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Separation of the Monotone NC Hierarchy
, 1999
"... We prove tight lower bounds, of up to n ffl , for the monotone depth of functions in monotoneP. As a result we achieve the separation of the following classes. 1. monotoneNC 6= monotoneP. 2. For every i 1, monotoneNC i 6= monotoneNC i+1 . 3. More generally: For any integer function D( ..."
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We prove tight lower bounds, of up to n ffl , for the monotone depth of functions in monotoneP. As a result we achieve the separation of the following classes. 1. monotoneNC 6= monotoneP. 2. For every i 1, monotoneNC i 6= monotoneNC 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 (fanin 2) monotone Boolean circuits of depth less than Const \Delta D(n) (for some constant Const). Only a separation of monotoneNC 1 from monotoneNC 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...
Combinatorics of Monotone Computations
 Combinatorica
, 1998
"... 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 realvalued nondecreasing functions on 6 d variables. This r ..."
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Cited by 8 (4 self)
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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 realvalued nondecreasing functions on 6 d variables. This resolves a problem, raised by Razborov in 1986, and yields, in a uniform and easy way, nontrivial 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 superpolynomial lower bounds for explicit Boolean functions, associated with bipartite Paley graphs and partial tdesigns. We then derive exponential lower bounds for cliquelike graph functions of Tardos, thus establishing an exponential gap between the monotone real and nonmonotone Boolean circuit complexities. Since we allow real gates, the criterion...
Higher Lower Bounds On Monotone Size
 Proc. of 32nd STOC (2000
, 2000
"... We prove a lower bound of 2\Omega i ( n log n ) 1 3 j on the monotone size of an explicit function in monotoneNP (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 ..."
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We prove a lower bound of 2\Omega i ( n log n ) 1 3 j on the monotone size of an explicit function in monotoneNP (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 cwise 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...
Monotone Complexity by Switching Lemma
"... We prove a lower bound of 2 log n ) on the monotone size of an explicit function in monotoneNP (where n is the number of input variables). This is higher than any previous lower bound on the monotone size of a function. Our lower bound is proved by the symmetric version of Razborov' ..."
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We prove a lower bound of 2 log n ) on the monotone size of an explicit function in monotoneNP (where n is the number of input variables). This is higher than any previous lower bound on the monotone size of a function. 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 cwise independent random variables.
COMBINATORICA Bolyai Society – SpringerVerlag COMBINATORICA 19 (1) (1999) 65–85 COMBINATORICS OF MONOTONE COMPUTATIONS STASYS JUKNA*
, 1996
"... 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 ≤ d, and (ii) arbitrary realvalued nondecreasing functions on ≤ d variables. This r ..."
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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 ≤ d, and (ii) arbitrary realvalued nondecreasing functions on ≤ d variables. This resolves a problem, raised by Razborov in 1986, and yields, in a uniform and easy way, nontrivial lower bounds for circuits computing explicit functions even when d →∞. 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 superpolynomial lower bounds for explicit Boolean functions, associated with bipartite Paley graphs and partial tdesigns. We then derive exponential lower bounds for cliquelike graph functions of Tardos, thus establishing an exponential gap between the monotone real and nonmonotone 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. 1.
SPECIAL ISSUE: ANALYSIS OF BOOLEAN FUNCTIONS Tight Bounds for Monotone Switching Networks via Fourier Analysis
, 2012
"... Abstract: We prove tight size bounds on monotone switching networks for the NPcomplete problem of kclique, and for an explicit monotone problem by analyzing a pyramid structure of height h for the Pcomplete problem of generation. This gives alternative proofs of the separations of mNC from mP a ..."
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Abstract: We prove tight size bounds on monotone switching networks for the NPcomplete problem of kclique, and for an explicit monotone problem by analyzing a pyramid structure of height h for the Pcomplete problem of generation. This gives alternative proofs of the separations of mNC from mP and of mNCi from mNCi+1, different from Raz–McKenzie (Combinatorica 1999). The enumerativecombinatorial and Fourier analytic techniques in this paper are very different from a large body of work on circuit depth lower bounds, and may be of independent interest. ACM Classification: F.1.3 AMS Classification: 68Q17, 68Q15, 68Q10 Key words and phrases: lower bounds, space complexity, parallel complexity, monotone complexity, switching networks, Fourier analysis