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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
Monotone expanders constructions and applications
"... The main purpose of this work is to formally define monotone expanders and motivate their study with (known and new) connections to other graphs and to several computational and pseudorandomness problems. In particular we explain how monotone expanders of constant degree lead to: 1. Constant degree ..."
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The main purpose of this work is to formally define monotone expanders and motivate their study with (known and new) connections to other graphs and to several computational and pseudorandomness problems. In particular we explain how monotone expanders of constant degree lead to: 1. Constant degree dimension expanders in finite fields, resolving a question of [BISW04]. 2. O(1)page and O(1)pushdown expanders, resolving a question of [GKS86], and leading to tight lower bounds on simulation time for certain Turing Machines. Bourgain [Bou09] gave recently an ingenious construction of such constant degree monotone expanders. The first application (1) above follows from a reduction in [DS08]. We give a short exposition of both construction and reduction. The new contributions of this paper are simple. First, we explain the observation leading to the second application (2) above, and some of its consequences. Second, we observe that a variant of the zigzag graph product preserves monotonicity, and use it to give a simple alternative construction of monotone expanders, with nearconstant degree. 1
Notes on Complexity Theory Last updated: December, 2011
"... Recall that one motivation for studying nonuniform computation is the hope that it might be easier to prove lower bounds in that setting. (This is somewhat paradoxical, as nonuniform algorithms are more powerful than uniform algorithms; nevertheless, since circuits are more “combinatorial” in natu ..."
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Recall that one motivation for studying nonuniform computation is the hope that it might be easier to prove lower bounds in that setting. (This is somewhat paradoxical, as nonuniform algorithms are more powerful than uniform algorithms; nevertheless, since circuits are more “combinatorial” in nature than uniform algorithms, there may still be justification for such hope.) The ultimate goal here would be to prove that N P ̸ ⊂ P /poly, which would imply P ̸ = N P. Unfortunately, after over two decades of attempts we are unable to prove anything close to this. Here, we show one example of a lower bound that we have been able to prove; we then discuss one “barrier ” that partly explains why we have been unable to prove stronger bounds.
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,
Key words and phrases: expander, zigzag, kpage graphs, pushdown graphs
, 2010
"... Abstract: The main purpose of this work is to formally define monotone expanders and motivate their study with (known and new) connections to other graphs and to several computational and pseudorandomness problems. In particular we explain how monotone expanders of constant degree lead to: 1. consta ..."
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Abstract: The main purpose of this work is to formally define monotone expanders and motivate their study with (known and new) connections to other graphs and to several computational and pseudorandomness problems. In particular we explain how monotone expanders of constant degree lead to: 1. constantdegree dimension expanders in finite fields, resolving a question of Barak, Impagliazzo, Shpilka, and Wigderson (2004); 2. O(1)page and O(1)pushdown expanders, resolving a question of Galil, Kannan, and Szemerédi (1986) and leading to tight lower bounds on simulation time for certain Turing Machines. Recently, Bourgain (2009) gave a rather involved construction of such constantdegree monotone expanders. The first application (1) above follows from a reduction due to Dvir and Shpilka (2007). We sketch Bourgain’s construction and describe the reduction. The new contributions of this paper are simple. First, we explain the observation leading to the second application (2) above, and some of its consequences. Second, we observe that
Mathematical Methods in Computer Science Lecture 6: Communication Complexity
"... Communication complexity was introduced by Yao in 1979. In this model, there are two parties, Alice and Bob, who have access to strings x and y, respectively, where x, y ∈ {0, 1} n. The aim is for them to compute a function f: {0, 1} n → {0, 1} with as little communication between them as possible. ..."
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Communication complexity was introduced by Yao in 1979. In this model, there are two parties, Alice and Bob, who have access to strings x and y, respectively, where x, y ∈ {0, 1} n. The aim is for them to compute a function f: {0, 1} n → {0, 1} with as little communication between them as possible. The minimum k for which there exists a protocol that uses at most k bits of