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Upper and Lower Bounds for Treelike Cutting Planes Proofs
 In 9th IEEE Symposium on Logic in Computer Science
, 1994
"... In this paper we study the complexity of Cutting Planes (CP) refutations, and treelike CP refutations. Treelike CP proofs are natural and still quite powerful. In particular, the propositional pigeonhole principle (PHP) has been shown to have polynomialsized treelike CP proofs. Our main result s ..."
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Cited by 42 (10 self)
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In this paper we study the complexity of Cutting Planes (CP) refutations, and treelike CP refutations. Treelike CP proofs are natural and still quite powerful. In particular, the propositional pigeonhole principle (PHP) has been shown to have polynomialsized treelike CP proofs. Our main result shows that a family of tautologies, introduced in this paper requires exponentialsized treelike CP proofs. We obtain this result by introducing a new method which relates the size of a CP refutation to the communication complexity of a related search problem. Because these tautologies have polynomialsized Frege proofs, it follows that treelike CP cannot polynomially simulate Frege systems. 1 Introduction An important open problem is to determine whether there exists a propositional proof system that admits short (polynomial size) proofs for all tautologies, or equivalently, whether or not NP equals coNP. In order to attack Research supported by NSF NYI grant CCR92570979 y Research su...
On Frege and Extended Frege Proof Systems
, 1993
"... We propose a framework for proving lower bounds to the size of EF  proofs (equivalently, to the number of proofsteps in Fproofs) in terms of boolean valuations . The concept is motivated by properties of propositional provability in models of bounded arithmetic and it is a finitisation of a parti ..."
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Cited by 21 (2 self)
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We propose a framework for proving lower bounds to the size of EF  proofs (equivalently, to the number of proofsteps in Fproofs) in terms of boolean valuations . The concept is motivated by properties of propositional provability in models of bounded arithmetic and it is a finitisation of a particular forcing construction explained also in the paper. It reduces the question of proving a lower bound to the question of constructing a partial boolean algebra and a map of formulas into that algebra with particular properties. We show that in principle one can obtain via this method optimal lower bounds (up to a polynomial increase). Introduction A propositional proof system is any polynomial time function P whose range is exactly the set of tautologies TAUT, cf. [17]. For ø a tautology any string ß such that P (ß) = ø is called a P proof of ø . Any usual propositional calculus, be it resolution or extended resolution, a Hilbert style system based on finitely many axiom schemes and inf...
Counting, Fanout, And The Complexity Of Quantum Acc
, 2002
"... q are distinct primes, QACC[q] is strictly more powerful than its classical counterpart, as is QAC 0 when fanout is allowed. This adds to the growing list of quantum complexity classes which are provably more powerful than their classical counterparts. We also develop techniques for proving upp ..."
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Cited by 16 (2 self)
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q are distinct primes, QACC[q] is strictly more powerful than its classical counterpart, as is QAC 0 when fanout is allowed. This adds to the growing list of quantum complexity classes which are provably more powerful than their classical counterparts. We also develop techniques for proving upper bounds for QACC in terms of related language classes. We dene classes of languages closely related to QACC[2] and show that restricted versions of them can be simulated by polynomialsize circuits. With further restrictions, language classes related to QACC[2] operators can be simulated by classical threshold circuits of polynomial size and constant depth. Keywords: quantum computation, quantum & circuit complexity, threshold circuit Communicated by : R Cleve & J Watrous 1. Introduction Advances in quantum computation
Functionalgebraic characterizations of log and polylog parallel time
 Computational Complexity
, 1994
"... Abstract. The main results of this paper are recursiontheoretic characterizations of two parallel complexity classes: the functions computable by uniform bounded fanin circuit families of log and polylog depth (or equivalently, the functions bitwise computable by alternating Turing machines in log ..."
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Cited by 14 (4 self)
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Abstract. The main results of this paper are recursiontheoretic characterizations of two parallel complexity classes: the functions computable by uniform bounded fanin circuit families of log and polylog depth (or equivalently, the functions bitwise computable by alternating Turing machines in log and polylog time). The present characterizations avoid the complex base functions, function constructors, and a priori size or depth bounds typical of previous work on these classes. This simplicity is achieved by extending the \tiered recursion &quot; techniques of Leivant and Bellantoni&Cook. Key words. Circuit complexity � subrecursion. Subject classi cations. 68Q15, 03D20, 94C99. 1.
A Bounded Arithmetic Theory for Constant Depth Threshold Circuits
, 1996
"... . We define an extension R 0 2 of the bounded arithmetic theory R 0 2 and show that the class of functions \Sigma b 1 definable in R 0 2 coincides with the computational complexity class TC 0 of functions computable by polynomial size, constant depth threshold circuits. 1 Introduction Th ..."
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Cited by 8 (4 self)
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. We define an extension R 0 2 of the bounded arithmetic theory R 0 2 and show that the class of functions \Sigma b 1 definable in R 0 2 coincides with the computational complexity class TC 0 of functions computable by polynomial size, constant depth threshold circuits. 1 Introduction The theories S i 2 , for i 2 N, of Bounded Arithmetic were introduced by Buss [3]. The language of these theories is the language of Peano Arithmetic extended by symbols for the functions b 1 2 xc, jxj := dlog 2 (x + 1)e and x#y := 2 jxj\Deltajyj . A quantifier of the form 8xt , 9x t with x not occurring in t is called a bounded quantifier. Furthermore, a quantifier of the form 8x jtj , 9x jtj is called sharply bounded. A formula is called (sharply) bounded if all quantifiers in it are (sharply) bounded. The class of bounded formulae is divided into an hierarchy analogous to the arithmetical hierarchy: The class of sharply bounded formulae is denoted \Sigma b 0 or \Pi b 0 . For i...
Relating the Provable Collapse of P to NC¹ and the Power of Logical Theories
"... We show that the following three statements are equivalent: QPV is conservative over QALV, QALV proves its open induction formulas, and QALV proves P=NC¹. Here QPV and QALV are first order theories whose function symbols range over polynomialtime and NC¹ functions, respectively. ..."
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Cited by 6 (3 self)
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We show that the following three statements are equivalent: QPV is conservative over QALV, QALV proves its open induction formulas, and QALV proves P=NC¹. Here QPV and QALV are first order theories whose function symbols range over polynomialtime and NC¹ functions, respectively.
On the Complexity of Quantum ACC
 in Fifteenth Annual Conference on Computational Complexity Theory, IEEE Computer
, 2000
"... For any q> 1, let MODq be a quantum gate that determines if the number of 1’s in the input is divisible by q. We show that for any q, t> 1, MODq is equivalent to MODt (up to constant depth). Based on the case q = 2, Moore [8] has shown that quantum analogs of AC (0) , ACC[q], and ACC, denoted ..."
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Cited by 5 (1 self)
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For any q> 1, let MODq be a quantum gate that determines if the number of 1’s in the input is divisible by q. We show that for any q, t> 1, MODq is equivalent to MODt (up to constant depth). Based on the case q = 2, Moore [8] has shown that quantum analogs of AC (0) , ACC[q], and ACC, denoted QAC (0) wf, QACC[2], QACC respectively, define the same class of operators, leaving q> 2 as an open question. Our result resolves this question, implying that QAC (0) wf = QACC[q] = QACC for all q. We also prove the first upper bounds for QACC in terms of related language classes. We define classes of languages EQACC, NQACC (both for arbitrary complex amplitudes) and BQACCQ (for rational number amplitudes) and show that they are all contained in TC (0). To do this, we show that a TC (0) circuit can keep track of the amplitudes of the state resulting from the application of a QACC operator using a constant width polynomial size tensor sum. In order to accomplish this, we also show that TC (0) can perform iterated addition and multiplication in certain field extensions.
On the b 1 bitcomprehension rule
 Logic Colloquium 98
, 2000
"... Summary. The theory � b 1CR of Bounded Arithmetic axiomatized by the � b 1bitcomprehension rule is defined and shown to be strongly related to the complexity class TC 0. The � b 1definable functions of � b 1CR are those in uniform TC 0, and the � b 2definable functions are computable by counte ..."
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Cited by 5 (0 self)
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Summary. The theory � b 1CR of Bounded Arithmetic axiomatized by the � b 1bitcomprehension rule is defined and shown to be strongly related to the complexity class TC 0. The � b 1definable functions of � b 1CR are those in uniform TC 0, and the � b 2definable functions are computable by counterexample computations using TC 0functions. The latter is used to show that a collapse of stronger theories to � b 1CR implies that NP is contained in nonuniform TC 0. 1
Functional Characterizations of Uniform Logdepth and Polylogdepth Circuit Families
"... We characterize the classes of functions computable by uniform logdepth (NC 1) and polylogdepth circuit families as closures of a set of base functions. (The former is equivalent to ALOGTIME, the latter to polylogarithmic space.) The closures involve the "safe" composition of Bellantoni ..."
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Cited by 4 (0 self)
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We characterize the classes of functions computable by uniform logdepth (NC 1) and polylogdepth circuit families as closures of a set of base functions. (The former is equivalent to ALOGTIME, the latter to polylogarithmic space.) The closures involve the "safe" composition of Bellantoni and Cook as well as asafe "divide and conquer" recursion; a simple change to the definition of the latter distinguishes between log and polylog depth. The proofs proceed, in one direction, by showing that safe composition and divideandconquer recursion preserve growth rate and circuit depth bounds, and in the other, by simulating alternating Turing machines with divideandconquer recursion.