On the structure of subrecursive degrees

Basu, Sanat K.

doi:10.1016/s0022-0000(70)80042-3

Cited by 12 publications

(7 citation statements)

References 4 publications

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“…Thus, concerning (i), the inclusion from left to right is immediate from the definition of locally transitive upper bound, and the reverse inclusion follows by definition of l.u.b. Using (i), we then infer that the two sets denoted by the join expressions on both sides of the equation in (ii) have identical upper cones, while for the lower cones this is immediate from (6). Concerning (iii), it is sufficient to show that the lower (respectively, upper) cones of A, A ⊕ ∅, and A ⊕ ω are identical.…”

Section: Faithful Relationsmentioning

confidence: 93%

“…Axiomatic approaches to structural properties of bounded reducibilities have been presented previously by, among others, Basu [6], Mehlhorn [17,18], Mueller [23], and Schmidt [29]. Our generalized approach is closely related to the work of the latter three authors, while Basu's axiom system is designed to be used in connection with reducibilities between functions and is too general if applied to reducibilities between sets.…”

Section: Overview and Related Workmentioning

confidence: 99%

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Lattice Embeddings for Abstract Bounded Reducibilities

Merkle¹

2002

SIAM J. Comput.

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We give an abstract account of resource-bounded reducibilities as exemplified by the polynomially time-or logarithmically space-bounded reducibilities of Turing, truth-table, and many-one type. We introduce a small set of axioms that are satisfied for most of the specific resourcebounded reducibilities appearing in the literature. Some of the axioms are of a more algebraic nature, such as the requirement that the reducibility under consideration is a reflexive relation, while others are formulated in terms of recursion theory and, for example, are related to delayed computations of arbitrary recursive sets. The main technical result shown is that for any reducibility that satisfies these axioms, every countable distributive lattice can be embedded into any proper interval of the structure induced on the recursive sets. This extends a corresponding result for polynomially time-bounded reducibilities due to Ambos-Spies [Inform. and Control, 65 (1985), pp. 63-84], as well as a result on embeddings of partial orderings for axiomatically described reducibilities due to Mehlhorn [

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Section: Faithful Relationsmentioning

confidence: 93%

Section: Overview and Related Workmentioning

confidence: 99%

Lattice Embeddings for Abstract Bounded Reducibilities

Merkle¹

2002

SIAM J. Comput.

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“…It is easy to see that the class £ 3 remains the same if we use 2 X in place of E% in the definition. (E%(x) = Ef (2) where E\(x) = x 2 + 2.) Thus it follows that the class of functions elementary in / is the closure of {0,<S,Zf,2…”

Section: General Preliminaries and Definitionsmentioning

confidence: 99%

“…The structure of subrecursive honest degrees is studied, explicitly or implicitly, in Meyer and Ritchie [11], Basu [2], Machtey [8] [9] [10], Simmons [16], and Kristiansen [5] [6]. Machtey shows that the structure of elementary honest degrees is a lattice with strong density properties, for instance between any degrees a, b such that a < b there are two incomparable degrees.…”

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confidence: 99%

Information content and computational complexity of recursive sets

Kristiansen¹

1996

Lecture Notes in Logic

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An honest function is, roughly speaking, a unary, recursive, and strictly increasing function with a very simple graph. Thus if / is an honest function, then the growth of / reflects the computational complexity of /. The honest elementary degrees are the degree structure induced on the honest functions by the reducibility relation "being (Kalmar) elementary in". (Other subrecursive reducibility relations will also work, for example "being primitive recursive in", but not "being polynomial time computable in the length of input\ "Being polynomial time computable in the input * This paper is in its final form, and no similar paper has been or is being submitted elsewhere.

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“…Remember that every computable total function (or total Turing machine) defines a subrecursive class which is a proper subclass of others subrecursive classes (and of the class of all recursive functions) [5,9,23,28]. A subrecursive class is one defined by a proper subset of the set of all problems with Turing degree 0.…”

Section: Introductionmentioning

confidence: 99%

Relativizing an incompressible number and an incompressible function through subrecursive extensions of Turing machines

Abrahão

2016

Preprint

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We show in this article that uncomputability is also a relative property of subrecursive classes built on a recursive relative incompressible function, which acts as a higher-order "yardstick" of irreducible information for the respective subrecursive class. We define the concept of a Turing submachine, and a recursive relative version for the Busy Beaver function and for the halting probability (or Chaitin's constant) Ω; respectively the Busy Beaver Plus (BBP) function and a time-bounded halting probability. Therefore, we prove that the computable BBP function defined on any Turing submachine is neither computable nor compressible by any program running on this submachine. In addition, we build a Turing submachine that can use lower approximations to its own time-bounded halting probability to calculate the values of its Busy Beaver Plus function, in the "same" manner that universal Turing machines use approximations to Ω to calculate Busy Beaver values. Thus, the algorithmic information carried by the BBP function is relatively incompressible (and uncomputable) at the same time that it still is occasionally reached by submachines. We point that this phenomenon enriches the research on the relativization and simulation of uncomputability and irreducible information.

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On the structure of subrecursive degrees

Cited by 12 publications

References 4 publications

Lattice Embeddings for Abstract Bounded Reducibilities

Lattice Embeddings for Abstract Bounded Reducibilities

Information content and computational complexity of recursive sets

Relativizing an incompressible number and an incompressible function through subrecursive extensions of Turing machines

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