Results 1  10
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116
Monotone Complexity
, 1990
"... We give a general complexity classification scheme for monotone computation, including monotone spacebounded 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 ..."
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Cited by 2350 (12 self)
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We give a general complexity classification scheme for monotone computation, including monotone spacebounded 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 logspace) is not closed under complementation, in contrast to Immerman's and Szelepcs 'enyi's nonmonotone result [Imm88, Sze87] that NL = coNL; this is a simple extension of the monotone circuit depth lower bound of Karchmer and Wigderson [KW90] for stconnectivity. 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...
Quantum Circuit Complexity
, 1993
"... We study a complexity model of quantum circuits analogous to the standard (acyclic) Boolean circuit model. It is shown that any function computable in polynomial time by a quantum Turing machine has a polynomialsize quantum circuit. This result also enables us to construct a universal quantum compu ..."
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Cited by 278 (1 self)
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We study a complexity model of quantum circuits analogous to the standard (acyclic) Boolean circuit model. It is shown that any function computable in polynomial time by a quantum Turing machine has a polynomialsize quantum circuit. This result also enables us to construct a universal quantum computer which can simulate, with a polynomial factor slowdown, a broader class of quantum machines than that considered by Bernstein and Vazirani [BV93], thus answering an open question raised in [BV93]. We also develop a theory of quantum communication complexity, and use it as a tool to prove that the majority function does not have a linearsize quantum formula. Keywords. Boolean circuit complexity, communication complexity, quantum communication complexity, quantum computation AMS subject classifications. 68Q05, 68Q15 1 This research was supported in part by the National Science Foundation under grant CCR9301430. 1 Introduction One of the most intriguing questions in computation theroy ...
On the power of smalldepth threshold circuits
 Proceedings 31st Annual IEEE Symposium on Foundations of Computer Science
, 1990
"... Abstract. Weinvestigate the power of threshold circuits of small depth. In particular, we give functions that require exponential size unweighted threshold circuits of depth 3 when we restrict the bottom fanin. We also prove that there are monotone functions fk that can be computed in depth k and li ..."
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Cited by 103 (2 self)
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Abstract. Weinvestigate the power of threshold circuits of small depth. In particular, we give functions that require exponential size unweighted threshold circuits of depth 3 when we restrict the bottom fanin. We also prove that there are monotone functions fk that can be computed in depth k and linear size ^ � _circuits but require exponential size to compute by a depth k; 1 monotone weighted threshold circuit. Key words. Circuit complexity, monotone circuits, threshold circuits, lower bounds Subject classi cations. 68Q15, 68Q99 1.
Interpolation Theorems, Lower Bounds for Proof Systems, and Independence Results for Bounded Arithmetic
"... A proof of the (propositional) Craig interpolation theorem for cutfree sequent calculus yields that a sequent with a cutfree proof (or with a proof with cutformulas of restricted form; in particular, with only analytic cuts) with k inferences has an interpolant whose circuitsize is at most k. We ..."
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Cited by 86 (2 self)
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A proof of the (propositional) Craig interpolation theorem for cutfree sequent calculus yields that a sequent with a cutfree proof (or with a proof with cutformulas of restricted form; in particular, with only analytic cuts) with k inferences has an interpolant whose circuitsize is at most k. We give a new proof of the interpolation theorem based on a communication complexity approach which allows a similar estimate for a larger class of proofs. We derive from it several corollaries: 1. Feasible interpolation theorems for the following proof systems: (a) resolution. (b) a subsystem of LK corresponding to the bounded arithmetic theory S 2 2 (ff). (c) linear equational calculus. (d) cutting planes. 2. New proofs of the exponential lower bounds (for new formulas) (a) for resolution ([15]). (b) for the cutting planes proof system with coefficients written in unary ([4]). 3. An alternative proof of the independence result of [43] concerning the provability of circuitsize lower bounds ...
On Data Structures and Asymmetric Communication Complexity
 JOURNAL OF COMPUTER AND SYSTEM SCIENCES
, 1994
"... In this paper we consider two party communication complexity when the input sizes of the two players differ significantly, the "asymmetric" case. Most of previous work on communication complexity only considers the total number of bits sent, but we study tradeoffs between the number of bits the ..."
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Cited by 85 (9 self)
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In this paper we consider two party communication complexity when the input sizes of the two players differ significantly, the "asymmetric" case. Most of previous work on communication complexity only considers the total number of bits sent, but we study tradeoffs between the number of bits the first player sends and the number of bits the second sends. These
Private vs. Common Random bits in Communication Complexity
 Information Processing Letters
, 1995
"... We investigate the relative power of the common random string model vs. the private random string model in communication complexity. We show that the two model are essentially equal. Keywords: communication complexity, randomness, theory of computation. Communication complexity is a model of comp ..."
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Cited by 84 (0 self)
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We investigate the relative power of the common random string model vs. the private random string model in communication complexity. We show that the two model are essentially equal. Keywords: communication complexity, randomness, theory of computation. Communication complexity is a model of computation where two parties, each with an input, want to mutually compute a Boolean function that is defined on pairs of inputs. Formally, let f : X \Theta Y 7! f0; 1g be a Boolean function. The communication problem for f is the following twoplayer game. Player A gets x 2 X and player B gets y 2 Y . Their goal is to compute f(x; y). They have unlimited computational power and a full description of f , but they don't know each other's input. They determine the output value by exchanging messages. Let n, the length of the input, be log(jXjjY j). A protocol for computing f is a pair of algorithms (one for each player) according to which the players send binary messages. A protocol proceeds in ...
Monotone Circuits for Matching Require Linear Depth
"... We prove that monotone circuits computing the perfect matching function on nvertex graphs require\Omega\Gamma n) depth. This implies an exponential gap between the depth of monotone and nonmonotone circuits. ..."
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Cited by 77 (8 self)
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We prove that monotone circuits computing the perfect matching function on nvertex graphs require\Omega\Gamma n) depth. This implies an exponential gap between the depth of monotone and nonmonotone circuits.
Lower Bounds for Cutting Planes Proofs with Small Coefficients
, 1995
"... We consider smallweight Cutting Planes (CP ) proofs; that is, Cutting Planes (CP ) proofs with coefficients up to P oly(n). We use the well known lower bounds for monotone complexity to prove an exponential lower bound for the length of CP proofs, for a family of tautologies based on the cl ..."
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Cited by 77 (19 self)
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We consider smallweight Cutting Planes (CP ) proofs; that is, Cutting Planes (CP ) proofs with coefficients up to P oly(n). We use the well known lower bounds for monotone complexity to prove an exponential lower bound for the length of CP proofs, for a family of tautologies based on the clique function. Because Resolution is a special case of smallweight CP , our method also gives a new and simpler exponential lower bound for Resolution. We also prove the following two theorems : (1) Treelike CP proofs cannot polynomially simulate nontreelike CP proofs. (2) Treelike CP proofs and BoundeddepthFrege proofs cannot polynomially simulate each other. Our proofs also work for some generalizations of the CP proof system. In particular, they work for CP with a deduction rule, and also for proof systems that allow any formula with small communication complexity, and any set of sound rules of inference. 1 Introduction One of the most fundamental questions in pro...
Exponential Separation of Quantum and Classical Communication Complexity
, 1999
"... Communication complexity has become a central complexity model. In that model, we count the amount of communication bits needed between two parties in order to solve certain computational problems. We show that for certain communication complexity problems quantum communication protocols are expo ..."
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Cited by 77 (2 self)
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Communication complexity has become a central complexity model. In that model, we count the amount of communication bits needed between two parties in order to solve certain computational problems. We show that for certain communication complexity problems quantum communication protocols are exponentially faster than classical ones. More explicitly, we give an example for a communication complexity relation (or promise problem) P such that: 1. The quantum communication complexity of P is O(log m). 2. The classical probabilistic communication complexity of P is \Omega\Gamma m 1=4 = log m). (where m is the length of the inputs). This gives an exponential gap between quantum communication complexity and classical probabilistic communication complexity. Only a quadratic gap was previously known. Our problem P is of geometrical nature, and is a finite precision variation of the following problem: Player I gets as input a unit vector x 2 R n and two orthogonal subspaces M 0 ...
Reachability is harder for directed than for undirected finite graphs
 Journal of Symbolic Logic
, 1990
"... Abstract. Although it is known that reachability in undirected finite graphs can be expressed by an existential monadic secondorder sentence, our main result is that this is not the case for directed finite graphs (even in the presence of certain “builtin ” relations, such as the successor relatio ..."
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Cited by 71 (8 self)
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Abstract. Although it is known that reachability in undirected finite graphs can be expressed by an existential monadic secondorder sentence, our main result is that this is not the case for directed finite graphs (even in the presence of certain “builtin ” relations, such as the successor relation). The proof makes use of EhrenfeuchtFrai’sse games, along with probabilistic arguments. However, we show that for directed finite graphs with degree at most k, reachability is expressible by an existential monadic secondorder sentence. $1. Introduction. If s and t denote distinguished points in a directed (resp. undirected) graph, then we say that a graph is (s, t)connected if there is a directed (undirected) path from s to t. We sometimes refer to the problem of deciding whether a given directed (undirected) graph with two given points sand t is (s, t)connected as the directed (undirected) reachability problem.