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Kolmogorov Complexity Theory over the Reals
[article]

2008
*
arXiv
*
pre-print

*Kolmogorov*

*Complexity*constitutes an integral part of computability

*theory*, information

*theory*, and computational

*complexity*

*theory*-- in

*the*discrete setting of bits and Turing machines. ...

*Over*

*real*numbers, on

*the*other hand,

*the*BSS-machine (aka

*real*-RAM) has been established as a major model of computation. ... Finally we owe to Klaus Meer for pointing us to

*the*seminal work of Montaña and Pardo who first introduced

*real*

*Kolmogorov*

*Complexity*. ...

##
###
Kolmogorov Complexity Theory over the Reals

2008
*
Electronical Notes in Theoretical Computer Science
*

*Kolmogorov*

*Complexity*constitutes an integral part of computability

*theory*, information

*theory*, and computational

*complexity*

*theory*-in

*the*discrete setting of bits and Turing machines. ...

*Over*

*real*numbers, on

*the*other hand,

*the*BSS-machine (aka

*real*-RAM) has been established as a major model of computation. ... Finally we owe to Klaus Meer for pointing us to

*the*seminal work of Montaña and Pardo who first introduced

*real*

*Kolmogorov*

*Complexity*. ...

##
###
On the Kolmogorov complexity of continuous real functions

2013
*
Annals of Pure and Applied Logic
*

An asymptotic bound on

doi:10.1016/j.apal.2012.11.003
fatcat:xjky2cr3vrcldlgikehn7uqxz4
*the**Kolmogorov**complexities*of total single-valued computable*real*functions will be presented as well. ... Later,*the*definition was extended to*real*numbers based on*the*asymptotic behaviour of*the*sequence of*the**Kolmogorov**complexities*of*the*finitely-representable objects-such as rational numbers-used to ... Montaña and Pardo [17] and Ziegler and Koolen [25] have studied*Kolmogorov**complexity**over*sequences of*real*numbers based on*the**theory*of*real*Turing machines. ...##
###
Page 10262 of Mathematical Reviews Vol. , Issue 2004m
[page]

2004
*
Mathematical Reviews
*

*The*paper addresses a question on

*the*

*Kolmogorov*

*complexity*of

*reals*. ... Using prefix-free

*Kolmogorov*

*complexity*<j as a reducibil- ity on

*the*set of all

*reals*, it proves that there are continuum many H-degrees in

*the*random

*reals*. ...

##
###
Complexities Approach to Two Problems In Number Theory
[article]

2016
*
arXiv
*
pre-print

problems in

arXiv:1610.04026v3
fatcat:5udl5eurnzh63c6ysh6b7kta6y
*the*non-logical discipline.Futhermore,we show that resource-bounded*Kolmogorov**Complexity*and computational*complexity*can at least provide tips or principles to mathematical problems in*the*... By*Kolmogorov**Complexity*,two number-theoretic problems are solved in different way than before,one problem is Maxim Kontsevich and Don Bernard Zagier's Problem 3 Exhibit at least one number which does ... question by*Kolmogorov**Complexity*. ...##
###
Is Randomness Native to Computer Science? Ten Years After
[chapter]

2011
*
Randomness Through Computation
*

I remember that

doi:10.1142/9789814327756_0019
fatcat:lxjyg53fzzb2nlqb6i4oofiicy
*Kolmogorov*built*the*mathematical*theory*of*the*so-called*Kolmogorov**complexity*out of this paradox. ... Shannon proved that*the*functions*over**the**reals*which are computed by GPACs are exactly*the*solutions of algebraic differential systems. ...##
###
On the Kolmogorov Complexity of Continuous Real Functions
[chapter]

2011
*
Lecture Notes in Computer Science
*

An asymptotic bound on

doi:10.1007/978-3-642-21875-0_9
fatcat:6pbcuw4ydvbqtox7hkkczztu3q
*the**Kolmogorov**complexities*of total single-valued computable*real*functions will be presented as well. ... Later,*the*definition was extended to*real*numbers based on*the*asymptotic behaviour of*the*sequence of*the**Kolmogorov**complexities*of*the*finitely-representable objects-such as rational numbers-used to ... I would also like to thank Douglas Cenzer and Willem Fouché for informing me of their related work on*the*subject of algorithmic randomness*over*function spaces. ...##
###
Page 5977 of Mathematical Reviews Vol. , Issue 94j
[page]

1994
*
Mathematical Reviews
*

Chapter 2 (“Algorithmic

*complexity*”) is dedicated to*the**theory*of*the*most general form of*Kolmogorov**complexity*, called in*the*book*the*plain*Kolmogorov**complex*- ity. ...*Kolmogorov**complexity*is*the**theory*which explains such dis- crepancies and explores*the*properties and capabilities of objects like*the*ones above. ...##
###
Page 4394 of Mathematical Reviews Vol. , Issue 2002F
[page]

2002
*
Mathematical Reviews
*

Summary: “We consider for a

*real*number a*the**Kolmogorov**complexities*of its expansions with respect to different bases. ... Philip Ross Watson (Bradford) 2002f:68067 68Q30 Staiger, Ludwig (D-MLU-II; Halle an der Saale)*The**Kolmogorov**complexity*of*real*numbers. ...##
###
Computability, Complexity and Randomness (Dagstuhl Seminar 12021)

2012
*
Dagstuhl Reports
*

Research on

doi:10.4230/dagrep.2.1.19
dblp:journals/dagstuhl-reports/BecherBDM12
fatcat:bkapirz4vfgkzcexiutm33k74q
*the*notions of information and randomness has drawn on methods and ideas from computability*theory*and cumputational*complexity*, as well as core mathematical subjects like measure*theory*and ... information*theory*. ... Understanding*the**complexity*of*the*reachability problem is a central concern of computational*complexity**theory*. ...##
###
Is Randomness "Native" to Computer Science?
[article]

2008
*
arXiv
*
pre-print

We survey

arXiv:0801.0289v1
fatcat:7mwcyjmw75fppmgkjpzshxyapi
*the*Kolmogorov's approach to*the*notion of randomness through*the**Kolmogorov**complexity**theory*. ...*The*original motivation of*Kolmogorov*was to give up a quantitative definition of information. In this*theory*, an object is randomness in*the*sense that it has a large information content. ... Such Borel sets are, in fact, coded in any inner submodel of set*theory*. -Solovay random*reals**over*a submodel of set*theory*are*reals*outside every measure zero Borel set coded in that submodel. ...##
###
Review of Algorithmic Randomness and Complexity by Downey and Hirschfeldt

2013
*
ACM SIGACT News
*

In terms of

doi:10.1145/2447712.2447721
fatcat:ruecbllbf5d6nlwnglakqwb7zy
*Kolmogorov**complexity*and computability*theory*, one might say that highly random sequences contain a lot of information but not much useful information. ... Downey and Hirschfeldt examine other topics as well, such as Kummer's Gap Theorem on*the**Kolmogorov**complexity*of c.e sets, and his theorem that*the*set of*Kolmogorov*random strings are truth-table complete ...##
###
IS RANDOMNESS "NATIVE" TO COMPUTER SCIENCE?
[chapter]

2004
*
Current Trends in Theoretical Computer Science
*

*The*best reference to

*the*subject is Li-Vitanyi's book [29] (caution: they denote C, K what is here -and in most papers -denoted K, H). ... Such Borel sets are, in fact, coded in any inner submodel of set

*theory*. -Solovay random

*reals*

*over*a submodel of set

*theory*are

*reals*outside every measure zero Borel set coded in that submodel. ...

*The*Invariance Theorem A: Now, comes

*the*fundamental result of

*the*

*theory*. We shall state it uniquely for

*Kolmogorov*

*complexity*but it also holds for conditional

*Kolmogorov*

*complexity*. ...

##
###
On hausdorff and topological dimensions of the kolmogorov complexity of the real line

1994
*
Journal of computer and system sciences (Print)
*

We investigate

doi:10.1016/s0022-0000(05)80073-x
fatcat:vgmphpoc6nc4xhwatn3glv2qkq
*the**Kolmogorov**complexity*of*real*numbers. Let K be*the**Kolmogorov**complexity*function; we determine*the*Hausdorff dimension and*the*topological dimension of*the*graph of K. ... Since these dimensions are different,*the*graph of*the**Kolmogorov**complexity*function of*the**real*line forms a fractal in*the*sense of Mandelbrot. ... We thank Professors Allen Back, David Henderson, and Tom Rishel for interesting discussions on dimension*theory*and analysis. ...##
###
Computational power of neural networks: a characterization in terms of Kolmogorov complexity

1997
*
IEEE Transactions on Information Theory
*

*The*computational power of recurrent neural networks is shown to depend ultimately on

*the*

*complexity*of

*the*

*real*constants (weights) of

*the*network. ...

*The*

*complexity*, or information contents, of

*the*weights is measured by a variant of resource-bounded

*Kolmogorov*

*complexity*, taking into account

*the*time required for constructing

*the*numbers. ... ACKNOWLEDGMENT

*The*authors wish to thank E. Sontag for his encouragement and support via many discussions and D. Harel for asking

*the*question that led them to Corollary 5.4. ...

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