We give a general complexity classification scheme for monotone computation, including monotone space-bounded and Turing machine models not previously considered. We propose monotone complexity classes including mAC i , mNC i , mLOGCFL, mBWBP , mL, mNL, mP , mBPP and mNP . We define a simple notion of monotone reducibility and exhibit complete problems. This provides a framework for stating existing results and asking new questions. We show that mNL (monotone nondeterministic log-space) is not closed under complementation, in contrast to Immerman's and Szelepcs 'enyi's nonmonotone result [Imm88, Sze87] that NL = co-NL; this is a simple extension of the monotone circuit depth lower bound of Karchmer and Wigderson [KW90] for st-connectivity. We also consider mBWBP (monotone bounded width branching programs) and study the question of whether mBWBP is properly contained in mNC 1 , motivated by Barrington's result [Bar89] that BWBP = NC 1 . Although we cannot answer t...

We introduce a linear algebraic model of computation, the Span Program, and prove several upper and lower bounds on it. These results yield the following applications in complexity and cryptography: • SL ⊆ ⊕L (a weak Logspace analogue of N P ⊆ ⊕P). • The first super-linear size lower bounds on branching programs that count. • A broader class of functions which posses information-theoretic secret sharing schemes. The proof of the main connection, between span programs and counting branching programs, uses a variant of Razborov’s general approximation method. 1

How can molecules compute? In his early studies of reversible computation, Bennett imagined an enzymatic Turing Machine which modified a hetero-polymer (such as DNA) to perform computation with asymptotically low energy expenditures. Adleman's recent experimental demonstration of a DNA computation, using an entirely different approach, has led to a wealth of ideas for how to build DNA-based computers in the laboratory, whose energy efficiency, information density, and parallelism may have potential to surpass conventional electronic computers for some purposes. In this thesis, I examine one mechanism used in all designs for DNA-based computer -- the self-assembly of DNA by hybridization and formation of the double helix -- and show that this mechanism alone in theory can perform universal computation. To do so, I borrow an important result in the mathematical theory of tiling: Wang showed how jigsaw-shaped tiles can be designed to simulate the operation of any Turing Machine. I propose...

We survey lower bounds established for the complexity of computing explicitly given Boolean functions by switching-and-rectifier networks, branching programs and switching networks. We first consider the unrestricted case and then proceed to various restricted models. Among these are monotone networks, bounded-width devices , oblivious devices and read-k times only devices. 1 Introduction The main goal of the Boolean complexity theory is to prove lower bounds on the complexity of computing "explicitly given" Boolean functions in interesting computational models. By "explicitly given" researchers usually mean "belonging to the class NP ". This is a very plausible interpretation since on the one hand this class contains the overwhelming majority of interesting Boolean functions and on the other hand it is small enough to prevent us from the necessity to take into account counting arguments. To illustrate the second point, let me remind the reader that already the class \Delta p 2 ,...

In [Li1] and [Ad2] a formal model for molecular computing was proposed, which makes focused use of affinity purification. The use of PCR was suggested to expand the range of feasible computations, resulting in a second model. In this note, we give a precise characterization of these two models in terms of recognized computational complexity classes, namely branching programs (BP) and nondeterministic branching programs (NBP) respectively. This allows us to give upper and lower bounds on the complexity of desired computations. Examples are given of computable and uncomputable problems, given limited time. 1 Introduction Molecular computation, as introduced by [Ad1], provides a new approach to solving combinatorial inverse problems, where we are interested in computing f \Gamma1 (1) for n-bit strings x and boolean function f . Instances of NP-complete problems can be expressed in this form; for example 3-SAT. Adleman's technique involves using individual DNA strands to represent poten...

We use algebraic techniques to obtain superlinear lower bounds on the size of boundedwidth branching programs to solve a number of problems. In particular, we show that any bounded-width branching program computing a nonconstant threshold function has length \Omega\Gamma n log log n); improving on the previous lower bounds known to apply to all such threshold functions. We also show that any program over a finite solvable monoid computing products in a nonsolvable group has length\Omega\Gamma n log log n): This result is a step toward proving the conjecture that the circuit complexity class ACC 0 is properly contained in NC 1 : A preliminary version of this paper appeared in the Proceedings of the 1991 Structure in Complexity Theory Symposium. 1. The Main Results In this paper we describe a general algebraic technique for obtaining superlinear lower bounds on the length of bounded-width branching programs to solve certain problems. Our method is based on the interpretation, ...

This work is settled in the area of complexity theory for restricted variants of branching programs. Today, branching programs can be considered one of the standard nonuniform models of computation. One reason for their popularity is that they allow to describe computations in an intuitively straightforward way and promise to be easier to analyze than the traditional models. In complexity theory, we are mainly interested in upper and lower bounds on the size of branching programs. Although proving superpolynomial lower bounds on the size of general branching programs still remains a challenging open problem, there has been considerable success in the study of lower bound techniques for various restricted variants, most notably perhaps read-once branching programs and OBDDs (ordered binary decision diagrams). Surprisingly, OBDDs have also turned out to be extremely useful in practical applications as a data structure for Boolean functions. So far, research has concentrated on determinis...

Ordered binary decision diagrams (OBDDs) and several more general BDD models have turned out to be representations of Boolean functions which are useful in applications like verication, timing analysis, test pattern generation or combinatorial optimization. The hidden weighted bit function (HWB) is of particular interest, since it seems to be the simplest function with exponential OBDD size. The complexity of this function with respect to dierent circuit models, formulas, and various BDD models is discussed. Supported by DFG grant We 1066/8-1. 1. INTRODUCTION If one likes to have short representations of Boolean functions, circuits are the most powerful model. But if one likes to work with these representations, one additionally needs eÆcient algorithms for certain problems, among them satisability test, equivalence test, and synthesis, i. e., the combination of two or more representations by a Boolean operation. For this purpose ordered binary decision diagrams (OBDDs) introd...

Circuit size, branching program size, and formula size of Boolean functions, denoted by C(f), BP(f), and L(f), are the most important complexity measures for Boolean functions. Often also the formula size L (f) over the restricted basis f; ; :g is considered. It is well-known that C(f) 3 BP(f), BP(f) L (f), L (f) L(f) 2 , and C(f) L(f) \Gamma 1. These estimates are optimal. But the inequality BP(f) L(f) 2 can be improved to BP(f) 1:360 L(f) fi , where fi = log 4 (3 + p 5) ! 1:195. 1

Abstract Let x1,..., xk be n-bit numbers and T 2 N. Assume that P1,..., Pk are players such that Pi knows all of the numbers except xi. The players want to determine if Pkj=1 xj = T bybroadcasting as few bits as possible. Chandra, Furst, and Lipton obtained an upper bound of O(pn) bits for the k = 3 case, and a lower bound of!(1) for k> = 3 when T = \Theta (2n). We obtain(1) for general k> = 3 an upper bound of k + O(n1/(k-1)), (2) for k = 3, T = \Theta (2n), a lowerbound of \Omega (log log n), (3) a generalization of the protocol to abelian groups, (4) lower boundson the multiparty communication complexity of some regular languages, (5) lower bounds on branching programs, and (6) empirical results for the k = 3 case. 1 Introduction Multiparty communication complexity was first defined by Chandra, Furst, and Lipton [8] and used to obtain lower bounds on branching programs. Since then it has been used to get additional lower bounds and tradeoffs for branching programs [1, 5], lower bounds on problems in data structures [5], time-space tradeoffs for restricted Turing machines [1], and unconditional pseudorandom generators for logspace [1]. Def 1.1 Let f: {{0, 1}n}k! {0, 1}. Assume, for 1 < = i < = k, Pi has all of the inputs except xi. Let d(f) be the total number of bits broadcast in the optimal deterministic protocol for f. This is called the multiparty communication complexity of f. The scenario is called the forehead model.