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The Arithmetical Hierarchy of Real Numbers
[chapter]

1999
*
Lecture Notes in Computer Science
*

A

doi:10.1007/3-540-48340-3_3
fatcat:5cymfubplnautgoe7dhtd5hj4e
*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. ...##
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Weakly Computable Real Numbers

2000
*
Journal of Complexity
*

class

doi:10.1006/jcom.2000.0561
fatcat:ezmfmipc4rdzzj76mcuv2rqd3e
*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*. ...##
###
Hierarchies of function classes defined by the first-value operator

2007
*
RAIRO - Theoretical Informatics and Applications
*

For effective sequences

doi:10.1051/ita:2007031
fatcat:vc5lfnzgnvf6fft7eatxxo56e4
*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 ...##
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Hierarchies of Function Classes Defined by the First-Value Operator

2005
*
Electronical Notes in Theoretical Computer Science
*

For effective sequences

doi:10.1016/j.entcs.2004.06.034
fatcat:4yjg4fcxabfk7dp5y67p4k4djm
*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 ...##
###
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. ...##
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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.” ...##
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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. ...##
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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 ...##
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Andrei Ershov and the Soviet Information Age

2013
*
Kritika: Explorations in Russian and Eurasian History
*

Sciences

doi:10.1353/kri.2013.0046
fatcat:56bhj64gfjam5mpkiuhjpqbw3y
*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 ...##
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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*of*“*reals*” (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 ...##
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On the Information Carried by Programs about the Objects They Compute

2015
*
Symposium on Theoretical Aspects of Computer Science
*

We characterize this additional information to be any upper bound on the Kolmogorov complexity

doi:10.4230/lipics.stacs.2015.447
dblp:conf/stacs/HoyrupR15
fatcat:if7avyxdifeofgsoixcajy3xaa
*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. ...##
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On the information carried by programs about the objects they compute
[article]

2014
*
arXiv
*
pre-print

We characterize this additional information to be any upper bound on the Kolmogorov complexity

arXiv:1409.6977v1
fatcat:vbgwi7djondi3dxhovxzwjhqg4
*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. ...##
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Mathematics based on incremental learning—Excluded middle and inductive inference

2006
*
Theoretical Computer Science
*

Mathematical or logical concepts seem to be one

doi:10.1016/j.tcs.2005.10.019
fatcat:bjcpur2pxzeihdscoer2fz6ty4
*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. ...##
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A hierarchy of Turing degrees of divergence bounded computable real numbers

2006
*
Journal of Complexity
*

In this paper we give a

doi:10.1016/j.jco.2006.04.009
fatcat:tdxqjubzlvdylbgzbxvyrzmpyu
*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*. ...##
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On the Information Carried by Programs About the Objects they Compute

2016
*
Theory of Computing Systems
*

We note that Čeitin's version

doi:10.1007/s00224-016-9726-9
fatcat:nw4jw2qgv5dgpjopsig4dnwlvi
*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 ...
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