We prove in this paper that it is much harder to evaluate depth--2, size--N circuits with MOD m gates than with MOD p gates by k--party communication protocols: we show a k--party protocol which communicates O(1) bits to evaluate circuits with MOD p gates, while evaluating circuits with MOD m gates needs\Omega\Gamma N) bits, where p denotes a prime, and m a composite, non-prime power number. As a corollary, for all m, we show a function, computable with a depth--2 circuit with MODm gates, but not with any depth--2 circuit with MOD p gates. Obviously, the k--party protocols are not weaker than the k 0 --party protocols, for k 0 ? k. Our results imply that if there is a prime p between k and k 0 : k ! p k 0 , then there exists a function which can be computed by a k 0 --party protocol with a constant number of communicated bits, while any k--party protocol needs linearly many bits of communication. This result gives a hierarchy theorem for multi--party protocols. 1 1.

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Several recent results in circuit complexity theory have used a representation of Boolean functions by polynomials over finite fields. Our current inability to extend these results to superficially similar situations may be related to properties of these polynomials which do not extend to polynomials over general finite rings or finite abelian groups. Here we pose a number of conjectures on the behavior of such polynomials over rings and groups, and present some partial results toward proving them. 1. Introduction 1.1. Polynomials and Circuit Complexity The representation of Boolean functions as polynomials over the finite field Z 2 = f0; 1g dates back to early work in switching theory [?]. A formal language L can be identified with the family of functions f i : Z i 2 ! Z 2 , where f i (x 1 ; : : : ; x i ) = 1 iff x 1 : : : x i 2 L. Each of these functions can be written as a polynomial in the variables x 1 ; : : : ; x n . We can consider algebraic formulas or circuits with...

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Consider the following communication problem. Alice gets a word x 2 f0; 1g n and Bob gets a word y 2 f0; 1g n . Alice and Bob are told that x 6= y. Their goal is to find an index 1 i n such that x i 6= y i (the index i should be known to both of them). This problem is one of the most basic communication problems. It arises naturally from the correspondence between circuit depth and communication complexity discovered by Karchmer and Wigderson. We present three protocols using which Alice and Bob can solve the problem by exchanging at most n + 2 bits. One of this protocols is due to Rudich and Tardos. These protocols improve the previous upper bound of n + log n, obtained by Karchmer. We also show that any protocol for solving the problem must exchange, in the worst case, at least n+ 1 bits. This improves a simple lower bound of n \Gamma 1 obtained by Karchmer. Our protocols, therefore, are at most one bit away from optimality. The three n + 2 bit protocols use two completely d...

We study the relationship between the complexity of languages, in Yao's 2-party communication game and its extensions, and the algebraic properties of finite monoids that can recognize them.

plus an arbitrary linear function of n input variables. Keywords: Circuit complexity, modular circuits, composite modulus 1 Introduction Boolean circuits are one of the most interesting models of computation. They are widely examined in VLSI design, in general computability theory and in complexity theory context as well as in the theory of parallel computation. Almost all of the strongest and deepest lower bound results for the computational complexity of finite functions were proved using the Boolean circuit model of computation ([13], [22], [9], [14], [15], or see [20] for a survey). Even these famous and sophisticated lower bound results were proven for very restricted circuit classes. Bounded depth and polynomial size is one of the most natural restrictions. Ajtai [1], Furst, Saxe, and Sipser [5] proved that no polynomial sized, constant depth circuit can compute the PARITY function. Yao [22] and Hastad [9] generalized this result

grateful to Pavel Pudlák for suggesting the use of the Hales-Jewett Theorem. We study languages with bounded communication complexity in the multiparty “input on the forehead model ” with worst-case partition. In the two-party case, languages with bounded complexity are exactly those recognized by programs over commutative monoids [20]. This can be used to show that these languages all lie in shallow ACC 0. In contrast, we use different coding techniques to show that there are languages of arbitrarily large circuit complexity which can be recognized in constant communication by k players for k ≥ 3. However, if a language has a neutral letter and bounded communication complexity in the k-party game for some fixed k then the language is in fact regular and we give an algebraic characterization of regular languages with this property. We also prove that a symmetric language has bounded k-party complexity for some fixed k iff it has bounded two party complexity. 1

Ordered binary decision diagrams (OBDDs) are a model for representing Boolean functions. There is also a more powerful variant called parity OBDDs. The size of the representation of a given function depends in both these models on an ordering of the variables. It is known that there are functions such that almost all orderings of its variables yield an OBDD of polynomial size, but there are also some exceptional orderings, for which the size is exponential. We prove that for parity OBDDs, the size for a random ordering and the size for the worst ordering are polynomially related. More exactly, for every " ? 0 there is a number c ? 0 such that the following holds. If a Boolean function f is such that a random ordering of the variables yields a parity OBDD for f of size at most s with probability at least ", then every ordering of the variables yields a parity OBDD for f of size at most s c . 1 Introduction Parity OBDDs were introduced by Gergov and Meinel [4] and simplified by Waack ...

y programs over commutative finite monoids. Our results may be seen as a first step towards an analog of Szegedy's result for k-party communication complexity. We study the multiparty communication complexity of regular languages with an emphasis on star-free languages, and submit evidence of a log gap in the case of 2-party communication complexity of regular languages. Section II is devoted to definitions and notation that will be used. In Section III, we give general results linking the communication complexity of regular languages and the monoids recognizing them. Section IV gives bounds on the communication complexity of specific regular languages. Finally, Section V discusses the results and proposes questions whose solutions would complete our work. ckground nd de nitions 2.1 o unication co ple it We give here some formal definitions together with a few results in communication complexity. For a more detailed introduction, we refer the