Results 1  10
of
13
TimeSpace Lower Bounds for Satisfiability
 JACM
, 2005
"... We establish the first polynomial timespace lower bounds for satisfiability on general models of computation. We show that for any constant c less than the golden ratio there exists a positive constant d such that no deterministic randomaccess Turing machine can solve satisfiability in time n c an ..."
Abstract

Cited by 25 (7 self)
 Add to MetaCart
We establish the first polynomial timespace lower bounds for satisfiability on general models of computation. We show that for any constant c less than the golden ratio there exists a positive constant d such that no deterministic randomaccess Turing machine can solve satisfiability in time n c and space n d, where d approaches 1 when c does. On conondeterministic instead of deterministic machines, we prove the same for any constant c less than √ 2. Our lower bounds apply to nondeterministic linear time and almost all natural NPcomplete problems known. In fact, they even apply to the class of languages that can be solved on a nondeterministic machine in linear time and space n 1/c. Our proofs follow the paradigm of indirect diagonalization. We also use that paradigm to prove timespace lower bounds for languages higher up in the polynomialtime hierarchy.
TimeSpace Tradeoffs for Nondeterministic Computation
 In Proceedings of the 15th IEEE Conference on Computational Complexity
, 2000
"... We show new tradeoffs for satisfiability and nondeterministic linear time. Satisfiability cannot be solved on general purpose randomaccess Turing machines in time n 1.618 and space n o(1) . This improves recent results of Fortnow and of Lipton and Viglas. In general, for any constant a less tha ..."
Abstract

Cited by 24 (2 self)
 Add to MetaCart
We show new tradeoffs for satisfiability and nondeterministic linear time. Satisfiability cannot be solved on general purpose randomaccess Turing machines in time n 1.618 and space n o(1) . This improves recent results of Fortnow and of Lipton and Viglas. In general, for any constant a less than the golden ratio, we prove that satisfiability cannot be solved in time n a and space n b for some positive constant b. Our techniques allow us to establish this result for b < 1 2 ( a+2 a 2  a). We can do better for a close to the golden ratio, for example, satisfiability cannot be solved by a randomaccess Turing machine using n 1.46 time and n .11 space. We also show tradeoffs for nondeterministic linear time computations using sublinear space. For example, there exists a language computable in nondeterministic linear time and n .619 space that cannot be computed in deterministic n 1.618 time and n o(1) space. Higher up the polynomialtime hierarchy we can get be...
Interactive Proof Systems And Alternating TimeSpace Complexity
 Theoretical Computer Science
, 1991
"... . We show a rough equivalence between alternating timespace complexity and a publiccoin interactive proof system with the verifier having a polynomial related timespace complexity. Special cases include ffi All of NC has interactive proofs with a logspace polynomialtime publiccoin verifier va ..."
Abstract

Cited by 20 (4 self)
 Add to MetaCart
. We show a rough equivalence between alternating timespace complexity and a publiccoin interactive proof system with the verifier having a polynomial related timespace complexity. Special cases include ffi All of NC has interactive proofs with a logspace polynomialtime publiccoin verifier vastly improving the best previous lower bound of LOGCFL for this model [7]. ffi All languages in P have interactive proofs with a polynomialtime publiccoin verifier using o(log 2 n) space. ffi All exponentialtime languages have interactive proof systems with publiccoin polynomialspace exponentialtime verifiers. To achieve better bounds, we show how to reduce a ktape alternating Turing machine to a 1tape alternating Turing machine with only a constant factor increase in time and space. 1. Introduction In 1981, Chandra, Kozen and Stockmeyer [4] introduced alternating Turing machines, an extension of nondeterministic computation where the Turing 1 Supported by NSF Grant CCR900993...
The Power Of Interaction
, 1991
"... : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : vii Chapter 1. INTRODUCTION : : : : : : : : : : : : : : : : : : : : : : : : : 1 2. PRELIMINARIES : : : : : : : : : : : : : : : : : : : : : : : : : 4 2.1 Basic Definitions : : : : : : : : : : : : : : : : : : : : : : : 4 2.1.1 Basics : : ..."
Abstract

Cited by 10 (0 self)
 Add to MetaCart
: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : vii Chapter 1. INTRODUCTION : : : : : : : : : : : : : : : : : : : : : : : : : 1 2. PRELIMINARIES : : : : : : : : : : : : : : : : : : : : : : : : : 4 2.1 Basic Definitions : : : : : : : : : : : : : : : : : : : : : : : 4 2.1.1 Basics : : : : : : : : : : : : : : : : : : : : : : : : 4 2.1.2 Boolean Formulas : : : : : : : : : : : : : : : : : 4 2.1.3 Arithmetic Formulas and Expressions : : : : : : 5 2.2 Computational Models : : : : : : : : : : : : : : : : : : : : 9 2.2.1 Deterministic Computation : : : : : : : : : : : : 9 2.2.2 Probabilistic Computation : : : : : : : : : : : : 11 2.2.3 NonDeterministic Computation : : : : : : : : : 12 2.2.4 Alternating Computations : : : : : : : : : : : : 13 2.2.5 Interactive Proof Systems : : : : : : : : : : : : : 13 2.2.6 Multiple Prover Interactive Proof Systems : : : 15 2.2.7 Computation relative to an Oracle : : : : : : : : 15 2.3 Complexity Classes : : : : : : : : : : : : : : : : : : : : ...
A new parallel vector model, with exact characterizations of NC k
 in Proc. 11th Annual Symposium on Theoretical Aspects of Computer Science
, 1994
"... This paper develops a new and natural parallel vector model, and shows that for all k ≥ 1, the languages recognizable in O(log k n) time and polynomial work in the model are exactly those in NC k. Some improvements to other simulations in parallel models and reversal complexity are given. 1 ..."
Abstract

Cited by 5 (4 self)
 Add to MetaCart
This paper develops a new and natural parallel vector model, and shows that for all k ≥ 1, the languages recognizable in O(log k n) time and polynomial work in the model are exactly those in NC k. Some improvements to other simulations in parallel models and reversal complexity are given. 1
Parallelism Always Helps
 SIAM J. Comput
, 1997
"... . It is shown that every unitcost randomaccess machine (RAM) that runs in time T can be simulated by a concurrentread exclusivewrite parallel randomaccess machine (CREW PRAM) in time O(T 1/2 log T ). The proof is constructive; thus it gives a mechanical way to translate any sequential algori ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
. It is shown that every unitcost randomaccess machine (RAM) that runs in time T can be simulated by a concurrentread exclusivewrite parallel randomaccess machine (CREW PRAM) in time O(T 1/2 log T ). The proof is constructive; thus it gives a mechanical way to translate any sequential algorithm designed to run on a unitcost RAM into a parallel algorithm that runs on a CREW PRAM and obtain a nearly quadratic speedup. One implication is that there does not exist any recursive function that is "inherently not parallelizable." Key words. computational complexity, time complexity, randomaccess machine, parallel randomaccess machine, simulation, speedup AMS subject classifications. 68Q05, 68Q10, 68Q15, 03D10, 03D15 PII. S0097539794265402 1. Introduction. 1.1. Motivation. For some problems, the direct parallelization of a sequential algorithm gives a faster parallel algorithm. An example is matrix multiplication. The bruteforce sequential algorithm for matrix multiplication runs ...
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 ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
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.
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 ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
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
On Multiplayer NonCooperative Games of Incomplete Information: Part 2  Lower Bounds
, 1991
"... We extend the alternating machine (ATM) of Chandra, Kozen and Stockmeyer [CKS81], the private and the blind alternating machines of Reif [Reif84] to model multiplayer games of incomplete information. We use these machines to provide matching lower bounds for our decision algorithms described in ..."
Abstract
 Add to MetaCart
We extend the alternating machine (ATM) of Chandra, Kozen and Stockmeyer [CKS81], the private and the blind alternating machines of Reif [Reif84] to model multiplayer games of incomplete information. We use these machines to provide matching lower bounds for our decision algorithms described in the first part of this pair of papers [APR91a]. We apply multiple person alternation to other machine types. We show that multiplayer games of incomplete information can be undecidable in general, unless the information is hierarchically arranged (as defined later in this paper). In hierarchical multiplayer games, each additional clique (subset of players with same information) increases the complexity of the outcome problem by a further exponential. Consequently, if a multiplayer game of incomplete information with k cliques has a space bound of S(n), then its outcome is k repeated exponentials harder than games of complete information with the same space bound S(n). This paper prov...