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Page 1185 of Mathematical Reviews Vol. 44, Issue 6 [page]

1972 Mathematical Reviews  
He does this by constructing a minimal degree d, for each ae 2" in a way that makes these minimal degrees very independent of one another. J. R. Shoenfield (Durham, N.C.) 6493 Herman, G.  ...  A countable set of notations O* and hierarchy of level 2 jumps J,5(a € O%) is defined and the usual properties, such as uniqueness, are established.  ... 

Page 944 of Mathematical Reviews Vol. 37, Issue 5 [page]

1969 Mathematical Reviews  
Hensel, Gustav; Putnam, Hilary 5096 On the notational independence of various hierarchies of degrees of unsolvability. J. Symbolic Logic 30 (1965), 69-86.  ...  Marek (Warsaw) Hirose, Ken 5095 An investigation on degrees of unsolvability. J. Math. Soc.  ... 

Book Review: Computability and unsolvability

J. C. E. Dekker
1959 Bulletin of the American Mathematical Society  
Part III continues the development started in Part I. It is concerned with the Kleene hierarchy, computable functional, the recursion theorems, and degrees of unsolvability.  ...  The unsolvability of the word problem for groups (shown by Novikoff and, independently, Boone) is stated, but not proved.  ... 
doi:10.1090/s0002-9904-1959-10329-3 fatcat:ycmrl3br5rgqli3j5lnvoj6cie

Quantitative and qualitative information in computations

Robert P. Daley
1980 Information and Control  
One is the minimal-program complexity of the sequences and represents a quantitative information content, and the other is the degree of unsolvability of the underlying set and represents a qualitative  ...  The major conclusion from this work is that with few exceptions these measures of information content are unrelated. Various tradeotis between these measures are also demonstrated.  ...  define the complexity classes, By deg A we will denote the degree of unsolvability of the set A, and A' will denote the jump of the set A.  ... 
doi:10.1016/s0019-9958(80)90615-4 fatcat:f62popptmjfcpprgkjlh6njg4u

JSL volume 30 issue 4 Cover and Back matter

1965 Journal of Symbolic Logic (JSL)  
CRESSWELL 65 On the notational independence of various hierarchies of degrees of unsolvability.  ...  Trial and error predicates and the solution to a problem of Mostowski. An algebraic study of Diodorean modal systems. By R. A. BULL . . 58 On the logic of incomplete answers. By M. J.  ... 
doi:10.1017/s0022481200055845 fatcat:bxhbwrkfrrfsdfmdd7lrvqzth4

Degrees of Transducibility [chapter]

Jörg Endrullis, Jan Willem Klop, Aleksi Saarela, Markus Whiteland
2015 Lecture Notes in Computer Science  
This preorder induces equivalence classes, called degrees, and a partial order on the degrees. For Turing Machines, this structure of degrees is well-studied and known as degrees of unsolvability.  ...  In contrast with the degrees of unsolvability, very little is known about the structure of degrees obtained from finite state transducers or Mealy Machines.  ...  The ensuing hierarchy of degrees of unsolvability has been widely studied in the 60's and 70's of the last century and later. We use the notion of degrees of unsolvability as a guiding analogy.  ... 
doi:10.1007/978-3-319-23660-5_1 fatcat:57hl6axxanbi7gqagwhifm5dru

Secure and Scalable Key Aggregation Scheme for Cloud Storage

YoHan Park, YoungHo Park
2015 Journal of the Korea Industrial Information Systems Research  
And recently, many companies support cloud storages to meet the requirements of the customers.  ...  Especially in the business field where various companies collaborate, data sharing is an essential functionality to enhance their work performance.  ...  However many unsolved security problems in CS restrict the extension of cloud technology to various IT areas. The basic security problem is originated by the remote place to store user's data.  ... 
doi:10.9723/jksiis.2015.20.2.011 fatcat:7osiaxzvtjeypet23u5e3qpwmy

Hyperarithmetical Sets [chapter]

Yiannis N. Moschovakis
2016 Outstanding Contributions to Logic  
The first, key question was how to measure the unsolvability of a set of natural numbers. 1A. Post's degrees of unsolvability.  ...  In modern notation, the Borel hierarchy {Σ 0 : < Ω 1 } (on N ) is defined by setting 3C.  ...  To see that it is the least model of (∆ 1 1 -Comp), assume that F satisfies (∆ 1 1 -Comp) and prove by induction on that ∆ ⊆ F using Theorem 3I.1. (2) Suppose that (∀s)(∃α ∈ HYP)(∃ ∈ HYP)ϕ(s, , α) with  ... 
doi:10.1007/978-3-319-41842-1_5 fatcat:qbbfx2gan5ddjcjx6vrnswr2v4

Page 6027 of Mathematical Reviews Vol. , Issue 2000i [page]

2000 Mathematical Reviews  
Summary: “We show that the length of the naturally occurring jump hierarchy of the infinite time Turing degrees is precisely a, and construct continuum many incomparable such degrees which are minimal  ...  and closed under composition and “bounded recursion on notation”.  ... 

On a subrecursive hierarchy and primitive recursive degrees

Paul Axt
1959 Transactions of the American Mathematical Society  
Pursuit of this interest led therefore to the devising of various hierarchies of recursive functions.  ...  In [4, §6] the notion of primitive recursive degree is introduced in analogy to that of degree of recursive unsolvability (general recursive degree) of Kleene and Post [5] .  ... 
doi:10.1090/s0002-9947-1959-0126377-3 fatcat:7tshveruejbfjostbqhhed2fp4

On a Subrecursive Hierarchy and Primitive Recursive Degrees

Paul Axt
1959 Transactions of the American Mathematical Society  
Pursuit of this interest led therefore to the devising of various hierarchies of recursive functions.  ...  In [4, §6] the notion of primitive recursive degree is introduced in analogy to that of degree of recursive unsolvability (general recursive degree) of Kleene and Post [5] .  ... 
doi:10.2307/1993169 fatcat:gxkidhbui5cxhd2f2rzc4htdo4

Degrees of computability

Norman Shapiro
1956 Transactions of the American Mathematical Society  
In the theory of recursive functions, several decision problems have been proved unsolvable.  ...  A decision problem arises when we are confronted with a set TV of mathematical objects and a subset S of 7Y. We desire a mechanical procedure which will operate on elements of TV, deter-  ...  In all cases, the notation chosen to represent elements of N will be a natural one.  ... 
doi:10.1090/s0002-9947-1956-0085187-3 fatcat:a64bfycbcrdylczg2vklfvs37a

Mass problems associated with effectively closed sets

Stephen G. Simpson
2011 Tohoku mathematical journal  
One of these results answers a question which arises naturally from the Kolmogorov interpretation.  ...  The purpose of this paper is to summarize recent investigations into the lattice of Muchnik degrees of nonempty effectively closed sets in Euclidean space. Let Ew be this lattice.  ...  consistent axiomatizable extension of one of these. 6 Indeed, the recursive homeomorphism type of C(T ) is independent of the choice of T .  ... 
doi:10.2748/tmj/1325886278 fatcat:nsfrryez4rdrxjikkwclfokely

Emil Post [chapter]

UrquharT. Alasdair
2009 Handbook of the History of Logic  
The biographical essay [Gleiser, 1980] is a very informative source on Post's life. The description of Post's papers was obtained from the web site of the American Philosophical Society, which is  ...  ACKNOWLEDGEMENTS My greatest debt is to Martin Davis, whose editorial work and writings on Post were indispensable in preparing this chapter.  ...  This definition results in a hierarchy of degrees of unsolvability, with respect to Turing reducibility.  ... 
doi:10.1016/s1874-5857(09)70016-0 fatcat:erwzbkejovc7vfrx2xbqqmtybe

Standardization and Böhm Trees for Λμ-Calculus [chapter]

Alexis Saurin
2010 Lecture Notes in Computer Science  
the failure of separation in Parigot's original λµ-calculus.  ...  Then, we study innite Λµ-Böhm trees, which are Böhm-like trees for Λµ-calculus; this allows to strengthen the separation results that we established previously for Λµ-calculus and to shed a new light on  ...  Acknowledgments: The author wishes to thank Simona Ronchi della Rocca as well as Kazushige Terui for helpful discussions on the material of this paper.  ... 
doi:10.1007/978-3-642-12251-4_11 fatcat:niemqq2cf5aczbwgdvortwjoqa
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