Consider two sets of test inputs, a positive (the output is 1) and a negative one. Gi...

Our main result is a combinatorial lower bounds criterion for a general model of monotone circuits, where we allow as gates: (i) arbitrary monotone Boolean functions whose minterms or maxterms (or both) have length 6 d, and (ii) arbitrary real-valued non-decreasing functions on 6 d variables. This resolves a problem, raised by Razborov in 1986, and yields, in a uniform and easy way, non-trivial lower bounds for circuits computing explicit functions even when d !1. The proof is relatively simple and direct, and combines the bottlenecks counting method of Haken with the idea of finite limit due to Sipser. We demonstrate the criterion by super-polynomial lower bounds for explicit Boolean functions, associated with bipartite Paley graphs and partial t-designs. We then derive exponential lower bounds for clique-like graph functions of Tardos, thus establishing an exponential gap between the monotone real and non-monotone Boolean circuit complexities. Since we allow real gates, the criterion...

We prove a lower bound of 2\Omega i ( n log n ) 1 3 j on the monotone size of an explicit function in monotone-NP (where n is the number of input variables). This is higher than any previous lower bound on the monotone size of a function. The previous best being a lower bound of about 2 \Omega\Gamma n 1 4 ) for Andreev's function, proved in [AlBo87]. Our lower bound is proved by the symmetric version of Razborov's method of approximations. However, we present this method in a new and simpler way: Rather than building approximator functions for all the gates in a circuit, we use a gate elimination argument that is based on a Monotone Switching Lemma. The bound applies for a family of functions, each defined by a construction of a small probability space of c-wise independent random variables. 1 Introduction 1.1 Background and Previous Work A monotone function is one that can be computed by a monotone circuit i.e., a circuit with only AND and OR gates. The monoton...