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 construct an oracle A such that P A = \PhiP A and NP A = EXP A : This relativized world has several amazing properties: ffl The oracle A gives the first relativized world where one can solve satisfiability on formulae with at most one assignment yet P 6= NP. ffl The oracle A is the first where P A = UP A 6= NP A = coNP A : ffl The construction gives a much simpler proof than Fenner, Fortnow and Kurtz of a relativized world where all NP-complete sets are polynomial-time isomorphic. It is the first such computable oracle. ffl Relative to A we have a collapse of \PhiEXP A ` ZPP A ` P A /poly. We also create a different relativized world where there exists a set L in NP that is NP- complete under reductions that make one query to L but not under traditional many-one reductions. This contrasts with the result of Buhrman, Spaan and Torenvliet showing that these two completeness notions for NEXP coincide. 1 Introduction Valiant and Vazirani [VV86] show the sur...

We say an integer polynomial p, on Boolean inputs, weakly m-represents a Boolean function f if p is non-constant and is zero (mod m), whenever f is zero. In this paper we prove that if a polynomial weakly m-represents the Mod q function on n inputs, where q and m are relatively prime and m is otherwise arbitrary, then the degree of the polynomial is \Omega\Gamma n). This generalizes previous results of Barrington, Beigel and Rudich (Computational Complexity 4 (1994), pp. 367-382) and Tsai (Structures 1993, pp. 96-101), which held only for constant or slowly growing m. In addition, the proof technique given here is quite different. We use a method (adapted from Barrington and Straubing, (Computational Complexity 4 (1994), pp. 325-338) in which the inputs are represented as complex q th roots of unity. In this representation it is possible to compute the Fourier transform using some elementary properties of the algebraic integers. As a corollary of the main theorem and the proof of Tod...

. Define the MODm-degree of a boolean function F to be the smallest degree of any polynomial P , over the ring of integers modulo m, such that for all 0-1 assignments ~x , F (~x) = 0 iff P (~x) = 0. By exploring the periodic property of the binomial coefficients modulo m, two new lower bounds on the MODm -degree of the MOD l and :MODm functions are proved, where m is any composite integer and l has a prime factor not dividing m. Both bounds improve from sub-linear to \Omega\Gamma n). With the periodic property, a simple proof of a lower bound on the MODm -degree with symmetric multilinear polynomial of the OR function is given. It is also proved that the majority function has a lower bound n 2 and the MidBit function has a lower bound p n. Key words. boolean function complexity, circuit complexity, computational complexity AMS subject classifications. 68Q05, 68Q15, 68Q25, 94C10 1. Introduction. Proving lower bounds on explicitly given boolean functions is a notoriously difficul...

We observe that a combination of known top-down and bottom-up lower bound techniques of circuit complexity may yield new circuit lower bounds.

The counting class C= P, which captures the notion of "exact counting", while extremely powerful under various nondeterministic reductions, is quite weak under polynomial-time deterministic reductions. We discuss the analogies between NP and co-C = P, which allow us to derive many interesting results for such deterministic reductions to co-C = P. We exploit these results to obtain some interesting oracle separations. Most importantly, we show that there exists an oracle A such that \PhiP A 6` P C=P A and BPP A 6` P C=P A . Therefore, techniques that would prove that C=P and PP are polynomial-time Turing equivalent, or that C=P is polynomial-time Turing hard for the polynomial-time hierarchy, would not relativize. 1 Introduction The class C=P (see section 2 for precise definitions) is an extremely powerful counting class which captures the notion of "exact counting". It can be characterized by nondeterministic machines which accept if and only if the number of accepting path...

Abstract We demonstrate the applicability of the polynomial degree bound technique to notions such as the nonexistence of Turing-hard sets in some relativized world, (non)uniform gap-definability, and relativized separations. This way, we settle certain open questions of Hemaspaandra, Ramachandran & Zimand [HRZ95] and Fenner, Fortnow & Kurtz [FFK94], extend results of Hemaspaandra, Jain & Vereshchagin [HJV93] and construct oracles achieving desired results.