The function class #P lacks an important closure property: it is not closed under subtraction. To remedy this problem, we introduce the function class GapP as a natural alternative to #P. GapP is the closure of #P under subtraction, and has all the other useful closure properties of #P as well. We show that most previously studied counting classes, including PP, C=P, and Mod k P, are "gap-definable," i.e., definable using the values of GapP functions alone. We show that there is a smallest gap-definable class, SPP, which is still large enough to contain Few. We also show that SPP consists of exactly those languages low for GapP, and thus SPP languages are low for any gap-definable class. These results unify and improve earlier disparate results of Cai & Hemachandra [7] and Kobler, Schoning, Toda, & Tor'an [15]. We show further that any countable collection of languages is contained in a unique minimum gap-definable class, which implies that the gap-definable classes form a lattice un...

The representation of functions as low-degree polynomials over various rings has provided many insights in the theory of small-depth circuits. We survey some of the closure properties, upper bounds, and lower bounds obtained via this approach. 1. Introduction There is a long history of using polynomials in order to prove complexity bounds. Minsky and Papert [39] used polynomials to prove early lower bounds on the order of perceptrons. Razborov [46] and Smolensky [49] used them to prove lower bounds on the size of AND-OR circuits. Other lower bounds via polynomials are due to [50, 4, 10, 51, 9, 55]. Paturi and Saks [44] discovered that rational functions could be used for lower bounds on the size of threshold circuits. Toda [53] used polynomials to prove upper bounds on the power of the polynomial hierarchy. This led to a series of upper bounds on the power of the polynomial hierarchy [54, 52], AC 0 [2, 3, 52, 19], and ACC [58, 20, 30, 37], and related classes [21, 42]. Beigel and Gi...

We give the first polynomial time algorithm to learn any function of a constant number of halfspaces under the uniform distribution to within any constant error parameter. We also give the first quasipolynomial time algorithm for learning any function of a polylog number of polynomial-weight halfspaces under any distribution. As special cases of these results we obtain algorithms for learning intersections and thresholds of halfspaces. Our uniform distribution learning algorithms involve a novel non-geometric approach to learning halfspaces; we use Fourier techniques together with a careful analysis of the noise sensitivity of functions of halfspaces. Our algorithms for learning under any distribution use techniques from real approximation theory to construct low degree polynomial threshold functions.

The study of the complexity of sets encompasses two complementary aims: (1) establishing -- usually via explicit construction of algorithms -- that sets are feasible, and (2) studying the relative complexity of sets that plausibly might be feasible but are not currently known to be feasible (such as the NP-complete sets and the PSPACE-complete sets). For the study of the complexity of closure properties, a recent urry of results [21, 33, 49, 6, 7, 16] has established an analog of (1); these papers explicitly demonstrate many closure properties possessed by PP and C=P (and the proofs implicitly give closure properties of the function class #P). The present paper presents and develops, for function classes such as #P, SpanP, OptP, and MidP, an analog of (2): a general theory of the complexity of closure properties. In particular, we show that subtraction is hard for the closure properties of each of these classes: each is closed under subtraction if and only if it is closed under every polynom...

. We introduce a technique of arithmetization of the process of computation in order to obtain novel characterizations of certain complexity classes via multivariate polynomials. A variety of concepts and tools of elementary algebra, such as the degree of polynomials and interpolation, becomes thereby available for the study of complexity classes. The theory to be described provides a unified framework from which powerful recent results follow naturally. The central result is a characterization of ]P in terms of arithmetic straight line programs. The consequences include a simplified proof of Toda's Theorem that PH ` P ]P ; and an infinite class of natural and potentially inequivalent functions, checkable in the sense of Blum et al. Similar characterizations of PSPACE are also given. The arithmetization technique was independently discovered by Adi Shamir. While this simultaneous discovery was driven by applications to interactive proofs, the present paper demonstrates the applicabil...

Beigel, Reingold and Spielman [BRS] showed that PP is closed under intersection and a variety of special cases of truth-table closure. We extend the techniques in [BRS] to show PP is closed under general polynomial-time truth-table reductions. 1 Introduction In the seminal paper on probabilistic computation, Gill [G] defined the class PP, the class of problems decidable by a probabilistic polynomial-time Turing machine that need only accept a string with probability at least one-half. Gill left open the question as to whether PP is closed under intersection. Recently Beigel, Reingold and Spielman [BRS] showed that in fact PP is closed under intersection. They also showed PP is closed under a variety of other reductions including polynomial-time conjunctive and disjunctive reductions, bounded-depth Boolean formula reductions, O(logn) Turing reductions, threshold reductions, symmetric reductions, and multilinear reductions. However they left open the question as to whether PP is closed ...

Several recent nonrelativizing results in the area of interactive proofs have caused many people to review the importance of relativization. In this paper we take a look at how complexity theorists use and misuse oracle results. We pay special attention to the new interactive proof systems and program checking results and try to understand why they do not relativize. We give some new results that may help us to understand these questions better.

We give an algorithm to learn constant-depth polynomial-size circuits augmented with majority gates under the uniform distribution using random examples only. For circuits which contain a polylogarithmic number of majority gates the algorithm runs in quasipolynomial time. This is the first algorithm for learning a more expressive circuit class than the class AC° of constant-depth polynomial-size circuits, a class which was shown to be learnable in quasipolynomial time by Linial, Mansour and Nisan in 1989. Our approach combines an extension of some of the Fourier analysis from Linial et al. with hypothesis boosting. We also show that under a standard cryptographic assumption our algorithm is essentially optimal with respect to both running time and expressiveness (number of majority gates) of the circuits being learned.

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I study the class of problems efficiently solvable by a quantum computer, given the ability to “postselect” on the outcomes of measurements. I prove that this class coincides with a classical complexity class called PP, or Probabilistic Polynomial-Time. Using this result, I show that several simple changes to the axioms of quantum mechanics would let us solve PP-complete problems efficiently. The result also implies, as an easy corollary, a celebrated theorem of Beigel, Reingold, and Spielman that PP is closed under intersection, as well as a generalization of that theorem due to Fortnow and Reingold. This illustrates that quantum computing can yield new and simpler proofs of major results about classical computation.