In his seminal paper on probabilistic Turing machines, Gill [13] asked whether the class PP is closed under intersection and union. We give a positive answer to this question. We also show that PP is closed under a variety of polynomial-time truth-table reductions. Consequences in complexity theory include the definite collapse and (assuming P<F NaN> 6= PP) separation of certain query hierarchies over PP. Similar techniques allow us to combine several threshold gates into a single threshold gate. Consequences in the study of circuits include the simulation of circuits with a small number of threshold gates by circuits having only a single threshold gate at the root (perceptrons), and a lower bound on the number of threshold gates needed to compute the parity function. 1. Introduction The class PP was defined in 1972 by John Gill [13, 14] and independently by Janos Simon [26] in 1974. PP is the class of languages accepted by a polynomial-time bounded nondeterministic Turing machine t...

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...

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...

Chang and Kadin have shown that if the difference hierarchy over NP collapses to level k, then the polynomial hierarchy (PH) is equal the kth level of the difference hierarchy over \Sigma p 2 . We simplify their proof and obtain a slightly stronger conclusion: If the difference hierarchy over NP collapses to level k, then PH collapses to i P NP (k\Gamma1)-tt j NP , the class of sets recognized in polynomial time with k \Gamma 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 p m -complete sets and is closed under p conj - and NP m -reductions (alternatively, closed under p disj - and co-NP m -reductions), if the difference hierarchy over C collapses to level k, then PH C = i P NP (k\Gamma1)-tt j C . Then we show that the exact counting class C=P is closed under p disj - and co-NP m - reductions. Consequently, if the difference hiera...

In this paper we extend a key result of Nisan and Wigderson [17] to the nondeterministic setting: for all ff ? 0 we show that if there is a language in E = DTIME(2 O(n) ) that is hard to approximate by nondeterministic circuits of size 2 ffn , then there is a pseudorandom generator that can be used to derandomize BP \Delta NP (in symbols, BP \Delta NP = NP). By applying this extension we are able to answer some open questions in [14] regarding the derandomization of the classes BP \Delta \Sigma P k and BP \Delta \Theta P k under plausible measure theoretic assumptions. As a consequence, if \Theta P 2 does not have p-measure 0, then AM " coAM is low for \Theta P 2 . Thus, in this case, the graph isomorphism problem is low for \Theta P 2 . By using the NisanWigderson design of a pseudorandom generator we unconditionally show the inclusion MA ` ZPP NP and that MA " coMA is low for ZPP NP . 1 Introduction In recent years, following the development of resource-bounded meas...

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.

We investigate function classes h#Pi f which are defined as the closure of #P under the operation f and a set of known closure properties of #P, e.g. summation over an exponential range. First, we examine operations f under which #P is closed (i.e., h#Pi f = #P) in every relativization. We obtain the following complete characterization of these operations: #P is closed under f in every relativization if and only if f is a finite sum of binomial coefficients over constants. Second, we characterize operations f with respect to their power in the counting context in the unrelativized case. For closure properties f of #P, we have h#Pi f = #P. The other end of the range is marked by operations f for which h#Pi f corresponds to the counting hierarchy. We call these operations counting hard and give general criteria for hardness. For many operations f we show that h#Pi f corresponds to some subclass C of the counting hierarchy. This will then imply that #P is closed under f if and only if ...

We present two results for computational models that allow error probabilities close to 1/2. First, most computational complexity classes have an analogous class in communication complexity. The class PP in fact has two, a version with weakly restricted bias called PP cc, and a version with unrestricted bias called UPP cc. Ever since their introduction by Babai, Frankl, and Simon in 1986, it has been open whether these classes are the same. We show that PP cc � UPP cc. Our proof combines a query complexity separation due to Beigel with a technique of Razborov that translates the acceptance probability of quantum protocols to polynomials. Second, we study how small the bias of minimal-degree polynomials that sign-represent Boolean functions needs to be. We show that the worst-case bias is at worst doubleexponentially small in the sign-degree (which was very recently shown to be optimal by Podolski), while the averagecase bias can be made single-exponentially small in the sign-degree (which we show to be close to optimal). 1

Recent results byToda, Vinay, Damm, and Valianthave shown that the complexity of the determinantischaracterized by the complexity of counting the number of accepting computations of a nondeterministic logspace-bounded machine. #This class of functions is known as #L.# By using that characterization and by establishing a few elementary closure properties, we giveavery simple proof of a theorem of Jung, showing that probabilistic logspace-bounded #PL# machines lose none of their computational power if they are restricted to run in polynomial time.

. Suppose that a Boolean function f is computed by a constant depth circuit with 2 m AND-, OR-, and NOT-gates---and m majoritygates. We prove that f is computed by a constant depth circuit with 2 m O(1) AND-, OR-, and NOT-gates---and a single majority-gate, which is at the root. One consequence is that if a Boolean function f is computed by an AC 0 circuit plus polylog majority-gates, then f is computed by a probabilistic perceptron having polylog order. Another consequence is that if f agrees with the parity function on three-fourths of all inputs, then f cannot be computed by a constant depth circuit with 2 n o(1) AND-, OR-, and NOT-gates, and n o(1) majority-gates. Key words. Majority-gate; threshold-gate; symmetric gate; circuit; parity. Subject classifications. 68Q15. 1. Introduction One of the goals of complexity theory is to find ways to reduce the use of one resource. Typically this entails a modest increase in some other resources. Recently, quasipolynomial size...