The threshold order of a Boolean function
Discrete Applied Mathematics
On the degree of polynomials that approximate symmetric Boolean functions (preliminary version)
STOC '92 Proceedings of the twenty-fourth annual ACM symposium on Theory of computing
Polynomial threshold functions, AC0 functions, and spectral norms
SIAM Journal on Computing
Rank-r decision trees are a subclass of r-decision lists
Information Processing Letters
Surveys in combinatorics, 1993
PP is closed under intersection
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Classification by polynomial surfaces
Discrete Applied Mathematics
On the degree of Boolean functions as real polynomials
Computational Complexity - Special issue on circuit complexity
Perceptrons, PP, and the polynomial hierarchy
Computational Complexity - Special issue on circuit complexity
Journal of Computer and System Sciences - Special issue: 26th annual ACM symposium on the theory of computing & STOC'94, May 23–25, 1994, and second annual Europe an conference on computational learning theory (EuroCOLT'95), March 13–15, 1995
An efficient membership-query algorithm for learning DNF with respect to the uniform distribution
Journal of Computer and System Sciences
A note on quantum black-box complexity of almost all Boolean functions
Information Processing Letters
STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
Quantum lower bounds by polynomials
Journal of the ACM (JACM)
Learning Intersections and Thresholds of Halfspaces
FOCS '02 Proceedings of the 43rd Symposium on Foundations of Computer Science
New degree bounds for polynomial threshold functions
Proceedings of the thirty-fifth annual ACM symposium on Theory of computing
Representing boolean functions using polynomials: more can offer less
ISNN'11 Proceedings of the 8th international conference on Advances in neural networks - Volume Part III
Unbounded-error quantum query complexity
Theoretical Computer Science
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In this paper we give new extremal bounds on polynomial threshold function (PTF) representations of Boolean functions. Our results include the following:*Almost every Boolean function has PTF degree at most n2+O(nlogn). Together with results of Anthony and Alon, this establishes a conjecture of Wang and Williams [C. Wang, A.C. Williams, The threshold order of a Boolean function, Discrete Appl. Math. 31 (1991) 51-69] and Aspnes, Beigel, Furst, and Rudich [J. Aspnes, R. Beigel, M. Furst, S. Rudich, The expressive power of voting polynomials, Combinatorica 14 (2) (1994) 1-14] up to lower order terms. *Every Boolean function has PTF density at most (1-1O(n))2^n. This improves a result of Gotsman [C. Gotsman, On Boolean functions, polynomials and algebraic threshold functions, Technical Report TR-89-18, Department of Computer Science, Hebrew University, 1989]. *Every Boolean function has weak PTF density at most o(1)2^n. This gives a negative answer to a question posed by Saks [M. Saks, Slicing the hypercube, in: London Math. Soc. Lecture Note Ser., vol. 187, 1993, pp. 211-257]. *PTF degree @?log"2m@?+1 is necessary and sufficient for Boolean functions with sparsity m. This answers a question of Beigel [R. Beigel, personal communication, 2000]. We also give new extremal bounds on polynomials which approximate Boolean functions in the @?"~ norm.