A complexity theory based on Boolean algebra
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The monotone circuit complexity of Boolean functions
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Journal of the ACM (JACM)
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STOC '96 Proceedings of the twenty-eighth annual ACM symposium on Theory of computing
Lower bounds for monotone span programs
Computational Complexity
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SIAM Journal on Computing
The complexity of matrix rank and feasible systems of linear equations
Computational Complexity
On arithmetic branching programs
Journal of Computer and System Sciences
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Journal of Combinatorial Theory Series A
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FOCS '95 Proceedings of the 36th Annual Symposium on Foundations of Computer Science
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FOCS '95 Proceedings of the 36th Annual Symposium on Foundations of Computer Science
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STOC '82 Proceedings of the fourteenth annual ACM symposium on Theory of computing
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STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
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SCN '08 Proceedings of the 6th international conference on Security and Cryptography for Networks
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TCC '09 Proceedings of the 6th Theory of Cryptography Conference on Theory of Cryptography
Binary Covering Arrays and Existentially Closed Graphs
IWCC '09 Proceedings of the 2nd International Workshop on Coding and Cryptology
On the Readability of Monotone Boolean Formulae
COCOON '09 Proceedings of the 15th Annual International Conference on Computing and Combinatorics
Complexity Lower Bounds using Linear Algebra
Foundations and Trends® in Theoretical Computer Science
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IWCC'11 Proceedings of the Third international conference on Coding and cryptology
On the readability of monotone Boolean formulae
Journal of Combinatorial Optimization
On the size of monotone span programs
SCN'04 Proceedings of the 4th international conference on Security in Communication Networks
On the incompressibility of monotone DNFs
FCT'05 Proceedings of the 15th international conference on Fundamentals of Computation Theory
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We give a characterization of span program size by a combinatorial-algebraic measure. The measure we consider is a generalization of a measure on covers which has been used to prove lower bounds on formula size and has also been studied with respect to communication complexity.In the monotone case our new methods yield nΩ(log n) lower bounds for the monotone span program complexity of explicit Boolean functions in n variables over arbitrary fields, improving the previous lower bounds on monotone span program size. Our characterization of span program size implies that any matrix with superpolynomial separation between its rank and cover number can be used to obtain superpolynomial lower bounds on monotone span program size. We also identify a property of bipartite graphs that is sufficient for constructing Boolean functions with large monotone span program complexity.