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The Arithmetical Hierarchy of Real Numbers [chapter]

Xizhong Zheng, Klaus Weihrauch
1999 Lecture Notes in Computer Science  
A real number x is called ∆ 0 2 if its binary expansion corresponds to a ∆ 0 2 -set of natural numbers.  ...  Depending on how fast the sequences converge, ∆ 0 2 -reals have different levels of effectiveness. This leads to various hierarchies of ∆ 0 2 reals.  ...  Ershov's Hierarchy Ershov's hierarchy [6] of ∆ 0 2 -subsets of natural numbers can be transferred to reals straightforwardly if any real number is appointed to a set.  ... 
doi:10.1007/3-540-48340-3_3 fatcat:5cymfubplnautgoe7dhtd5hj4e

Weakly Computable Real Numbers

Klaus Ambos-Spies, Klaus Weihrauch, Xizhong Zheng
2000 Journal of Complexity  
class of weakly computable real numbers.  ...  It is also shown that the class of weakly computable real numbers extends properly the class of semi-computable real numbers and the class of recursively approximable real numbers extends properly the  ...  Then we show that the semi-computable real numbers can have binary sets up to |-r.e. in the Ershov's hierarchy.  ... 
doi:10.1006/jcom.2000.0561 fatcat:ezmfmipc4rdzzj76mcuv2rqd3e

Hierarchies of function classes defined by the first-value operator

Armin Hemmerling
2007 RAIRO - Theoretical Informatics and Applications  
For effective sequences of computable discrete functions, we obtain a hierarchy connected with Ershov's one within ∆ 0 2 .  ...  The non-effective version over real functions is connected with the degrees of discontinuity and yields a hierarchy related to Hausdorff's difference hierarchy in the Borel class ∆ B 2 .  ...  The Hausdorff-Ershov hierarchy for real functions The Hausdorff-Ershov hierarchy for functions is obtained by a suitable combination of ingredients both of Ershov's hierarchy for discrete functions and  ... 
doi:10.1051/ita:2007031 fatcat:vc5lfnzgnvf6fft7eatxxo56e4

Hierarchies of Function Classes Defined by the First-Value Operator

Armin Hemmerling
2005 Electronical Notes in Theoretical Computer Science  
For effective sequences of computable discrete functions, we obtain a hierarchy connected with Ershov's one within ∆ 0 2 .  ...  The non-effective version over real functions is connected with the degrees of discontinuity and yields a hierarchy related to Hausdorff's difference hierarchy in the Borel class ∆ B 2 .  ...  The Hausdorff-Ershov hierarchy for real functions The Hausdorff-Ershov hierarchy for functions is obtained by a suitable combination of ingredients both of Ershov's hierarchy for discrete functions and  ... 
doi:10.1016/j.entcs.2004.06.034 fatcat:4yjg4fcxabfk7dp5y67p4k4djm

Page 1712 of Mathematical Reviews Vol. , Issue 84e [page]

1984 Mathematical Reviews  
Finally, he considers the object of (Dedekind) real numbers in & and a subobject whose global elements correspond to Banach- Mazur computable real numbers.  ...  all finite limits and colimits as well as those higher type objects that exist in Ershov’s category; the embedding sends the set of natural numbers (enumerated by the identity function) to N.  ... 

Page 34 of Mathematical Reviews Vol. , Issue 87a [page]

1987 Mathematical Reviews  
The constructed class of hierarchies contains the hyperarithmetical hierarchy, various relativized versions of Ershov’s hierarchy, and hierarchies of limit enumerable functions.  ...  The density and dis- creteness properties proved for this class of hierarchies offer hope that a maximal class of hierarchies of hyperarithmetical sets and functions has been selected.”  ... 

Page 3012 of Mathematical Reviews Vol. , Issue 2002E [page]

2002 Mathematical Reviews  
The existence of a Q-sequence is equivalent to the existence of a Q-set of reals. Martin’s axiom implies the existence of such sets.  ...  Prather (San Antonio, TX) 2002¢:03063 03D25 03D15 Gale, Amy (NZ-VCTR-SMC; Wellington); Downey, Rod (NZ-VCTR-SMC; Wellington) On genericity and Ershov’s hierarchy. (English summary) ML@Q Math. Log.  ... 

Page 1204 of Mathematical Reviews Vol. , Issue 94c [page]

1994 Mathematical Reviews  
function that maps every state (natural number) to its “time”.  ...  Ershov’s language of £-expressions serves as the sublanguage of terms, and we construct a denotational semantics for it.” 94c:03036 03B70 03B25 03B45 03D15 68Q60 Alur, Rajeev (1-STF-C; Stanford, CA); Henzinger  ... 

Andrei Ershov and the Soviet Information Age

Gregory Afinogenov
2013 Kritika: Explorations in Russian and Eurasian History  
Sciences hierarchy; Akademgorodok itself was a massive monument to the state's commitment to supporting both basic and applied research.  ...  A considerable number of Yamaha machines were imported to compensate, but they were too expensive and their numbers were insufficient. 63 Since even Moscow schools were undersupplied, and the situation  ... 
doi:10.1353/kri.2013.0046 fatcat:56bhj64gfjam5mpkiuhjpqbw3y

Page 1826 of Mathematical Reviews Vol. , Issue 91D [page]

1991 Mathematical Reviews  
Such reductions are, of course, not transitive but nevertheless the authors show a number of interesting connections with (for example) Ershov’s difference hierarchy.  ...  The first of the examples, called B, has the set “w ofreals” (in the sense of descriptive set theory) as its only object and has as its mor- phisms the restrictions (in domain and range) to “w of recursive  ... 

On the Information Carried by Programs about the Objects They Compute

Mathieu Hoyrup, Cristóbal Rojas, Marc Herbstritt
2015 Symposium on Theoretical Aspects of Computer Science  
We characterize this additional information to be any upper bound on the Kolmogorov complexity of the object.  ...  We then use this relationship to obtain several results characterizing the computational and topological structure of Markov-semidecidable sets.  ...  In the case X = N with the discrete topology, the effective Borel hierarchy is exactly the arithmetical hierarchy, the class D n of effective difference hierarchy is exactly the class of n-c.e. sets.  ... 
doi:10.4230/lipics.stacs.2015.447 dblp:conf/stacs/HoyrupR15 fatcat:if7avyxdifeofgsoixcajy3xaa

On the information carried by programs about the objects they compute [article]

Mathieu Hoyrup, Cristobal Rojas
2014 arXiv   pre-print
We characterize this additional information to be any upper bound on the Kolmogorov complexity of the object.  ...  We then use this relationship to obtain several results characterizing the computational and topological structure of Markov-semidecidable sets.  ...  Given an index e of s ∈ S, enumerate U ⊤ and enumerate the set of functions f such that ϕ e (e) does not halt in exactly f (e) steps. The latter set of functions is effectively open, uniformly in e.  ... 
arXiv:1409.6977v1 fatcat:vbgwi7djondi3dxhovxzwjhqg4

Mathematics based on incremental learning—Excluded middle and inductive inference

Susumu Hayashi
2006 Theoretical Computer Science  
Mathematical or logical concepts seem to be one of the main research targets of learning theory and its applications.  ...  Although they study mathematical objects, the objective of the studies is learning. In this paper, a mathematics of which foundation itself is learning theoretic will be introduced.  ...  A formulation of existence of step functions such as y = x] over real numbers is equivalent to 0 2 -LEM.  ... 
doi:10.1016/j.tcs.2005.10.019 fatcat:bjcpur2pxzeihdscoer2fz6ty4

A hierarchy of Turing degrees of divergence bounded computable real numbers

Robert Rettinger, Xizhong Zheng
2006 Journal of Complexity  
In this paper we give a hierarchy theorem for Turing degrees of different classes of f-bc reals.  ...  A real number x is f-bounded computable (f-bc, for short) for a function f if there is a computable sequence (x s ) of rational numbers which converges to x f-bounded effectively in the sense that, for  ...  Concerning the f-Cauchy computability there is a hierarchy theorem similar to Ershov's Hierarchy theorem [4] of subsets of natural numbers.  ... 
doi:10.1016/j.jco.2006.04.009 fatcat:tdxqjubzlvdylbgzbxvyrzmpyu

On the Information Carried by Programs About the Objects they Compute

Mathieu Hoyrup, Cristóbal Rojas
2016 Theory of Computing Systems  
We note that Čeitin's version of this result shows that over the real line, Markov-computable functions and Type-2-computable functions coincide.  ...  can we characterize Ershov's open sets ?  ...  In the case X = N with the discrete topology, the effective Borel hierarchy is exactly the arithmetical hierarchy, the class D n of the effective difference hierarchy is exactly the class of n-c.e. sets  ... 
doi:10.1007/s00224-016-9726-9 fatcat:nw4jw2qgv5dgpjopsig4dnwlvi
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