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NPcomplete problems and physical reality
 ACM SIGACT News Complexity Theory Column, March. ECCC
, 2005
"... Can NPcomplete problems be solved efficiently in the physical universe? I survey proposals including soap bubbles, protein folding, quantum computing, quantum advice, quantum adiabatic algorithms, quantummechanical nonlinearities, hidden variables, relativistic time dilation, analog computing, Mal ..."
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Can NPcomplete problems be solved efficiently in the physical universe? I survey proposals including soap bubbles, protein folding, quantum computing, quantum advice, quantum adiabatic algorithms, quantummechanical nonlinearities, hidden variables, relativistic time dilation, analog computing, MalamentHogarth spacetimes, quantum gravity, closed timelike curves, and “anthropic computing. ” The section on soap bubbles even includes some “experimental ” results. While I do not believe that any of the proposals will let us solve NPcomplete problems efficiently, I argue that by studying them, we can learn something not only about computation but also about physics. 1
Algorithmic randomness, quantum physics, and incompleteness
 Proceedings of the Conference “Machines, Computations and Universality” (MCU’2004), number 3354 in Lecture Notes in Computer Science
, 2006
"... When a distinguished but elderly scientist states that something is possible, he is almost certainly right. When he states that something is impossible, he is almost certainly wrong. Arthur C. Clarke ..."
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When a distinguished but elderly scientist states that something is possible, he is almost certainly right. When he states that something is impossible, he is almost certainly wrong. Arthur C. Clarke
Limits on the Computational Power of Random Strings
"... Let C(x) andK(x) denote plain and prefix Kolmogorov complexity, respectively, and let RC and RK denote the sets of strings that are “random ” according to these measures; both RK and RC are undecidable. Earlier work has shown that every set in NEXP is in NP relative to both RK and RC, and that every ..."
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Let C(x) andK(x) denote plain and prefix Kolmogorov complexity, respectively, and let RC and RK denote the sets of strings that are “random ” according to these measures; both RK and RC are undecidable. Earlier work has shown that every set in NEXP is in NP relative to both RK and RC, and that every set in BPP is polynomialtime truthtable reducible to both RK and RC [ABK06a, BFKL10]. (All of these inclusions hold, no matter which “universal ” Turing machine one uses in the definitions of C(x) andK(x).) Since each machine U gives rise to a slightly different measure CU or KU, these inclusions can be stated as: • BPP ⊆ DEC ∩ ⋂ U
Curiouser and curiouser: The link between incompressibility and complexity
 In Proc. Computability in Europe (CiE), LNCS
, 2012
"... Abstract. This talk centers around some audacious conjectures that attempt to forge firm links between computational complexity classes and the study of Kolmogorov complexity. More specifically, let R denote the set of Kolmogorovrandom strings. Let BPP denote the class of problems that can be solve ..."
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Abstract. This talk centers around some audacious conjectures that attempt to forge firm links between computational complexity classes and the study of Kolmogorov complexity. More specifically, let R denote the set of Kolmogorovrandom strings. Let BPP denote the class of problems that can be solved with negligible error by probabilistic polynomialtime computations, and let NEXP denote the class of problems solvable in nondeterministic exponential time. Conjecture 1: NEXP = NP R. Conjecture 2: BPP is the class of problems nonadaptively polynomialtime reducible to R. These conjectures are not only audacious; they are obviously false! R is not a decidable set, and thus it is absurd to suggest that the class of problems reducible to it constitutes a complexity class. The absurdity fades if, for example, we interpret “NP R ” to be “the class of problems that are NPTuring reducible to R, no matter which universal machine we use in defining Kolmogorov complexity”. The lecture will survey the body of work (some of it quite recent) that suggests that, when interpreted properly, the conjectures may actually be true. 1
Kolmogorov Complexity, Circuits, and the Strength of Formal Theories of Arithmetic
"... Can complexity classes be characterized in terms of efficient reducibility to the (undecidable) set of Kolmogorovrandom strings? Although this might seem improbable, a series of papers has recently provided evidence that this may be the case. In particular, it is known that there is a class of prob ..."
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Can complexity classes be characterized in terms of efficient reducibility to the (undecidable) set of Kolmogorovrandom strings? Although this might seem improbable, a series of papers has recently provided evidence that this may be the case. In particular, it is known that there is a class of problems C defined in terms of polynomialtime truthtable reducibility to RK (the set of Kolmogorovrandom strings) that lies between BPP and PSPACE [4, 3]. In this paper, we investigate improving this upper bound from PSPACE to PSPACE ∩ P/poly. More precisely, we present a collection of true statements in the language of arithmetic, (each provable in ZF) and show that if these statements can be proved in certain extensions of Peano arithmetic, then BPP ⊆C⊆PSPACE ∩ P/poly. We conjecture that C is equal to P, and discuss the possibility this might be an avenue for trying to prove the equality of BPP and P.
On strings with trivial Kolmogorov complexity
 Int J Software Informatics
"... Abstract The Kolmogorov complexity of a string is the length of the shortest program that generates it. A binary string is said to have trivial Kolmogorov complexity if its complexity is at most the complexity of its length. Intuitively, such strings carry no more information than the information th ..."
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Abstract The Kolmogorov complexity of a string is the length of the shortest program that generates it. A binary string is said to have trivial Kolmogorov complexity if its complexity is at most the complexity of its length. Intuitively, such strings carry no more information than the information that is inevitably coded into their length (which is the same as the information coded into a sequence of 0s of the same length). We study the set of these trivial sequences from a computational perspective, and with respect to plain and prefixfree Kolmogorov complexity. This work parallels the well known study of the set of nonrandom strings (which was initiated by Kolmogorov and developed by Kummer, Muchnik, Stephan, Allender and others) and points to several directions for further research.
On the computational power of random strings
 Annals of Pure and Applied Logic
"... Abstract. There are two fundamental computably enumerable sets associated with any Kolmogorov complexity measure. These are the set of nonrandom strings and the overgraph. This paper investigates the computational power of these sets. It follows work done by Kummer, Muchnik and Positselsky, and All ..."
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Abstract. There are two fundamental computably enumerable sets associated with any Kolmogorov complexity measure. These are the set of nonrandom strings and the overgraph. This paper investigates the computational power of these sets. It follows work done by Kummer, Muchnik and Positselsky, and Allender and coauthors. Muchnik and Positselsky asked whether there exists an optimal monotone machine whose overgraph is not ttcomplete. This paper answers this question in the negative by proving that the overgraph of any optimal monotone machine, or any optimal process machine, is ttcomplete. The monotone results are shown for both descriptional complexity Km and KM, the complexity measure derived from algorithmic probability. A distinction is drawn between two definitions of process machines that exist in the literature. For one class of process machines, designated strict process machines, it is shown that there is a universal machine whose set of nonrandom strings is not ttcomplete. 1.
RANDOM STRINGS AND TRUTHTABLE DEGREES OF TURING COMPLETE C.E. SETS
"... Abstract. We investigate the truthtable degrees of (co)c.e. sets, in particular, sets of random strings. It is known that the set of random strings with respect to any universal prefixfree machine is Turing complete, but that truthtable completeness depends on the choice of universal machine. W ..."
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Abstract. We investigate the truthtable degrees of (co)c.e. sets, in particular, sets of random strings. It is known that the set of random strings with respect to any universal prefixfree machine is Turing complete, but that truthtable completeness depends on the choice of universal machine. We show that for such sets of random strings, any finite set of their truthtable degrees do not meet to the degree 0, even within the c.e. truthtable degrees, but when taking the meet over all such truthtable degrees, the infinite meet is indeed 0. The latter result proves a conjecture of Allender, Friedman and Gasarch. We also show that there are two Turing complete c.e. sets whose truthtable degrees form a minimal pair. 1.
Algorithmic Randomness and Computability
"... Abstract. We examine some recent work which has made significant progress in out understanding of algorithmic randomness, relative algorithmic randomness and their relationship with algorithmic computability and relative algorithmic computability. ..."
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Abstract. We examine some recent work which has made significant progress in out understanding of algorithmic randomness, relative algorithmic randomness and their relationship with algorithmic computability and relative algorithmic computability.
Reductions to the set of random strings: The resourcebounded case
 in Proc. 37th International Symposium on Mathematical Foundations of Computer Science (MFCS ’12), 2012, Lecture Notes in Computer Science
"... ABSTRACT. This paper is motivated by a conjecture [All12, ADF+13] that BPP can be characterized in terms of polynomialtime nonadaptive reductions to the set of Kolmogorovrandom strings. In this paper we show that an approach laid out in [ADF+13] to settle this conjecture cannot succeed without sig ..."
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ABSTRACT. This paper is motivated by a conjecture [All12, ADF+13] that BPP can be characterized in terms of polynomialtime nonadaptive reductions to the set of Kolmogorovrandom strings. In this paper we show that an approach laid out in [ADF+13] to settle this conjecture cannot succeed without significant alteration, but that it does bear fruit if we consider timebounded Kolmogorov complexity instead. We show that if a setA is reducible in polynomial time to the set of timetbounded Kolmogorovrandom strings (for all large enough time bounds t), then A is in P/poly, and that if in addition such a reduction exists for any universal Turing machine one uses in the definition of Kolmogorov complexity, then A is in PSPACE. 1.