## Some Computability-Theoretical Aspects of Reals and Randomness (2001)

Venue: | the Lect. Notes Log. 18, Assoc. for Symbol. Logic |

Citations: | 24 - 7 self |

### BibTeX

@INPROCEEDINGS{Downey01somecomputability-theoretical,

author = {Rodney G. Downey},

title = {Some Computability-Theoretical Aspects of Reals and Randomness},

booktitle = {the Lect. Notes Log. 18, Assoc. for Symbol. Logic},

year = {2001},

pages = {97--148}

}

### OpenURL

### Abstract

We study computably enumerable reals (i.e. their left cut is computably enumerable) in terms of their spectra of representations and presentations.

### Citations

1682 | An Introduction to Kolmogorov Complexity and its Applications
- Li, Vitányi
- 1997
(Show Context)
Citation Context ...erest to computable analysis (e.g. Weihrauch [59], PourEl [45], Pour-El and Richards [46], Ko [32]), and to algorithmic information theory (e.g. Chaitin [11], Calude [6], Martin-Lof [44], Li-Vitanyi [=-=-=-41]), is the collection of computably enumerable reals. These are the reals such the lower cut L() consisting of rationals less than forms a computably enumerable set. Thesrst part of these notes co... |

1157 |
M.: On computable numbers, with an application to the Entscheidungsproblem
- Turing
- 1936
(Show Context)
Citation Context ...ave a unique dyadic expansion. Much of modern computability theory is concerned with understanding the computational complexity of sets of positive integers, yet, even in the original paper of Turing =-=[57-=-], a central topic is interest in eectiveness considerations for reals. Of particular interest to computable analysis (e.g. Weihrauch [59], PourEl [45], Pour-El and Richards [46], Ko [32]), and to alg... |

473 | Recursively Enumerable Sets and Degrees - Soare - 1987 |

332 |
The definition of random sequences
- Martin-Löf
- 1966
(Show Context)
Citation Context ...Of particular interest to computable analysis (e.g. Weihrauch [59], PourEl [45], Pour-El and Richards [46], Ko [32]), and to algorithmic information theory (e.g. Chaitin [11], Calude [6], Martin-Lof [=-=-=-44], Li-Vitanyi [41]), is the collection of computably enumerable reals. These are the reals such the lower cut L() consisting of rationals less than forms a computably enumerable set. Thesrst part ... |

330 | A theory of program size formally identical to information theory
- Chaitin
(Show Context)
Citation Context ...ess considerations for reals. Of particular interest to computable analysis (e.g. Weihrauch [59], PourEl [45], Pour-El and Richards [46], Ko [32]), and to algorithmic information theory (e.g. Chaitin =-=-=-[11], Calude [6], Martin-Lof [44], Li-Vitanyi [41]), is the collection of computably enumerable reals. These are the reals such the lower cut L() consisting of rationals less than forms a computably... |

66 | Randomness and recursive enumerability
- Kučera, Slaman
(Show Context)
Citation Context ...s follows by the Chaitin denition of randomness and by Theorem 25. Finally Kucera and Slaman showed that domination provides a precise characterization of randomness. Theorem 26 (Kucera and Slaman [37]). Suppose that is random and c.e.. Then for all c.e. realss,sS . Proof. Suppose that is random andsis a c.e. real. We need to show thatsS . We enumerate a Martin-Lof test Fn : n 2 ! in stage... |

59 | Information-theoretical characterizations of recursive infinite strings, Theor - Chaitin - 1976 |

59 | Calibrating randomness
- Downey, Hirschfeldt, et al.
(Show Context)
Citation Context ...ial Schnorr test is not really one until he enumerates to within of the claimed measure of V 1 ; V 2 :::V n for somesxed n.) Actually this theorem follows from some recent work of Downey and Griths [=-=14]-=-. Downey and Griths have shown that every Schnorr random c.e. real is of high c.e. degree. However, they also use a relatively dicult 0 00 agrument to prove that there exist incomplete Schnorr random ... |

53 |
of a paper (or series of papers) on Chaitin’s work ... done for the most part during the period of Sept
- Solovay, Draft
- 1974
(Show Context)
Citation Context ...material of the calibration of relative randomness using notions such as Solovay reducibility and some new reducibilities rH and sw reducibilities. Notation is more or less standard and follows Soare =-=[54]-=-. As these notes are aimed at graduate students, and one learns from actively engaging in the material, we will not always provide complete proofs, but will always provide sketches, referring the read... |

52 |
Computational Complexity of Real Functions
- Ko
- 1991
(Show Context)
Citation Context ...er of Turing [57], a central topic is interest in eectiveness considerations for reals. Of particular interest to computable analysis (e.g. Weihrauch [59], PourEl [45], Pour-El and Richards [46], Ko [=-=3-=-2]), and to algorithmic information theory (e.g. Chaitin [11], Calude [6], Martin-Lof [44], Li-Vitanyi [41]), is the collection of computably enumerable reals. These are the reals such the lower cut ... |

51 | Counting complexity
- Fortnow
- 1997
(Show Context)
Citation Context ... of randomness used if we demanded prex free complexity.) There is one good trade-o, namely now H(xy) H(x) +H(y) +O(1): For these and other facts we refer the reader to Li-Vitanyi [41] or Fortnow [25=-=]-=-.) The actual relationships between K and H are H(x) = K(x) +K(K(x)) +O(K(K(K(x))): K(x) = H(x) H(H(x)) +OH 3 (x): These are due to Solovay in the 1975 manuscript, and are nontrivial. 8 Complexity of ... |

46 | Degrees of Random Sets
- Kautz
- 1991
(Show Context)
Citation Context ... natural direction is to examine not necessarily c.e. reals, and perhaps the collection of all random reals. A counting argument shows that the set of random reals has measure 1. Kurtz [39] and Kautz =-=-=-[30] have a lot of material here. A nice observation more or less due to Martin-Lof [44], is that the random reals form a 0 2 class: an eective union of 0 1 classes. To see this let C k = fx 2 [0; 1... |

44 | A variant of the kolmogorov concept of complexity - Loveland - 1969 |

38 | Recursive real numbers - Rice - 1954 |

30 |
Array nonrecursive sets and multiple permitting arguments
- Downey, Jockusch, et al.
- 1990
(Show Context)
Citation Context ...f(k) many strings of K complexity at most g(k) and this contradict the fact that f(k) > 2 g(k)+1 1: Actually Kummer classied the degrees containing complex c.e. sets. From Downey, Jockusch, and Stob [=-=-=-22], a degree a is called array noncomputable i for all g tt ; 0 there is a function h T a not dominated by g. The anc degrees form a upwards closed class of the c.e. degrees including some low degr... |

30 | Lowness for the class of random sets
- Kučera, Terwijn
- 1999
(Show Context)
Citation Context ...acle, and hence get the class RAND X . Van Lambalgen asked if there is a Martin-Lof low set X : a set X such that RAND X = RAND: This question has an armative answer. Theorem 52 (Kucera and Terwijn [3=-=6-=-]). There is a c.e. set A that is Martin-Lof low. Proof. We give an alternative proof to that in [36]. It is clear that there is a primitive recursive function f , so that U A f(n is the universal Mar... |

30 |
Recursion theory and Dedekind cuts
- Soare
- 1969
(Show Context)
Citation Context ... 1 degree classes. In particular we will look at the recent work of Calude, Coles, Hertling, Khoussainov, Downey [13], Downey and Laforte [17], Ho [27], and Wu [59], as well as older of work of Soare =-=[50]-=- and others. Our main goal is to look at algorithmic randomness, especially with respect to computably enumerable reals. To this end we will next introduce the basic approaches to the study of algorit... |

26 | Randomness and reducibility
- Downey, Hirschfeldt, et al.
(Show Context)
Citation Context ...c and exist. Notice that c(s n i=1 q i ) > n i=1 (i)q i : Hence Ss: For the other direction, we need the following Lemmas. Thesrst is inplicit in Solovay's manuscript, but issrst proven in [20]. 20 Lemma 22. Let andsbe c.e. reals, and let 0 ; 1 ; : : : ands0 ;s1 ; : : : be computable increasing sequences of rationals converging to ands, respectively. ThensS if and only if there a... |

26 | Computational randomness and lowness
- Terwijn, Zambella
(Show Context)
Citation Context .... One really nice aspect of Schnorr randomness is that there is a complete characterization of Schnorr low sets. As usual, let D x denote the x-th canonicalsnite set. Theorem 54 (Terwijn and Zambella =-=[5-=-6]). A set X is Schnorr low i there is a computable function p, such that, for all functions g X , there is a function h where, for all n, (i) jH h(n) jsp(n); (ii) g(n) 2 D h(n) : 42 The proof is nont... |

23 | Kolmogorov complexity and instance complexity of recursively enumerable sets
- Kummer
- 1996
(Show Context)
Citation Context ...1). Solovay asked if this bound was attainable. Certainly for K-complexity no c.e. set has initial segment complexity on length n always greater than 2 log n O(1): In a very interesting paper, Kummer =-=[-=-38] proved that there are c.e. complex sets. Theorem 55 (Kummer [38]). There is a set A such that there is a constant c, with K(A n) 2 log n c for innitely many n. Proof. Kummer's proof runs as foll... |

19 | Weihrauch: “The Arithmetical Hierarchy of Real Numbers - Zheng, K - 2001 |

17 |
Complexity of programs to determine whether natural numbers not greater than n belong to a recursively enumerable set
- Barzdin
- 1968
(Show Context)
Citation Context ... 1 an is the minimum length of a string such that the universal machine U with oracle and argument m computes a 1 am for all m n.) We refer the reader to Li-Vitanyi [41] and Barzdin [4] for more details. A prex-free machine is one whose domain is prex-free. It is usual to take the machine as \self delimiting" which means that it has a one way read head which halts when the machine... |

17 |
Splitting theorems in recursion theory
- Downey, Stob
- 1993
(Show Context)
Citation Context ...e following properties. 8 (i) R(a 1 ) consists of every c.e. (m-) degree (ii) R(a 2 ) forms an atomless boolean algebra, which is nowhere dense in the c.e. degrees. For the proofs see Downey and Stob =-=[24]-=-. We also remark that the above has a number of other consequences regarding known limits to splittings. For instance; Corollary 12. If a c.e. real a has representations in each T -degree below that o... |

16 |
On the use of diagonally nonrecursive functions
- Kučera
- 1987
(Show Context)
Citation Context ...random i there is a k with x 2 C k . RAND = [ k C k : s a consequence, all the apparatus of 0 1 classes apply. There are, for instance, random reals of low degree by the low basis theorem. Kucera [34=-=, 35-=-] has a lot of material here, additionally relating these notions to genericity and other notions such as DNR functions. 13.2 Martin-Lof Lowness One very interesting area comes from relativizing the n... |

13 |
Randomness in computability theory. In Computability theory and its applications
- Ambos-Spies, Kučera
- 1999
(Show Context)
Citation Context ...ization of S is the following: Theorem 21 (Downey, Hirschfeldt, Nies [15]). For c.e. reals, Ssi for all c.e. sequences fq i : i 2 !g such thats= i q i ; there is a computable function : ! 7! [0; 1] and a constant c, such that, = c( i (i)q i ): Hence Ss, i there exists a c and a c.e. realssuch that cs= +s: Proof. (if) One direction is easy. Suppose that c and exist. Notice that c(s ... |

13 | Degree-theoretic aspects of computably enumerable reals
- Calude, Coles, et al.
- 1999
(Show Context)
Citation Context ...ons. Denition 2. We say that a c.e. sequence of rationals fq i : i 2 !g with monotonic limit represents . 6 Representations weresrst eectively analyzed by Calude, Coles, Hertling and Khoussainov [7]. We have seen that if a real is c.e. then it has a computable representation. If a real is computable then every representation must be computable (exercise). Suppose that a c.e. real is noncomputabl... |

12 |
Relatively recursive reals and real functions
- Ho
- 1999
(Show Context)
Citation Context ...lso thanks Andrew Arana for his skillful note-taking. 1 degree classes. In particular we will look at the recent work of Calude, Coles, Hertling, Khoussainov, Downey [13], Downey and Laforte [17], Ho =-=[27]-=-, and Wu [59], as well as older of work of Soare [50] and others. Our main goal is to look at algorithmic randomness, especially with respect to computably enumerable reals. To this end we will next i... |

12 |
A uni approach to the de of random sequences
- Schnorr
- 1971
(Show Context)
Citation Context ... such that, for all n H(A n) n O(1): It can be shown that if K(A n) n O(1) for innitely many n; (this being called Kolmogorov random) then the real is Chaitin random. Unfortunately Schnorr [49] proved that the converse does not hold. (It can, however, be shown that the set of languages which are Chaitin random but not Kolmogorov random has measure zero.) There are reals that are K-random bu... |

11 |
Weak recursive degrees and a problem of Spector, in Recursion Theory and Complexity
- Ishmukhametov
- 1999
(Show Context)
Citation Context ...e a notion of \multiple permitting" common to a 43 number of degree constructions. For example, the are the degrees A B such that A and B are c.e. and have no complete separating set. Ishmukhame=-=tov [2-=-8] has the following characterization of the (c.e.) array computable degrees. a is array computable i there is a computable function p such that, for all g T a, there is a computable function h such ... |

10 |
Information Theory and Randomness, an Algorithmic Perspective
- Calude
- 1994
(Show Context)
Citation Context ...tions for reals. Of particular interest to computable analysis (e.g. Weihrauch [59], PourEl [45], Pour-El and Richards [46], Ko [32]), and to algorithmic information theory (e.g. Chaitin [11], Calude [6], Martin-Lof [44], Li-Vitanyi [41]), is the collection of computably enumerable reals. These are the reals such the lower cut L() consisting of rationals less than forms a computably enumerable ... |

10 |
A.: Chaitin Numbers and Strong Reducibilities
- Calude, Nies
- 1997
(Show Context)
Citation Context ...eory. The proof of theorem 32 uses Nies's method of interpreting eectively dense boolean algebras, together with a technical construction of a certain class of (strongly) c.e. reals. Calude and Nies [=-=9-=-] have proven that the random reals are all wtt-complete. Very little else is known about the Solovay degrees of c.e. reals. 28 11 Other measures of relative randomness A reducibility on reals is a m... |

10 | Presentations of computably enumerable reals
- Downey, LaForte
(Show Context)
Citation Context ...ort. He also thanks Andrew Arana for his skillful note-taking. 1 degree classes. In particular we will look at the recent work of Calude, Coles, Hertling, Khoussainov, Downey [13], Downey and Laforte =-=[17]-=-, Ho [27], and Wu [59], as well as older of work of Soare [50] and others. Our main goal is to look at algorithmic randomness, especially with respect to computably enumerable reals. To this end we wi... |

10 |
Classification of degree classes associated with r.e
- Downey, Remmel
- 1989
(Show Context)
Citation Context ...em tt-reduces to the conjugacy problem ([29]), algebraic closure is related to Q-reducibility ([5, 18, 43, 61]) and wtt-degrees characterize the degrees of bases of a c.e. vector space (Downey-Remmel =-=[-=-23]).) We would like to prove that if A is half of a splitting of L() then A represents . But it is not dicult to prove that this is not true. We know that if A represents then there needs to be a co... |

10 | Cohesive sets and recursively enumerable dedekind cuts - Soare - 1969 |

8 | Recursively enumerable reals and Chaitin ! numbers - Calude, Hertling, et al. - 1998 |

7 |
On relative randomness. Ann
- Kučera
- 1993
(Show Context)
Citation Context ...random i there is a k with x 2 C k . RAND = [ k C k : s a consequence, all the apparatus of 0 1 classes apply. There are, for instance, random reals of low degree by the low basis theorem. Kucera [34=-=, 35-=-] has a lot of material here, additionally relating these notions to genericity and other notions such as DNR functions. 13.2 Martin-Lof Lowness One very interesting area comes from relativizing the n... |

7 |
Algebraisch abgeschlossen gruppen in Word Problems
- Ziegler
- 1980
(Show Context)
Citation Context ...t occur naturally tend to be stronger than T . For instance in asnitely presented group, the word problem tt-reduces to the conjugacy problem ([29]), algebraic closure is related to Q-reducibility ([=-=5, 18, 43, 6-=-1]) and wtt-degrees characterize the degrees of bases of a c.e. vector space (Downey-Remmel [23]).) We would like to prove that if A is half of a splitting of L() then A represents . But it is not dic... |

5 | personal communication - Stephan - 1994 |

4 |
Resource bounded measure and randomness
- Ambos-Spies, Mayordomo
- 1997
(Show Context)
Citation Context ...irect the reader towards the literature. I certainly do not claim completeness here. In particular, I will not even discuss the rich area of resource bounded randomness. (See e.g. Ambos-Spies et. al. =-=[-=-2]) Otherwise the reader has Ambos-Spies and Kucera [1], van Lambalgen [58], and Li-Vitanyi [41] as general references. 13.1 0 1 classes One natural direction is to examine not necessarily c.e. reals... |

4 | On the continued fraction representation of computable real numbers, Theor - Ko - 1986 |

4 |
A formal theory of inductive inference, part 1 and part 2
- Solomono
- 1964
(Show Context)
Citation Context ...ance, 0 10000000 can be described by saying that we should repeat 0 10000000 times. This can easily be described by an algorithm shorter than 10000000.) We formalize this notion,srst due to Solomono [=-=5-=-3], but independently to Kolmogorov and Chaitin, as follows. Let f : 2 7! 2 be a partial computable function. Then we can denote the Kolmogorov complexity of a string with respect to f via K f () = m... |

3 |
On algebraically closed groups, Algebra i
- Belegradek
- 1974
(Show Context)
Citation Context ...t occur naturally tend to be stronger than T . For instance in asnitely presented group, the word problem tt-reduces to the conjugacy problem ([29]), algebraic closure is related to Q-reducibility ([=-=5, 18, 43, 6-=-1]) and wtt-degrees characterize the degrees of bases of a c.e. vector space (Downey-Remmel [23]).) We would like to prove that if A is half of a splitting of L() then A represents . But it is not dic... |

3 | On the universal splitting property - Downey - 1997 |

3 |
On presentations of algebraic structures, to appear
- Downey
(Show Context)
Citation Context ...heir hospitality and support. He also thanks Andrew Arana for his skillful note-taking. 1 degree classes. In particular we will look at the recent work of Calude, Coles, Hertling, Khoussainov, Downey =-=[13]-=-, Downey and Laforte [17], Ho [27], and Wu [59], as well as older of work of Soare [50] and others. Our main goal is to look at algorithmic randomness, especially with respect to computably enumerable... |

3 | Enumerable sets and quasireducibility
- Downey, Laforte, et al.
- 1998
(Show Context)
Citation Context ...t occur naturally tend to be stronger than T . For instance in asnitely presented group, the word problem tt-reduces to the conjugacy problem ([29]), algebraic closure is related to Q-reducibility ([=-=5, 18, 43, 6-=-1]) and wtt-degrees characterize the degrees of bases of a c.e. vector space (Downey-Remmel [23]).) We would like to prove that if A is half of a splitting of L() then A represents . But it is not dic... |

3 | Aspects of Complexity (Short courses in complexity from the New - Downey, Hirschfeldt - 2001 |

3 |
From axiomatics to intrinsic characterization, some open problems in computable analysis
- Pour-El
(Show Context)
Citation Context ...egers, yet, even in the original paper of Turing [57], a central topic is interest in eectiveness considerations for reals. Of particular interest to computable analysis (e.g. Weihrauch [59], PourEl [=-=45-=-], Pour-El and Richards [46], Ko [32]), and to algorithmic information theory (e.g. Chaitin [11], Calude [6], Martin-Lof [44], Li-Vitanyi [41]), is the collection of computably enumerable reals. These... |

2 | Undecidability of Solovay and other degree structures for c.e. reals, in preparation - Downey, Hirschfeldt, et al. |

2 |
Jockusch Jr., Fine degrees of word problems in cancellation semigroups
- G
- 1980
(Show Context)
Citation Context ...e role in eective mathematics since reducibilities that occur naturally tend to be stronger than T . For instance in asnitely presented group, the word problem tt-reduces to the conjugacy problem ([2=-=9]-=-), algebraic closure is related to Q-reducibility ([5, 18, 43, 61]) and wtt-degrees characterize the degrees of bases of a c.e. vector space (Downey-Remmel [23]).) We would like to prove that if A is ... |

2 |
Measures of complexity of objects (axiomatic description
- Levin
- 1976
(Show Context)
Citation Context ... its length, and only then read the bits of . In this way the actually represents jj + log jj many bits of information. If one accepts this argument then one ought to circumvent this. Both Levin [40] and Chaitin [11] suggested prex free machines to circumvent this. It has also been pointed out that one can circumvent this by asking that the complexity measure be continuous. This gives rise to th... |