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Equivalence of Measures of Complexity Classes
"... The resourcebounded measures of complexity classes are shown to be robust with respect to certain changes in the underlying probability measure. Specifically, for any real number ffi ? 0, any uniformly polynomialtime computable sequence ~ fi = (fi 0 ; fi 1 ; fi 2 ; : : : ) of real numbers (biases ..."
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Cited by 70 (19 self)
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The resourcebounded measures of complexity classes are shown to be robust with respect to certain changes in the underlying probability measure. Specifically, for any real number ffi ? 0, any uniformly polynomialtime computable sequence ~ fi = (fi 0 ; fi 1 ; fi 2 ; : : : ) of real numbers (biases) fi i 2 [ffi; 1 \Gamma ffi], and any complexity class C (such as P, NP, BPP, P/Poly, PH, PSPACE, etc.) that is closed under positive, polynomialtime, truthtable reductions with queries of at most linear length, it is shown that the following two conditions are equivalent. (1) C has pmeasure 0 (respectively, measure 0 in E, measure 0 in E 2 ) relative to the cointoss probability measure given by the sequence ~ fi. (2) C has pmeasure 0 (respectively, measure 0 in E, measure 0 in E 2 ) relative to the uniform probability measure. The proof introduces three techniques that may be useful in other contexts, namely, (i) the transformation of an efficient martingale for one probability measu...
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.
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
The Kolmogorov Complexity of Random Reals
 Ann. Pure Appl. Logic
, 2003
"... We investigate the initial segment complexity of random reals. Let K(... ..."
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Cited by 5 (1 self)
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We investigate the initial segment complexity of random reals. Let K(...
Algorithmic complexity of recursive and inductive algorithms
 THEORETICAL COMPUTER SCIENCE
, 2004
"... The main goal of this paper is to compare recursive algorithms such as Turing machines with such superrecursive algorithms as inductive Turing machines. This comparison is made in a general setting of dual complexity measures such as Kolmogorov or algorithmic complexity. To make adequate comparison ..."
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Cited by 3 (1 self)
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The main goal of this paper is to compare recursive algorithms such as Turing machines with such superrecursive algorithms as inductive Turing machines. This comparison is made in a general setting of dual complexity measures such as Kolmogorov or algorithmic complexity. To make adequate comparison, we reconsider the standard axiomatic approach to complexity of algorithms. The new approach allows us to achieve a more adequate representation of static system complexity in the axiomatic context. It is demonstrated that for solving many problems inductive Turing machines have much lower complexity than Turing machines and other recursive algorithms. Thus, inductive Turing machines are not only more powerful, but also more efficient than Turing machines.
Comparison Between the Complexity of a Function and the Complexity of Its Graph
, 1998
"... This paper investigates in terms of Kolmogorov complexity the differences between the information necessary to compute a recursive function and the information contained in its graph. Our first result is that the complexity of the initial parts of the graph of a recursive function, although bounded, ..."
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Cited by 1 (1 self)
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This paper investigates in terms of Kolmogorov complexity the differences between the information necessary to compute a recursive function and the information contained in its graph. Our first result is that the complexity of the initial parts of the graph of a recursive function, although bounded, has almost never a limit. The second result is that the complexity of these initial parts approximate the complexity of the function itself in most cases (and in the average) but not always.
Comparison Between the Complexity of a Function and the Complexity of Its Graph
, 1998
"... This paper investigates in terms of Kolmogorov complexity the differences between the information necessary to compute a recursive function and the information contained in its graph. Our first result is that the complexity of the initial parts of the graph of a recursive function, although bounded, ..."
Abstract
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This paper investigates in terms of Kolmogorov complexity the differences between the information necessary to compute a recursive function and the information contained in its graph. Our first result is that the complexity of the initial parts of the graph of a recursive function, although bounded, has almost never a limit. The second result is that the complexity of these initial parts approximate the complexity of the function itself in most cases (and in the average) but not always.