What are the gains of the theory of algorithms

Uspensky, Vladimir; Semenov, A. L.

doi:10.1007/3-540-11157-3_27

Cited by 22 publications

(4 citation statements)

References 84 publications

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“…According to this definition, Gurevich's abstract state machines capture algorithms. In [19], it is claimed that the only algorithms are those realized by Kolmogorov machines and that therefore the concept of a Kolmogorov machine can be regarded as an adequate formal characterization of the concept of an algorithm (see also [25]).…”

Section: Discussionmentioning

confidence: 99%

Long multiplication by instruction sequences with backward jump instructions

Bergstra¹,

Middelburg²

2013

Preprint

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For each function on bit strings, its restriction to bit strings of any given length can be computed by a finite instruction sequence that contains only instructions to set and get the content of Boolean registers, forward jump instructions, and a termination instruction. Backward jump instructions are not necessary for this, but instruction sequences can be significantly shorter with them. We take the function on bit strings that models the multiplication of natural numbers on their representation in the binary number system to demonstrate this by means of a concrete example. The example is reason to discuss points concerning the halting problem and the concept of an algorithm.

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Section: Discussionmentioning

confidence: 99%

Long multiplication by instruction sequences with backward jump instructions

Bergstra¹,

Middelburg²

2013

Preprint

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“…In [16], it is claimed that the only algorithms are those expressed by Kolmogorov machines and that therefore the concept of a Kolmogorov machine can be regarded as an adequate formal characterization of the concept of an algorithm (see also [21]). With this the concept of a Kolmogorov machine is actually qualified as a replacement of the concept of an algorithm.…”

Section: Background On the Notion Of Algorithmic Samenessmentioning

confidence: 99%

On Algorithmic Equivalence of Instruction Sequences for Computing Bit String Functions

Bergstra

Middelburg

2015

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Every partial function from bit strings of a given length to bit strings of a possibly different given length can be computed by a finite instruction sequence that contains only instructions to set and get the content of Boolean registers, forward jump instructions, and a termination instruction. We look for an equivalence relation on instruction sequences of this kind that captures to a reasonable degree the intuitive notion that two instruction sequences express the same algorithm.

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“…For a discussion, see [3,15,16,20,21]. The mean, or expected, value of a random variable X will be denoted by E(X ).…”

Section: Preliminariesmentioning

confidence: 99%

The Descriptive Complexity of Brownian Motion

Fouché

2000

Advances in Mathematics

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A continuous function x on the unit interval is a generic Brownian motion when every probabilistic event which holds almost surely with respect to the Wiener measure is reflected in x, provided that the event has a suitably effective description. We show that a generic one-dimensional Brownian motion can be computed from an infinite binary string which is complex in the sense of Kolmogorov Chaitin. Conversely, one can construct a Kolmogorov Chaitin random string from the values at the rational numbers of a generic Brownian motion. In this way, we construct a recursive isomorphism between encoded versions of generic Brownian motions and Kolmogorov Chaitin random reals. Academic Press

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What are the gains of the theory of algorithms

Cited by 22 publications

References 84 publications

Long multiplication by instruction sequences with backward jump instructions

Long multiplication by instruction sequences with backward jump instructions

On Algorithmic Equivalence of Instruction Sequences for Computing Bit String Functions

The Descriptive Complexity of Brownian Motion

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