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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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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...
Monotone Circuits for Monotone Weighted Threshold Functions ∗
"... Weighted threshold functions with positive weights are a natural generalization of unweighted threshold functions. These functions are clearly monotone. However, the naive way of computing them is adding the weights of the satisfied variables and checking if the sum is greater than the threshold; th ..."
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Weighted threshold functions with positive weights are a natural generalization of unweighted threshold functions. These functions are clearly monotone. However, the naive way of computing them is adding the weights of the satisfied variables and checking if the sum is greater than the threshold; this algorithm is inherently nonmonotone since addition is a nonmonotone function. In this work we bypass this addition step and construct a polynomial size logarithmic depth unbounded fanin monotone circuit for every weighted threshold function, i.e., we show that weighted threshold functions are in mAC 1. (To the best of our knowledge, prior to our work no polynomial monotone circuits were known for weighted threshold functions.) Our monotone circuits are applicable for the cryptographic tool of secret sharing schemes. Using general results for compiling monotone circuits (Yao, 1989) and monotone formulae (Benaloh and Leichter, 1990) into secret sharing schemes, we get secret sharing schemes for every weighted threshold access structure. Specifically, we get: (1) informationtheoretic secret sharing schemes where the size of each share is quasipolynomial in the number of users, and (2) computational secret sharing schemes where the size of each share is polynomial in the number of users.
Monotone Circuits for Weighted Threshold Functions
 IN PROC. OF THE 20TH ACM SYMP. ON THE THEORY OF COMPUTING
, 2004
"... Weighted threshold functions with positive weights are a natural generalization of unweighted threshold functions. These functions are clearly monotone. However, the naive way of computing it is adding the weights of the satisfied variables and checking if the sum is greater than the threshold; th ..."
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Cited by 3 (2 self)
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Weighted threshold functions with positive weights are a natural generalization of unweighted threshold functions. These functions are clearly monotone. However, the naive way of computing it is adding the weights of the satisfied variables and checking if the sum is greater than the threshold; this algorithm is inherently nonmonotone since addition is a nonmonotone function. In this work we bypass this addition step and construct a polynomial size logarithmic depth unbounded fanin monotone circuit for every weighted threshold function, i. e., we show that weighted threshold functions are in mAC . (To the best of our knowledge, prior to our work no polynomial monotone circuits were known for weighted threshold functions.) Our monotone circuits are applicable for the cryptographic tool of secret sharing schemes. Using general results for compiling monotone circuits (Yao, 1989) and monotone formulae (Benaloh and Leichter, 1990) into secret sharing schemes, we get secret sharing schemes for every weighted threshold access structure. Specifically, we get: (1) informationtheoretic secret sharing schemes where the size of each share is quasipolynomial in the number of users, and (2) computation secret sharing schemes where the size of each share is polynomial in the number of users.
WHierarchies Defined by Symmetric Gates
 THEORY COMPUT SYST
"... The classes of the Whierarchy are the most important classes of intractable problems in parameterized complexity. These classes were originally defined via the weighted satisfiability problem for Boolean circuits. Here, besides the Boolean connectives we consider connectives such as majority, not ..."
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The classes of the Whierarchy are the most important classes of intractable problems in parameterized complexity. These classes were originally defined via the weighted satisfiability problem for Boolean circuits. Here, besides the Boolean connectives we consider connectives such as majority, notallequal, and unique. For example, a gate labelled by the majority connective outputs TRUE if more than half of its inputs are TRUE. For any finite set C of connectives we construct the corresponding W(C)hierarchy. We derive some general conditions which guarantee that the Whierarchy and the W(C)hierarchy coincide levelwise. If C only contains the majority connective then the first levels of the hierarchies coincide. We use this to show that a variant of the parameterized vertex cover problem, the majority vertex cover problem, is W[1]complete.
A Generalization of Spira’s Theorem and Circuits with Small Segregators or Separators
"... Abstract. Spira [28] showed that any Boolean formula of size s can be simulated in depth O(log s). We generalize Spira’s theorem and show that any Boolean circuit of size s with segregators of size f(s) can be simulated in depth O(f(s) log s). If the segregator size is at least s ε for some constant ..."
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Abstract. Spira [28] showed that any Boolean formula of size s can be simulated in depth O(log s). We generalize Spira’s theorem and show that any Boolean circuit of size s with segregators of size f(s) can be simulated in depth O(f(s) log s). If the segregator size is at least s ε for some constant ε> 0, then we can obtain a simulation of depth O(f(s)). This improves and generalizes a simulation of polynomialsize Boolean circuits of constant treewidth k in depth O(k 2 log n) by Jansen and Sarma [17]. Since the existence of small balanced separators in a directed acyclic graph implies that the graph also has small segregators, our results also apply to circuits with small separators. Our results imply that the class of languages computed by nonuniform families of polynomialsize circuits that have constant size segregators equals nonuniform NC 1. Considering space bounded Turing machines to generate the circuits, for f(s) log 2 sspace uniform families of Boolean circuits our smalldepth simulations are also f(s) log 2 sspace uniform. As a corollary, we show that the Boolean Circuit Value problem for circuits with constant size segregators (or separators) is in deterministic SP ACE(log 2 n). Our results also imply that the Planar Circuit Value problem, which is known to be PComplete [16], can be solved in deterministic SP ACE ( √ n log n). Key words: Boolean circuits, circuit size, circuit depth, Spira’s theorem, Turing machines, space complexity 1
The Size and Depth of Boolean Circuits: A Dissertation Proposal
, 2011
"... In this thesis, we study the relationship between size and depth for Boolean circuits. Over four decades, very few results were obtained for either special or general Boolean circuits since Spira gave the first related result. Spira showed in 1971 that any Boolean formula of size s can be simulated ..."
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In this thesis, we study the relationship between size and depth for Boolean circuits. Over four decades, very few results were obtained for either special or general Boolean circuits since Spira gave the first related result. Spira showed in 1971 that any Boolean formula of size s can be simulated in depth O(log s). (A Boolean formula is a treelike circuit, that is the fanout of every gate is 1.) Spira’s result means that an arbitrary Boolean expression can be replaced by an equivalent ”balanced ” expression, that can be evaluated very efficiently in parallel. For general Boolean circuits, the strongest known result is that Boolean circuits of size s can be simulated in depth O(s / log s). This result was first proved by Paterson and Valiant in 1976, and later proved by Dymond and Tompa in 1985 using another method. There are many consequences if the simulation for general circuits can be improved in a uniform setting, including implications about the relationship between deterministic time and space in the Turing machine model, deterministic time of Turing machines versus parallel time in the PRAM model,
Circuit Complexity

"... Combinational circuits or shortly circuits are a model of the lowest level of computer hardware which is of interest from the point of view of computer science. Circuit complexity has a longer history than complexity theory. Complexity measures like circuit size and depth model sequential time, ..."
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Combinational circuits or shortly circuits are a model of the lowest level of computer hardware which is of interest from the point of view of computer science. Circuit complexity has a longer history than complexity theory. Complexity measures like circuit size and depth model sequential time, hardware cost, parallel time, and even storage space. This chapter contains an overview on the research area called complexity of boolean functions. The complexity measures of circuits are discussed and compared with other complexity measures. As an example, the design of efficient circuits is discussed for arithmetic functions. The limits of known lowerbound techniques are discussed.