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15
Almost Everywhere High Nonuniform Complexity
, 1992
"... . We investigate the distribution of nonuniform complexities in uniform complexity classes. We prove that almost every problem decidable in exponential space has essentially maximum circuitsize and spacebounded Kolmogorov complexity almost everywhere. (The circuitsize lower bound actually exceeds ..."
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Cited by 177 (36 self)
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. We investigate the distribution of nonuniform complexities in uniform complexity classes. We prove that almost every problem decidable in exponential space has essentially maximum circuitsize and spacebounded Kolmogorov complexity almost everywhere. (The circuitsize lower bound actually exceeds, and thereby strengthens, the Shannon 2 n n lower bound for almost every problem, with no computability constraint.) In exponential time complexity classes, we prove that the strongest relativizable lower bounds hold almost everywhere for almost all problems. Finally, we show that infinite pseudorandom sequences have high nonuniform complexity almost everywhere. The results are unified by a new, more powerful formulation of the underlying measure theory, based on uniform systems of density functions, and by the introduction of a new nonuniform complexity measure, the selective Kolmogorov complexity. This research was supported in part by NSF Grants CCR8809238 and CCR9157382 and in ...
Recursively Enumerable Reals and Chaitin Ω Numbers
"... A real is called recursively enumerable if it is the limit of a recursive, increasing, converging sequence of rationals. Following Solovay [23] and Chaitin [10] we say that an r.e. real dominates an r.e. real if from a good approximation of from below one can compute a good approximation of from b ..."
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Cited by 34 (3 self)
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A real is called recursively enumerable if it is the limit of a recursive, increasing, converging sequence of rationals. Following Solovay [23] and Chaitin [10] we say that an r.e. real dominates an r.e. real if from a good approximation of from below one can compute a good approximation of from below. We shall study this relation and characterize it in terms of relations between r.e. sets. Solovay's [23]like numbers are the maximal r.e. real numbers with respect to this order. They are random r.e. real numbers. The halting probability ofa universal selfdelimiting Turing machine (Chaitin's Ω number, [9]) is also a random r.e. real. Solovay showed that any Chaitin Ω number islike. In this paper we show that the converse implication is true as well: any Ωlike real in the unit interval is the halting probability of a universal selfdelimiting Turing machine.
An Observation on Probability versus Randomness with Applications to Complexity Classes
 MATHEMATICAL SYSTEMS THEORY
, 1993
"... Every class C of languages satisfying a simple topological condition is shown to have probability one if and only if it contains some language that is algorithmically random in the sense of MartinLof. This result is used to derive separation properties of algorithmically random oracles and to gi ..."
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Cited by 20 (7 self)
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Every class C of languages satisfying a simple topological condition is shown to have probability one if and only if it contains some language that is algorithmically random in the sense of MartinLof. This result is used to derive separation properties of algorithmically random oracles and to give characterizations of the complexity classes P, BPP, AM, and PH in terms of reducibility to such oracles. These characterizations lead to results like: P = NP if and only if there exists an algorithmically random set that is P btt hard for NP.
Recursive computational depth
 Information and Computation
, 1999
"... In the 1980's, Bennett introduced computational depth as a formal measure of the amount of computational history that is evident in an object's structure. In particular, Bennett identi ed the classes of weakly deep and strongly deep sequences, and showed that the halting problem is strongly deep. Ju ..."
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Cited by 18 (2 self)
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In the 1980's, Bennett introduced computational depth as a formal measure of the amount of computational history that is evident in an object's structure. In particular, Bennett identi ed the classes of weakly deep and strongly deep sequences, and showed that the halting problem is strongly deep. Juedes, Lathrop, and Lutz subsequently extended this result by de ning the class of weakly useful sequences, and proving that every weakly useful sequence is strongly deep. The present paper investigates re nements of Bennett's notions of weak and strong depth, called recursively weak depth (introduced by Fenner, Lutz and Mayordomo) and recursively strong depth (introduced here). It is argued that these re nements naturally capture Bennett's idea that deep objects are those which \contain internal evidence of a nontrivial causal history. " The fundamental properties of recursive computational depth are developed, and it is shown that the recursively weakly (respectively, strongly) deep sequences form a proper subclass of the class of weakly (respectively, strongly) deep sequences. The abovementioned theorem of Juedes, Lathrop, and Lutz is then strengthened by proving that every weakly useful sequence is recursively strongly deep. It follows from these results that not every strongly deep sequence is weakly useful, thereby answering a question posed by Juedes.
Circuit size relative to pseudorandom oracles, Theoretical Computer Science A 107
, 1993
"... Circuitsize complexity is compared with deterministic and nondeterministic time complexity in the presence of pseudorandom oracles. The following separations are shown to hold relative to every pspacerandom oracle A, and relative toalmost every oracle A 2 ESPACE. (i) NP A is not contained in SIZE ..."
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Cited by 15 (4 self)
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Circuitsize complexity is compared with deterministic and nondeterministic time complexity in the presence of pseudorandom oracles. The following separations are shown to hold relative to every pspacerandom oracle A, and relative toalmost every oracle A 2 ESPACE. (i) NP A is not contained in SIZE A (2 n)foranyreal < 1 3. (ii) E A is not contained in SIZE A ( 2n n). Thus, neither NP A nor E A is contained in P A /Poly. In fact, these separations are shown to hold for almost every n. Since a randomly selected oracle is pspacerandom with probability one, (i) and (ii) immediately imply the corresponding random oracle separations, thus improving a result of Bennett and Gill [9] and answering open questions of Wilson [47]. 1
Computational depth and reducibility
 Theoretical Computer Science
, 1994
"... This paper reviews and investigates Bennett's notions of strong and weak computational depth (also called logical depth) for in nite binary sequences. Roughly, an in nite binary sequence x is de ned to be weakly useful if every element of a nonnegligible set of decidable sequences is reducible to x ..."
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Cited by 12 (2 self)
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This paper reviews and investigates Bennett's notions of strong and weak computational depth (also called logical depth) for in nite binary sequences. Roughly, an in nite binary sequence x is de ned to be weakly useful if every element of a nonnegligible set of decidable sequences is reducible to x in recursively bounded time. It is shown that every weakly useful sequence is strongly deep. This result (which generalizes Bennett's observation that the halting problem is strongly deep) implies that every high Turing degree contains strongly deep sequences. It is also shown that, in the sense of Baire category, almost
A Characterization of C.E. Random Reals
 THEORETICAL COMPUTER SCIENCE
, 1999
"... A real # is computably enumerable if it is the limit of a computable, increasing, converging sequence of rationals; # is random if its binary expansion is a random sequence. Our aim is to offer a selfcontained proof, based on the papers [7, 14, 4, 13], of the following theorem: areal is c.e. and ra ..."
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Cited by 10 (0 self)
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A real # is computably enumerable if it is the limit of a computable, increasing, converging sequence of rationals; # is random if its binary expansion is a random sequence. Our aim is to offer a selfcontained proof, based on the papers [7, 14, 4, 13], of the following theorem: areal is c.e. and random if and only if it a Chaitin# real, i.e., the halting probability of some universal selfdelimiting Turing machine.
Some basic problems with complex systems
, 1999
"... From an engineering perspective, it is well known that there are numerous problems to predict and control complex systems. In addition, there are also problems to understand the concept of complexity from the perspective of physics and the epistemology of physics. Three outstanding topics in this re ..."
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Cited by 9 (4 self)
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From an engineering perspective, it is well known that there are numerous problems to predict and control complex systems. In addition, there are also problems to understand the concept of complexity from the perspective of physics and the epistemology of physics. Three outstanding topics in this regard are discussed: 1. the fundamental contextdependence of the de nition of complexity, 2. the relation between complexity and meaning, and 3. the restrictions on the applicability of limit theorems in the study of complex systems. 1
A Glimpse into Algorithmic Information Theory
 Logic, Language and Computation, Volume 3, CSLI Series
, 1999
"... This paper is a subjective, short overview of algorithmic information theory. We critically discuss various equivalent algorithmical models of randomness motivating a #randomness hypothesis". Finally some recent results on computably enumerable random reals are reviewed. 1 Randomness: An Informa ..."
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Cited by 6 (6 self)
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This paper is a subjective, short overview of algorithmic information theory. We critically discuss various equivalent algorithmical models of randomness motivating a #randomness hypothesis". Finally some recent results on computably enumerable random reals are reviewed. 1 Randomness: An Informal Discussion In which we discuss some di#culties arising in de#ning randomness. Suppose that one is watching a simple pendulum swing back and forth, recording 0 if it swings clockwise at a given instant and 1 if it swings counterclockwise. Suppose further that after some time the record looks as follows: 10101010101010101010101010101010: At this point one would like to deduce a #theory" from the experiment. 1 The #theory" should account for the data presently available and make #predictions" about future observations. How should one proceed? It is obvious that there are many #theories" that one could writedown for the given data, for example: 10101010101010101010101010101010000000000...
Resource Bounded Randomness and Computational Complexity
 Theoretical Computer Science
, 1997
"... We give a survey of resource bounded randomness concepts and show their relations to each other. Moreover, we introduce several new resource bounded randomness concepts corresponding to the classical randomness concepts. We show that the notion of polynomial time bounded Ko randomness is independent ..."
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Cited by 4 (2 self)
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We give a survey of resource bounded randomness concepts and show their relations to each other. Moreover, we introduce several new resource bounded randomness concepts corresponding to the classical randomness concepts. We show that the notion of polynomial time bounded Ko randomness is independent of the notions of polynomial time bounded Lutz, Schnorr and Kurtz randomness. Lutz has conjectured that, for a given time or space bound, the corresponding resource bounded Lutz randomness is a proper refinement of resource bounded Schnorr randomness. We answer this conjecture for the case of polynomial time bound in this paper. Moreover, we show that polynomial time bounded Schnorr randomness is a proper refinement of polynomial time bounded Kurtz randomness too. In contrast to this result, however, we also show that the notions of polynomial time bounded Lutz, Schnorr and Kurtz randomness coincide in the case of recursive sets, whence it suffices to study the notion of resource bounded Lu...