Monotone circuits for connectivity require super-logarithmic depth
STOC '88 Proceedings of the twentieth annual ACM symposium on Theory of computing
Improvements on Khrapchenko's theorem
Theoretical Computer Science
On submodular complexity measures
Poceedings of the London Mathematical Society symposium on Boolean function complexity
Poceedings of the London Mathematical Society symposium on Boolean function complexity
Fractional Covers and Communication Complexity
SIAM Journal on Discrete Mathematics
Better lower bounds for monotone threshold formulas
Journal of Computer and System Sciences - Special issue: papers from the 32nd and 34th annual symposia on foundations of computer science, Oct. 2–4, 1991 and Nov. 3–5, 1993
Communication complexity
The Shrinkage Exponent of de Morgan Formulas is 2
SIAM Journal on Computing
Quantum lower bounds by quantum arguments
Journal of Computer and System Sciences - Special issue on STOC 2000
Las Vegas is better than determinism in VLSI and distributed computing (Extended Abstract)
STOC '82 Proceedings of the fourteenth annual ACM symposium on Theory of computing
Polynomial Degree vs. Quantum Query Complexity
FOCS '03 Proceedings of the 44th Annual IEEE Symposium on Foundations of Computer Science
THE QUANTUM ADVERSARY METHOD AND CLASSICAL FORMULA SIZE LOWER BOUNDS
Computational Complexity
Theoretical Computer Science
Optimal lower bounds on regular expression size using communication complexity
FOSSACS'08/ETAPS'08 Proceedings of the Theory and practice of software, 11th international conference on Foundations of software science and computational structures
Breaking the rectangle bound barrier against formula size lower bounds
MFCS'10 Proceedings of the 35th international conference on Mathematical foundations of computer science
A stronger LP bound for formula size lower bounds via clique constraints
Theoretical Computer Science
On the structure of boolean functions with small spectral norm
Proceedings of the 5th conference on Innovations in theoretical computer science
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We introduce a new technique for proving formula size lower bounds based on matrix rank. A simple form of this technique gives bounds at least as large as those given by the method of Khrapchenko, originally used to prove an n2 lower bound on the parity function. Applying our method to the parity function, we are able to give an exact expression for the formula size of parity: if n = 2l + k, where 0 ≤ k l, then the formula size of parity on n bits is exactly 2l(2l+3k) = n2+k2l-k2. Such a bound cannot be proven by any of the lower bound techniques of Khrapchenko, Nečiporuk, Koutsoupias, or the quantum adversary method, which are limited by n2.