α‎-c.a. functions

2020 ◽  
pp. 23-54
Author(s):  
Rod Downey ◽  
Noam Greenberg
Keyword(s):  
Computable Function ◽  
Order Type ◽  
Truth Table ◽  
Main Issue ◽  
Weak Truth Table Degrees ◽  
Definition Of

This chapter discusses the notion of α‎-c.a. functions. The main issue is to properly define what is meant by a computable function o from N to α‎, which is required for the definition of α‎-computable approximations. Naturally, to deal with an ordinal α‎ computably, one needs a notation for this ordinal, or more generally, a computable well-ordering of order-type α‎. To form the basis of a solid hierarchy, the notion of α‎-c.a. should not depend on which well-ordering one takes, rather it should only depend on its order-type. Thus, one cannot consider all computable copies of α‎. Rather, one restricts one's self to a class of particularly well-behaved well-orderings, in a way that ensures that they are all computably isomorphic. Having defined α‎-c.a. functions, the chapter relates these functions to iterations of the bounded jump (the jump inside the weak truth-table degrees).

The theory of the recursively enumerable weak truth-table degrees is undecidable

1992 ◽  
Vol 57 (3) ◽  
pp. 864-874 ◽  
Author(s):  
Klaus Ambos-Spies ◽  
André Nies ◽  
Richard A. Shore
Keyword(s):  
Partial Order ◽  
Open Problem ◽  
Truth Table ◽  
Recursively Enumerable ◽  
Undecidable Theory ◽  
Weak Truth Table Degrees

AbstractWe show that the partial order of -sets under inclusion is elementarily definable with parameters in the semilattice of r.e. wtt-degrees. Using a result of E. Herrmann, we can deduce that this semilattice has an undecidable theory, thereby solving an open problem of P. Odifreddi.

Photography-based taxonomy is still really inadequate, unnecessary, and potentially harmful for biological sciences. A reply to Thorpe (2017)

Bionomina ◽  
2017 ◽  
Vol 12 (1) ◽  
pp. 48-51
Author(s):  
VICTOR G.D. ORRICO
Keyword(s):  
Biological Sciences ◽  
Type A ◽  
Main Issue ◽  
Type Preservation ◽  
Definition Of

A recent defense of the photo-based taxonomy is discussed. The main issue concerns the simplistic definition of “type”. A comparison with other material forms of type preservation highlights weaknesses inherent to the use of photographs as types. Additional arguments from Thorpe’s contribution are also reviewed and disputed.

The complexity of ascendant sequences in locally nilpotent groups

2014 ◽  
Vol 24 (02) ◽  
pp. 189-205 ◽  
Author(s):  
Chris J. Conidis ◽  
Richard A. Shore
Keyword(s):  
Nilpotent Groups ◽  
Low Complexity ◽  
Order Type ◽  
Usual Definition ◽  
Locally Nilpotent Groups ◽  
Locally Nilpotent ◽  
Normal Form Theorem ◽  
Algebraic Normal Form ◽  
Definition Of ◽  
Transfinite Recursion

We analyze the complexity of ascendant sequences in locally nilpotent groups, showing that if G is a computable locally nilpotent group and x0, x1, …, xN ∈ G, N ∈ ℕ, then one can always find a uniformly computably enumerable (i.e. uniformly [Formula: see text]) ascendant sequence of order type ω + 1 of subgroups in G beginning with 〈x0, x1, …, xN〉G, the subgroup generated by x0, x1, …, xN in G. This complexity is surprisingly low in light of the fact that the usual definition of ascendant sequence involves arbitrarily large ordinals that index sequences of subgroups defined via a transfinite recursion in which each step is incomputable. We produce this surprisingly low complexity sequence via the effective algebraic commutator collection process of P. Hall, and a related purely algebraic Normal Form Theorem of M. Hall for nilpotent groups.

Computability and λ-definability

1937 ◽  
Vol 2 (4) ◽  
pp. 153-163 ◽  
Author(s):  
A. M. Turing
Keyword(s):  
Computable Function ◽  
General Recursive Function ◽  
Technical Details ◽  
Intuitive Idea ◽  
Short Space ◽  
Definition Of ◽  
Definable Functions ◽  
Positive Integers ◽  
Computable Functions ◽  
Definable Function

Several definitions have been given to express an exact meaning corresponding to the intuitive idea of ‘effective calculability’ as applied for instance to functions of positive integers. The purpose of the present paper is to show that the computable functions introduced by the author are identical with the λ-definable functions of Church and the general recursive functions due to Herbrand and Gödel and developed by Kleene. It is shown that every λ-definable function is computable and that every computable function is general recursive. There is a modified form of λ-definability, known as λ-K-definability, and it turns out to be natural to put the proof that every λ-definable function is computable in the form of a proof that every λ-K-definable function is computable; that every λ-definable function is λ-K-definable is trivial. If these results are taken in conjunction with an already available proof that every general recursive function is λ-definable we shall have the required equivalence of computability with λ-definability and incidentally a new proof of the equivalence of λ-definability and λ-K-definability.A definition of what is meant by a computable function cannot be given satisfactorily in a short space. I therefore refer the reader to Computable pp. 230–235 and p. 254. The proof that computability implies recursiveness requires no more knowledge of computable functions than the ideas underlying the definition: the technical details are recalled in §5.

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