We consider the problem of approximating a Boolean function~: {O,

Abstract. We construct a predicate that is computable by a perceptron with linear size, order 1, and exponential weights, but which cannot be computed by any perceptron having subexponential (2 ~~ size, subpolynomial (n~ order and subexponential weights. A consequence is that there is an oracle relative to which pNP is not contained in PP.

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We consider worst case time bounds for several NP-complete problems, based on a constraint satisfaction (CSP) formulation of these problems: (a, b)-CSP instances consist of a set of variables, each with up to a possible values, and constraints disallowing certain b-tuples of variable values; a problem is solved by assigning values to all variables satisfying all constraints, or by showing that no such assignment exist. 3-SAT is equivalent to (2, 3)-CSP while 3-coloring and various related problems are special cases of (3, 2)-CSP; there is also a natural duality transformation from (a, b)-CSP to (b, a)-CSP. We show that n-variable (3, 2)-CSP instances can be solved in time O(1.3645 n ), that satisfying assignments to (d, 2)-CSP instances can be found in randomized expected time O((0.4518d) n ); that 3-coloring of n-vertex graphs can be solved in time O(1.3289 n ); that 3-list-coloring of n-vertex graphs can be solved in time O(1.3645 n ); that 3-edge-coloring of n-vertex graphs can be solved in time O(2 n/2 ), and that 3-satisfiability of a formula with t 3-clauses can be solved in time O(n O(1) + 1.3645 t ).

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Abstract.Chang and Kadin have shown that if the difference hierarchy over NP collapses to level k, then the polynomial hierarchy (PH) is equal to the kth level of the difference hierarchy over E~. We simplify their poof and obtain a slightly stronger conclusion: if the difference hierarchy over NP collapses to level k, then PH collapses to (P~P_ l)_tt) NP, the class of sets recognized in polynomial time with k-1 nonadaptive queries to a set in NP NP and an unlimited number of queries to a set in NP. We also extend the result to classes other than NP: For any class C that has

Abstract.We show that every language L in the class ACC can be recognized by depth-two deterministic circuits with a symmetricfunction gate at the root and 2 l~176 AND gates of fan-in log~ at the leaves, or equivalently~ there exist polynomials p~ (xl,.~ ,, x~) over Z of degree log~ and with coefficients of magnitude 2 l~176 and functions h,~ : Z ---, {0, 1} such that for each n and eo~ch x C {0, 1} n, XL(X) = h,.(p~(xt,...,xn)).This improves an earlier result of Yao (1985). We also analyze and improve modulus-amplifying polynomiMs constructed by Toda (1991) andYao (1985)i Subject classifications. 68Q05, 68Q15, 68Q25.

In his seminal paper on probabilistic Turing machines, Gill [9] asked whether the class PP is closed under intersection and union. We give a positive answer to this question. In fact, PP is closed under polynomial-time multilineal reductions. In circuits, this allows us to combine several threshold gates into a single threshold gate, while increasing depth by only a constant.