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An Improved TimeSpace Lower Bounds for Tautologies
"... We show that for all reals c and d such that c 2 d < 4 there exists a positive real e such that tautologies of length n cannot be decided by both a nondeterministic algorithm that runs in time n c, and a nondeterministic algorithm that runs in time n d and space n e. In particular, for every d < 3√ ..."
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We show that for all reals c and d such that c 2 d < 4 there exists a positive real e such that tautologies of length n cannot be decided by both a nondeterministic algorithm that runs in time n c, and a nondeterministic algorithm that runs in time n d and space n e. In particular, for every d < 3√ 4 there exists a positive e such that tautologies cannot be decided by a nondeterministic algorithm that runs in time n d and space n e.
A New TimeSpace Lower Bound for Nondeterministic Algorithms Solving Tautologies
, 2007
"... We show that for all reals c and d such that c 2 d < 4 there exists a positive real e such that tautologies cannot be decided by both a nondeterministic algorithm that runs in time n c, and a nondeterministic algorithm that runs in time n d and space n e. In particular, for every real d < 3 √ 4 ther ..."
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We show that for all reals c and d such that c 2 d < 4 there exists a positive real e such that tautologies cannot be decided by both a nondeterministic algorithm that runs in time n c, and a nondeterministic algorithm that runs in time n d and space n e. In particular, for every real d < 3 √ 4 there exists a positive real e such that tautologies cannot be decided by a nondeterministic algorithm that runs in time n d and space n e. 1
The Descriptive Complexity of the FixedPoints of Bounded Formulas
 Computer Science Logic '2000, 14th Annual Conference of the EACSL, volume 1862 of Lecture Notes in Computer Science
, 2000
"... . We investigate the complexity of the fixedpoints of bounded formulas in the context of finite set theory; that is, in the context of arbitrary classes of finite structures that are equipped with a builtin BIT predicate, or equivalently, with a builtin membership relation between hereditaril ..."
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. We investigate the complexity of the fixedpoints of bounded formulas in the context of finite set theory; that is, in the context of arbitrary classes of finite structures that are equipped with a builtin BIT predicate, or equivalently, with a builtin membership relation between hereditarily finite sets (input relations are allowed). We show that the iteration of a positive bounded formula converges in polylogarithmically many steps in the cardinality of the structure. This extends a previously known much weaker result. We obtain a number of connections with the rudimentary languages and deterministic polynomialtime. Moreover, our results provide a natural characterization of the complexity class consisting of all languages computable by boundeddepth, polynomialsize circuits, and polylogarithmictime uniformity. As a byproduct, we see that this class coincides with LH(P), the logarithmictime hierarchy with an oracle to deterministic polynomialtime. Finally, we dis...
The Complexity of ResourceBounded Propositional Proofs
, 2001
"... Propositional Proof Complexity is an active area of research whose main focus is the study of the length of proofs in propositional logic. There are several motivations for such a study, the main of which is probably its connection to the P vs NP problem in Computational Complexity. The experience ..."
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Propositional Proof Complexity is an active area of research whose main focus is the study of the length of proofs in propositional logic. There are several motivations for such a study, the main of which is probably its connection to the P vs NP problem in Computational Complexity. The experience
NonLinear Time Lower Bound for (Succinct) Quantified Boolean Formulas
"... Abstract. We give a reduction from arbitrary languages in alternating time t(n) to quantified Boolean formulas (QBF) describable in O(t(n)) bits. The reduction works for a reasonable succinct encoding of Boolean formulas and for several reasonable machine models, including multitape Turing machines ..."
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Abstract. We give a reduction from arbitrary languages in alternating time t(n) to quantified Boolean formulas (QBF) describable in O(t(n)) bits. The reduction works for a reasonable succinct encoding of Boolean formulas and for several reasonable machine models, including multitape Turing machines and logarithmiccost RAMs. By a simple diagonalization, it follows that our succinct QBF problem requires superlinear time on those models. To our knowledge this is the first known instance of a nonlinear time lower bound (with no space restriction) for solving a natural linear space problem on a variety of computational models.
Computational Aspects of FirstOrder Logic on Finite Structures
, 1999
"... Descriptive complexity aims to classify properties of finite structures according to the logical resources that are needed to express them. The ImmermanVardi Theorem states that a property of finite ordered structures is computable in polynomialtime if and only if it can be expressed in least fixe ..."
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Descriptive complexity aims to classify properties of finite structures according to the logical resources that are needed to express them. The ImmermanVardi Theorem states that a property of finite ordered structures is computable in polynomialtime if and only if it can be expressed in least fixedpoint logic. Here, least fixedpoint logic is the extension of firstorder logic with the ability to buildin inductive definitions. It is known that firstorder logic is a fairly weak expressive language in the context of descriptive complexity. For example, least fixedpoint logic is strictly more expressive than firstorder logic on the class of all finite ordered structures. The ordered conjecture, formulated by Kolaitis and Vardi in 1992, asks whether this is also the case for any infinite class of finite ordered structures. Although some significant progress has been made since its formulation, the ordered conjecture remains open. In fact, it has been shown that any way of resolving ...
A Status Report on the P versus NP Question
"... We survey some of the history of the most famous open question in computing: the P versus NP question. We summarize some of the progress that has been made to date, and assess the current situation. ..."
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We survey some of the history of the most famous open question in computing: the P versus NP question. We summarize some of the progress that has been made to date, and assess the current situation.
TimeSpace Efficient Simulations of Quantum Computations
, 2010
"... We give two time and spaceefficient simulations of quantum computations with intermediate measurements, one by classical randomized computations with unbounded error and the other by quantum computations that use an arbitrary fixed universal set of gates. Specifically, our simulations show that ev ..."
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We give two time and spaceefficient simulations of quantum computations with intermediate measurements, one by classical randomized computations with unbounded error and the other by quantum computations that use an arbitrary fixed universal set of gates. Specifically, our simulations show that every language solvable by a boundederror quantum algorithm running in time t and space s is also solvable by an unboundederror randomized algorithm running in time O(t · log t) and space O(s + log t), as well as by a boundederror quantum algorithm restricted to use an arbitrary universal set and running in time O(t · polylog t) and space O(s + log t), provided the universal set is closed under adjoint. We also develop a quantum model that is particularly suitable for the study of general computations with simultaneous time and space bounds. As an application of our randomized simulation, we obtain the first nontrivial lower bound for general quantum algorithms solving problems related to satisfiability. Our bound applies to MajSAT and MajMajSAT, which are the problems of determining the truth value of a given Boolean formula whose variables are fully quantified by one or two majority quantifiers, respectively. We prove that for every real d and every positive real δ there exists a real c> 1 such that either • MajMajSAT does not have a boundederror quantum algorithm running in time O(n c), or • MajSAT does not have a boundederror quantum algorithm running in time O(n d) and space O(n 1−δ). In particular, MajMajSAT does not have a boundederror quantum algorithm running in time O(n 1+o(1) ) and space O(n 1−δ) for any δ> 0. Our lower bounds hold for any reasonable uniform model of quantum computation, in particular for the model we develop. 1
Amplifying Circuit Lower Bounds Against Polynomial Time With Applications
 In IEEE Conference on Computational Complexity
"... We give a selfreduction for the Circuit Evaluation problem (CircEval), and prove the following consequences. • Amplifying SizeDepth Lower Bounds. If CircEval has Boolean circuits of n k size and n1−δ depth for some k and δ, then for every ε> 0, there is a δ ′> 0 such that CircEval has circuits of ..."
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We give a selfreduction for the Circuit Evaluation problem (CircEval), and prove the following consequences. • Amplifying SizeDepth Lower Bounds. If CircEval has Boolean circuits of n k size and n1−δ depth for some k and δ, then for every ε> 0, there is a δ ′> 0 such that CircEval has circuits of n1+ε size and n1−δ ′ depth. Moreover, the resulting circuits require only Õ(nε) bits of nonuniformity to construct. As a consequence, strong enough depth lower bounds for Circuit Evaluation imply a full separation of P and NC (even with a weak size lower bound). • Lower Bounds for Quantified Boolean Formulas. Let c, d> 1 and e < 1 satisfy c < (1 − e + d)/d. Either the problem of recognizing valid quantified Boolean formulas (QBF) is not solvable in TIME[n c], or the Circuit Evaluation problem cannot be solved with circuits of n d size and n e depth. This implies unconditional polynomialtime uniform circuit lower bounds for solving QBF. We also prove that QBF does not have n ctime uniform NC circuits, for all c < 2. 1