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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.
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.
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.
Randomness, Computation and Mathematics
"... This article examines some of the recent advances in our understanding of algorithmic randomness. It also discusses connections with various areas of mathematics, computer science and other areas of science. Some questions and speculations will be discussed. ..."
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This article examines some of the recent advances in our understanding of algorithmic randomness. It also discusses connections with various areas of mathematics, computer science and other areas of science. Some questions and speculations will be discussed.
REDUCTIONS TO THE SET OF RANDOM STRINGS:
, 2012
"... Vol. 10(3:5)2014, pp. 1–18 www.lmcsonline.org ..."
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Some Applications of Randomness in Computational Complexity
, 2013
"... In this dissertation we consider two different notions of randomness and their applications to problems in complexity theory. In part one of the dissertation we consider Kolmogorov complexity, a way to formalize a measure of the randomness of a single finite string, something that cannot be done u ..."
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In this dissertation we consider two different notions of randomness and their applications to problems in complexity theory. In part one of the dissertation we consider Kolmogorov complexity, a way to formalize a measure of the randomness of a single finite string, something that cannot be done using the usual distributional definitions. We let R be the set of random strings under this measure and study what resourcebounded machines can compute using R as an oracle. We show the surprising result that under proper definitions we can in fact define wellformed complexity classes using this approach, and that perhaps it is possible to exactly characterize standard classes such as BPP and NEXP in this way. In part two of the dissertation we switch gears and consider the use of randomness as a tool in propositional proof complexity, a subarea of complexity theory that addresses the NP vs. coNP problem. Here we consider the ability of various proof systems to efficiently refute randomly generated unsatisfiable 3CNF and 3XOR formulas. In particular, we show that certain restricted proof systems based on Ordered Binary Decision Diagrams requires exponentialsize refutations of these formulas. We also