Results 1  10
of
19
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...
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...
Models of Computation  Exploring the Power of Computing
"... Theoretical computer science treats any computational subject for which a good model can be created. Research on formal models of computation was initiated in the 1930s and 1940s by Turing, Post, Kleene, Church, and others. In the 1950s and 1960s programming languages, language translators, and oper ..."
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Cited by 57 (7 self)
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Theoretical computer science treats any computational subject for which a good model can be created. Research on formal models of computation was initiated in the 1930s and 1940s by Turing, Post, Kleene, Church, and others. In the 1950s and 1960s programming languages, language translators, and operating systems were under development and therefore became both the subject and basis for a great deal of theoretical work. The power of computers of this period was limited by slow processors and small amounts of memory, and thus theories (models, algorithms, and analysis) were developed to explore the efficient use of computers as well as the inherent complexity of problems. The former subject is known today as algorithms and data structures, the latter computational complexity. The focus of theoretical computer scientists in the 1960s on languages is reflected in the first textbook on the subject, Formal Languages and Their Relation to Automata by John Hopcroft and Jeffrey Ullman. This influential book led to the creation of many languagecentered theoretical computer science courses; many introductory theory courses today continue to reflect the content of this book and the interests of theoreticians of the 1960s and early 1970s. Although
Unprovability of Lower Bounds on the Circuit Size in Certain Fragments of Bounded Arithmetic
 in Izvestiya of the Russian Academy of Science, mathematics
, 1995
"... To appear in Izvestiya of the RAN We show that if strong pseudorandom generators exist then the statement “α encodes a circuit of size n (log ∗ n) for SATISFIABILITY ” is not refutable in S2 2 (α). For refutation in S1 2 (α), this is proven under the weaker assumption of the existence of generators ..."
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Cited by 54 (6 self)
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To appear in Izvestiya of the RAN We show that if strong pseudorandom generators exist then the statement “α encodes a circuit of size n (log ∗ n) for SATISFIABILITY ” is not refutable in S2 2 (α). For refutation in S1 2 (α), this is proven under the weaker assumption of the existence of generators secure against the attack by small depth circuits, and for another system which is strong enough to prove exponential lower bounds for constantdepth circuits, this is shown without using any unproven hardness assumptions. These results can be also viewed as direct corollaries of interpolationlike theorems for certain “split versions ” of classical systems of Bounded Arithmetic introduced in this paper.
SuperLogarithmic Depth Lower Bounds Via The Direct Sum In Communication Complexity
 PROCEEDINGS OF 6 TH STRUCTURES IN COMPLEXITY THEORY
, 1991
"... Is it easier to solve two communication problems together than separately? This question is related to the complexity of the composition of boolean functions. Based on this relationship, an approach to separating NC¹ from P is outlined. Furthermore, it is shown that the approach provides a new p ..."
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Cited by 35 (9 self)
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Is it easier to solve two communication problems together than separately? This question is related to the complexity of the composition of boolean functions. Based on this relationship, an approach to separating NC¹ from P is outlined. Furthermore, it is shown that the approach provides a new proof of the separation of monotone NC¹ from monotone P.
Separation of the Monotone NC Hierarchy
, 1999
"... We prove tight lower bounds, of up to n ffl , for the monotone depth of functions in monotoneP. As a result we achieve the separation of the following classes. 1. monotoneNC 6= monotoneP. 2. For every i 1, monotoneNC i 6= monotoneNC i+1 . 3. More generally: For any integer function D( ..."
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Cited by 35 (0 self)
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We prove tight lower bounds, of up to n ffl , for the monotone depth of functions in monotoneP. As a result we achieve the separation of the following classes. 1. monotoneNC 6= monotoneP. 2. For every i 1, monotoneNC i 6= monotoneNC i+1 . 3. More generally: For any integer function D(n), up to n ffl (for some ffl ? 0), we give an explicit example of a monotone Boolean function, that can be computed by polynomial size monotone Boolean circuits of depth D(n), but that cannot be computed by any (fanin 2) monotone Boolean circuits of depth less than Const \Delta D(n) (for some constant Const). Only a separation of monotoneNC 1 from monotoneNC 2 was previously known. Our argument is more general: we define a new class of communication complexity search problems, referred to below as DART games, and we prove a tight lower bound for the communication complexity of every member of this class. As a result we get lower bounds for the monotone depth of many functions. In...
An n! Lower Bound On Formula Size
, 1999
"... We introduce a new EhrenfeuchtFra ss e game for proving lower bounds on the size of firstorder formulas. Up until now such games have only been used to prove bounds on the operator depth of formulas, not their size. We use this game to prove that the CTL + formula Occurn # E[Fp 1 # Fp 2 # # ..."
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Cited by 21 (2 self)
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We introduce a new EhrenfeuchtFra ss e game for proving lower bounds on the size of firstorder formulas. Up until now such games have only been used to prove bounds on the operator depth of formulas, not their size. We use this game to prove that the CTL + formula Occurn # E[Fp 1 # Fp 2 # # Fpn ] which says that there is a path along which the predicates p 1 through pn occur in some order, requires size n! to express in CTL. Our lower bound is optimal. It follows that the succinctness of CTL + with respect to CTL is exactly #(n)!. Wilke had shown that the succinctness was at least exponential [Wil99]. We also use our games to prove an optimal# n) lower bound on the number of boolean variables needed for a weak reachability logic (RL w ) to polynomially embed the language LTL. The number of booleans needed for full reachability logic RL and the transitive closure logic FO 2 (TC) remain open [IV97, AI00]. 1
Characterizing nondeterministic circuit size
 In Proceedings of the 25th STOC
, 1993
"... Consider the following simple communication problem. Fix a universe U and a family Ω of subsets of U. Players I and II receive, respectively, an element a ∈ U and a subset A ∈ Ω. Their task is to find a subset B of U such that A∩B  is even and a ∈ B. With every Boolean function f we associate a co ..."
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Cited by 10 (4 self)
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Consider the following simple communication problem. Fix a universe U and a family Ω of subsets of U. Players I and II receive, respectively, an element a ∈ U and a subset A ∈ Ω. Their task is to find a subset B of U such that A∩B  is even and a ∈ B. With every Boolean function f we associate a collection Ωf of subsets of U = f −1 (0), and prove that its (one round) communication complexity completely determines the size of the smallest nondeterministic circuit for f. We propose a linear algebraic variant to the general approximation method of Razborov, which has exponentially smaller description. We use it to derive four different combinatorial problems (like the one above) that characterize NP. These are tight, in the sense that they can be used to prove superlinear circuit size lower bounds. Combined with Razborov’s method, they present a purely combinatorial framework in which to study the P vs. NP vs. co − NP question.
Lower Bounds for Monotone Real Circuit Depth and Formula Size and Treelike Cutting Planes
, 1998
"... Using a notion of real communication complexity recently introduced by J. Krajcek, we prove a lower bound on the depth of monotone real circuits and the size of monotone real formulas for stconnectivity. This implies a superpolynomial speedup of daglike over treelike Cutting Planes proofs. Key w ..."
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Cited by 5 (2 self)
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Using a notion of real communication complexity recently introduced by J. Krajcek, we prove a lower bound on the depth of monotone real circuits and the size of monotone real formulas for stconnectivity. This implies a superpolynomial speedup of daglike over treelike Cutting Planes proofs. Key words: computational complexity, monotone circuit, communication complexity, Cutting Planes proof Introduction A monotone real circuit is a circuit computing with real numbers in which every gate computes a nondecreasing binary real function. This class of circuits was introduced in [10]. We require that such a circuit outputs 0 or 1 on every input of 0's and 1's only. Hence, monotone real circuits are a generalization of monotone boolean circuits, which was shown to be strictly more powerful in [11]. The depth and size of a monotone real circuit are dened as usual, and we call it a formula if every gate has fanout at most 1. We generalize the lower bounds on the depth of monotone boolea...
Monotone circuits for connectivity have depth (log ) 2(1
 SIAM Journal on Computing
, 1998
"... We prove that a monotone circuit of size n d recognizing connectivity must have depth ((log n) 2 = log d). For formulas this implies depth ((log n) 2 = log log n). For ((log n) 2)which is optimal up to a conpolynomialsize circuits the bound becomes stant. Warning: Essentially this paper has been p ..."
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Cited by 4 (0 self)
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We prove that a monotone circuit of size n d recognizing connectivity must have depth ((log n) 2 = log d). For formulas this implies depth ((log n) 2 = log log n). For ((log n) 2)which is optimal up to a conpolynomialsize circuits the bound becomes stant. Warning: Essentially this paper has been published in SIAM Journal on Computing is hence subject to copyright restrictions. It is for personal use only. 1