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Every Computably Enumerable Random Real Is Provably Computably Enumerable Random
[article]

2009
*
arXiv
*
pre-print

We prove that

arXiv:0808.2220v5
fatcat:scpq4odewbfozlta5d556gaqxq
*every**computably**enumerable*(c.e.)*random**real**is**provable*in Peano Arithmetic (PA) to be c.e.*random*. ... Our positive result can be contrasted with the case of*computable*functions, where not*every**computable*function*is**provably**computable*in PA, or even more interestingly, with the fact that almost all ... In Section 8 we prove our main theorem:*every*c.e.*random**real**is**provably**random*(Theorem 22). ...##
###
Every computably enumerable random real is provably computably enumerable random

2009
*
Logic Journal of the IGPL
*

We prove that

doi:10.1093/jigpal/jzp015
fatcat:m2jjedrarvgkhgm6w42wdasasu
*every**computably**enumerable*(c.e.)*random**real**is**provable*in Peano Arithmetic (PA) to be c.e.*random*. ... Our positive result can be contrasted with the case of*computable*functions, where not*every**computable*function*is**provably**computable*in PA, or even more interestingly, with the fact that almost all ... Corollary 19*Every**provably*c.e. and Chaitin-*random**real**is**provably**random*. Proof. ...##
###
Simplicity via provability for universal prefix-free Turing machines

2011
*
Theoretical Computer Science
*

Universality,

doi:10.1016/j.tcs.2010.08.002
fatcat:pb5ur3vizjerxar5hyxaxzuqza
*provability*and simplicity are key notions in*computability*theory. There are various criteria of simplicity for universal Turing machines. ... Probably the most popular one*is*to count the number of states/symbols. This criterion*is*more complex than it may appear at a first glance. ...*Every*c.e.*random**real**is**provably**random*. We say that a universal machine U*is*PA-simple for*randomness*if PA ⊢ ''Ω U*is**random*''. ...##
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Simplicity via Provability for Universal Prefix-free Turing Machines

2009
*
Electronic Proceedings in Theoretical Computer Science
*

Universality

doi:10.4204/eptcs.1.2
fatcat:6yd53xh2xngidcrfgqelsnub7y
*is*one of the most important ideas in*computability*theory. There are various criteria of simplicity for universal Turing machines. ... Probably the most popular one*is*to count the number of states/symbols. This criterion*is*more complex than it may appear at a first glance. ... Theorem 9 [6] A c.e.*real**is**provably*-*random*iff it*is**provably*Chaitin-*random*. ...##
###
Representation of left-computable ε-random reals

2011
*
Journal of computer and system sciences (Print)
*

The main result

doi:10.1016/j.jcss.2010.08.001
fatcat:275bffzaifgb3lau5dtv7oeuei
*is*the extension of the representability theorem for left-*computable**random**reals*to the case of ε-*random**reals*: a*real**is*left-*computable*ε-*random*iff it*is*the halting probability of ... We also show that left-*computable*ε-*random**reals*are*provable*ε*random*in the Peano Arithmetic. The theory developed here parallels to a large extent the classical theory, but not completely. ... The proof in [2] can be adapted to show that*every*left-*computable*(ε, V )-*random**real**is**provable*(ε, V )-*random*in PA. ...##
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Page 7292 of Mathematical Reviews Vol. , Issue 2003j
[page]

2003
*
Mathematical Reviews
*

*Every*expansion of the

*real*line (R,<), as well as

*every*o- minimal expansion of (R,<), has the intermediate value property. ... In Section 2, it

*is*shown that

*every*countable ordinal fails to be quasiresolvent, and we establish the criterion for a model to be intrinsically

*enumerable*in terms of being quasiresolvent. ...

##
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Computability, Complexity and Randomness (Dagstuhl Seminar 12021)

2012
*
Dagstuhl Reports
*

The Dagstuhl seminar 12021 "

doi:10.4230/dagrep.2.1.19
dblp:journals/dagstuhl-reports/BecherBDM12
fatcat:bkapirz4vfgkzcexiutm33k74q
*Computability*, Complexity and*Randomness*" was aimed to meet people and ideas in these areas to share new results and discuss open problems. ... Research on 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 ... For instance, Brattka, Miller, and Nies translated the theorem "*every*non-decreasing function*is*almost everywhere differentiable" to the*computable*world, by showing that a*real*x*is**computably**random*...##
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Page 438 of Mathematical Reviews Vol. , Issue 91A
[page]

1991
*
Mathematical Reviews
*

An effectively

*enumerable*class U of total recursive functions*is*constructed such that for an arbitrary ¢ > 0 there*is*a 7 -numbering of U such that (1)*every*deterministic IIM needs a linear number of ... Here, ca- pacity*is*defined as the maximum number of*random*binary words that can be stored at*random*addresses so that the probability that a word*is*in error*is*arbitrarily small when it*is*retrieved ...##
###
Chaitin Ω numbers, Solovay machines, and Gödel incompleteness

2002
*
Theoretical Computer Science
*

*Computably*

*enumerable*(c.e.)

*reals*can be coded by Chaitin machines through their halting probabilities. ... c.e.

*random*

*real*

*is*the halting probability of a universal Chaitin machine for which ZFC cannot determine more than its initial block of 1 bits-as soon as you get a 0, it

*is*all over. ... See [13] for a recent study on

*computably*

*enumerable*

*reals*. A

*real*

*is*

*random*if its binary expansion

*is*a

*random*(inÿnite) sequence (cf. [7, 8, 1] ); the choice of base

*is*irrelevant (cf. ...

##
###
Turing machine

2009
*
Scholarpedia
*

*is*true but not

*provable*in . ... A

*computable*

*real*function can be approximated to any degree of precision by a

*computable*function with rational values. A function

*is*lower semicomputable iff the set

*is*

*computably*

*enumerable*. ...

##
###
Measures and their random reals
[article]

2013
*
arXiv
*
pre-print

We show that

arXiv:0802.2705v2
fatcat:pxa56lgcoja5fckkgrd3hjruum
*every*non-*computable**real**is*non-trivially*random*with respect to some measure. The probability measures constructed in the proof may have atoms. ... If one rules out the existence of atoms, i.e. considers only continuous measures, it turns out that*every*non-hyperarithmetical*real**is**random*for a continuous measure. ...*random**real*(so for*every**real*x, x ⊕ ∅ ′*is*Turing equivalent to a Martin-Löf*random**real*). ...##
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A note on Autoreducibility for Infinite Time Register Machines and parameter-free Ordinal Turing Machines
[article]

2014
*
arXiv
*
pre-print

We propose a notion of autoreducibility for infinite time

arXiv:1402.1063v1
fatcat:yjiioq7nfbeptpppqbenklfqvi
*computability*and explore it and its connection with a notion of*randomness*for infinite time machines. ... Consequently, no OT M -autoreducible*real**is*OT M -*random*, and hence,*every*OT M -*random**real**is*OT M -incompressible. ... Hence, at least in L, not*every*(parameter-free) OT M -*random**real**is*totally incompressible. ...##
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The Mathematical Foundations of Randomness
[chapter]

2016
*
The Frontiers Collection
*

We give a nontechnical account of the mathematical theory of

doi:10.1007/978-3-319-26300-7_3
fatcat:fa4s27j6vfa2nfbhhuolzzy35u
*randomness*. The theory of*randomness**is*founded on*computability*theory, and it*is*nowadays often referred to as algorithmic*randomness*. ... Founded on the theory of*computation*, the study of*randomness*has itself profoundly influenced*computability*theory in recent years. ... It can be shown that with this modification*random**reals*exist. 12 Moreover, almost*every**real*in [0, 1]*is**random*, in the sense that the set of nonrandom*reals**is*of effective measure 0. ...##
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Path Sampling: A Fast and Provable Method for Estimating 4-Vertex Subgraph Counts
[article]

2014
*
arXiv
*
pre-print

We provide a sampling algorithm that

arXiv:1411.4942v1
fatcat:n6tztzp46beq7mlaugyxrgw5xu
*provably*and accurately approximates the frequencies of all 4-vertex pattern subgraphs. ... Indeed, even a highly tuned*enumeration*code takes more than a day on a graph with millions of edges. ... While the algorithm*is**provably*correct for any graph, the fact that it gives a significant improvement*is*dependent on the structure of*real*-world graphs. ...##
###
Page 1824 of Mathematical Reviews Vol. 56, Issue 5
[page]

1978
*
Mathematical Reviews
*

“In intuitive terms, our two main results may be stated as follows: (i)

*every**computable*function has very good programs which are (nearly) optimal (among the*provably*equivalent programs); (ii)*every*... Theorem 1: There exists a set of words over the alphabet {0, 1,2} that*is*not*enumerable*by a non-deterministic store- memory machine, but such that there exists a probabilistic store- 68*COMPUTER*SCIENCE ...
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