Combinatorics of Monotone Computations

, and (ii) arbitrary real-valued non-decreasing functions on 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 . 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 also implies corresponding lower bounds for the length of cutting planes proof in the propositional calculus.

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Received: July 2, 1996/Revised: Revised June 7, 1998

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Jukna, S. Combinatorics of Monotone Computations. Combinatorica 19, 65–85 (1999).

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  • AMS Subject Classification (1991) Classes:  03D15, 68R05; 05D15, 05B30, 05C65