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

Cristian S. Calude, Nicholas J. Hay
2009 arXiv   pre-print
We prove that 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).  ... 
arXiv:0808.2220v5 fatcat:scpq4odewbfozlta5d556gaqxq

Every computably enumerable random real is provably computably enumerable random

C. S. Calude, N. J. Hay
2009 Logic Journal of the IGPL  
We prove that 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.  ... 
doi:10.1093/jigpal/jzp015 fatcat:m2jjedrarvgkhgm6w42wdasasu

Simplicity via provability for universal prefix-free Turing machines

Cristian S. Calude
2011 Theoretical Computer Science  
Universality, 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''.  ... 
doi:10.1016/j.tcs.2010.08.002 fatcat:pb5ur3vizjerxar5hyxaxzuqza

Simplicity via Provability for Universal Prefix-free Turing Machines

Cristian S. Calude
2009 Electronic Proceedings in Theoretical Computer Science  
Universality 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.  ... 
doi:10.4204/eptcs.1.2 fatcat:6yd53xh2xngidcrfgqelsnub7y

Representation of left-computable ε-random reals

Cristian S. Calude, Nicholas J. Hay, Frank Stephan
2011 Journal of computer and system sciences (Print)  
The main result 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 εrandom 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.  ... 
doi:10.1016/j.jcss.2010.08.001 fatcat:275bffzaifgb3lau5dtv7oeuei

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.  ... 

Computability, Complexity and Randomness (Dagstuhl Seminar 12021)

Veronica Becher, Laurent Bienvenu, Rodney Downey, Elvira Mayordomo, Marc Herbstritt
2012 Dagstuhl Reports  
The Dagstuhl seminar 12021 "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  ... 
doi:10.4230/dagrep.2.1.19 dblp:journals/dagstuhl-reports/BecherBDM12 fatcat:bkapirz4vfgkzcexiutm33k74q

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

Cristian S. Calude
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.  ... 
doi:10.1016/s0304-3975(01)00068-8 fatcat:cq2tbgwpxbdc7h3ktwdlpla5ru

Turing machine

Paul Vitanyi
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.  ... 
doi:10.4249/scholarpedia.6240 fatcat:knxz2b77gbdjbix5ckmzhnfloi

Measures and their random reals [article]

Jan Reimann, Theodore A. Slaman
2013 arXiv   pre-print
We show that 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).  ... 
arXiv:0802.2705v2 fatcat:pxa56lgcoja5fckkgrd3hjruum

A note on Autoreducibility for Infinite Time Register Machines and parameter-free Ordinal Turing Machines [article]

Merlin Carl
2014 arXiv   pre-print
We propose a notion of autoreducibility for infinite time 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.  ... 
arXiv:1402.1063v1 fatcat:yjiioq7nfbeptpppqbenklfqvi

The Mathematical Foundations of Randomness [chapter]

Sebastiaan A. Terwijn
2016 The Frontiers Collection  
We give a nontechnical account of the mathematical theory of 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.  ... 
doi:10.1007/978-3-319-26300-7_3 fatcat:fa4s27j6vfa2nfbhhuolzzy35u

Path Sampling: A Fast and Provable Method for Estimating 4-Vertex Subgraph Counts [article]

Madhav Jha, C. Seshadhri, Ali Pinar
2014 arXiv   pre-print
We provide a sampling algorithm that 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.  ... 
arXiv:1411.4942v1 fatcat:n6tztzp46beq7mlaugyxrgw5xu

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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