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Formalism and intuition in computability

2012
*
Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
*

In 1936,

doi:10.1098/rsta.2011.0335
pmid:22711859
fatcat:z2ae3dac65dmfmijd2sylefawq
*Turing**and*in 1944,*Post*introduced more intuitive models of*Turing*machines*and*generational systems. ... A third wave of intuition has arisen from 1996 to the present with a return to the original meaning of computability in the sense of*Turing**and*Gödel,*and*a return of 'recursive' to its original meaning ... The grant is Computability*theory**and*applications, funded by the Simons Foundation Program for Mathematics*and*the Physical Sciences. ...##
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Page 4018 of Mathematical Reviews Vol. , Issue 2003f
[page]

2003
*
Mathematical Reviews
*

Baer (Mill Valley, CA)
2003f:03053 03D15 03F30 68Q05 68Q15 Gordeew, Lew (D-TBNG-ILS; Tiibingen)

*Proof**theory**and**Post*-*Turing**analysis*. ... Summary: “Traditional*proof**theory**and**post*-*Turing*computabil- ity are parts of discrete mathematics, whereas traditional*analysis*deals with non-discrete objects like real numbers*and*related topo- logical ...##
###
Book Review: Complexity theory of real functions

1993
*
Bulletin of the American Mathematical Society
*

*analysis*, functional

*analysis*,

*and*physical

*theory*. ... (For example, the

*Post*production, a rule for manipulating strings of symbols, played a role in the

*theory*of programming languages, the work of Noam Chomsky on the grammatical

*analysis*of language,

*and*...

##
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Page 2279 of Mathematical Reviews Vol. , Issue 87e
[page]

1987
*
Mathematical Reviews
*

The uniform version of the

*Post*-*Turing*thesis, i.e. Je B Vz(Bp(x) U(e,z)) is inconsistent. ... 2279 Sieg, Wilfried (1-CLMB-Q) Reductions of*theories*for*analysis*. Foundations of logic*and*linguistics (Salzburg, 1983), 199-231, Plenum, New York-London, 1985. ...##
###
Review of 'Computation: Finite and Infinite Machines' (Minsky, Marvin; 1967)

1968
*
IEEE Transactions on Information Theory
*

This is a beautifully written

doi:10.1109/tit.1968.1054133
fatcat:rriovgno5jbdtl5akbzcy6azo4
*and*produced introduction to finite automata,*Turing*machines,*Post*symbol-manipulation systems,*and*related topics in computability. ... Giving the*proofs*due to himself*and*to J. ...##
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The History and Concept of Computability
[chapter]

1999
*
Studies in Logic and the Foundations of Mathematics
*

*Post*1944 explicitly added the empty set as an r.e. set (see [11, p. 308]), which Church

*and*Kleene had omitted. ... since then, except for the term "computably enumerable," recently introduced, because

*Turing*

*and*Gödel did not explicitly introduce a term for these corresponding sets, but just for the computable functions ... A

*theory*of such machines was given by

*Turing*1936

*and*in less detail by

*Post*1936." ...

##
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Turing oracle machines, online computing, and three displacements in computability theory

2009
*
Annals of Pure and Applied Logic
*

,

doi:10.1016/j.apal.2009.01.008
fatcat:66vngrvfhfcr3c2rq36p2pbaeq
*analysis*, model*theory*, algorithmic complexity*and*many more. ...*Post*gave no hint of a universal*Turing*machine. Most important,*Post*gave no*analysis*, as did*Turing*, of why the intuitively computable functions are computable in his formal system. ...##
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Computability and Recursion

1996
*
Bulletin of Symbolic Logic
*

After a careful historical

doi:10.2307/420992
fatcat:yq5vicyjf5aq7ercdrg2jql6zm
*and*conceptual*analysis*of computability*and*recursion we make several recommendations in section §7 about preserving the intensional differences between the concepts of "computability ... We consider the informal concept of "computability" or "effective calculability"*and*two of the formalisms commonly used to define it, "(*Turing*) computability"*and*"(general) recursiveness". ... A*theory*of such machines was given by*Turing*1936*and*in less detail by*Post*1936 ." ...##
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1936: Post, Turing and 'a kind of miracle' in mathematical logic

2004
*
Mathematical Gazette
*

In the 1930s several mathematicians, principally Alonzo Church (1903-1995), Stephen Kleene (1909-1994), Emil

doi:10.1017/s0025557200174170
fatcat:echz27tckzegjf52yco6b5izna
*Post*(1897-1954)*and*Alan*Turing*(1912-1954), began investigating the notion of effective calculability ... Acknowledgements I am indebted to the referee for making helpful suggestions on the first draft*and*to Becky Carter who typed it. ... Independently of*Turing*but aware of Church's Thesis,*Post*gives an*analysis*of the computing process very similar to that given by*Turing*but there is a subtle difference. ...##
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Page 6526 of Mathematical Reviews Vol. , Issue 2003i
[page]

2003
*
Mathematical Reviews
*

It is a lucid

*and*insightful combination of conceptual*analysis**and*historical exegesis of the work of pioneers like Church, Kleene,*Post*, Gédel*and*, above all,*Turing*. ... Instead, the*analysis*produces rigorous axiomatizations of the key notions,*and*allows*proofs*that the intuitive notions are coextensive with each other*and*with*Turing*computability, (general) recursiveness ...##
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Definable degrees and automorphisms of $\mathcal {D}$

1981
*
Bulletin of the American Mathematical Society
*

The

doi:10.1090/s0273-0979-1981-14871-0
fatcat:oac6v2hkjbe27oygxvbt7cysqi
*analysis*of the structure V of these degrees has been a central topic in recursion*theory*beginning with the papers of*Post*[1944]*and*Kleene*and**Post*[1954]. ... If the*Turing*degree x is not hyperarithmetic*and*x, 0 < t then there is a degree s such that t is a minimal cover of s*and*x ^ s.*PROOF*. ... The*analysis*of the structure V of these degrees has been a central topic in recursion*theory*beginning with the papers of*Post*[1944]*and*Kleene*and**Post*[1954] . ...##
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Book Review: Recursively enumerable sets and degrees: A study of computable functions and computably generated sets

1988
*
Bulletin of the American Mathematical Society
*

This is obviously true of the work of

doi:10.1090/s0273-0979-1988-15624-8
fatcat:qi44akcymvblddnoohditudh7a
*Post**and**Turing*referred to earlier. It is equally true of very recent work. ... The*Post*Production, a rule for manipulating strings of symbols, found its way into programming languages, the work of Noam Chomsky on the grammatical*analysis*of language,*and*other topics. ... In this volume the modern conceptions*and*ramifications of manifolds are treated. Perhaps no other concept in modern mathematics has been so ...##
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Degrees of Unsolvability
[chapter]

2014
*
Handbook of the History of Logic
*

We finish our history by quoting Gerald Sacks, who through his own work

doi:10.1016/b978-0-444-51624-4.50010-1
fatcat:clf7varewrh6lc4jq2c2sp6pgy
*and*the work of his students (including, Harrington, Robinson, Shore, Simpson, Slaman,*and*Thomason) shaped degree*theory*throughout ... a good part of its existence: "Farewell to higher recursion*theory*, but not to recursion*theory*; there is no way to say good-bye to recursion*theory*" ( [Sacks, 1999] ). ... Kleene's real interests were in generalized recursion*theory**and*[Kleene*and**Post*, 1954] is his only paper in the*Turing*degrees. ...##
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The jump is definable in the structure of the degrees of unsolvability

1990
*
Bulletin of the American Mathematical Society
*

The structural

doi:10.1090/s0273-0979-1990-15923-3
fatcat:d3g7bx5i6neufhcr5b5i32okie
*analysis*of the partial ordering 3 has been a major area of research in recursion*theory*since the pioneering paper of Kleene*and**Post*[14] . ... Recursion*theory*deals with computability on the natural numbers. ... The structural*analysis*of the partial ordering 3 has been a major area of research in recursion*theory*since the pioneering paper of Kleene*and**Post*[14] . ...##
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Incomputability after Alan Turing
[article]

2013
*
arXiv
*
pre-print

Incomputability as a mathematical notion arose from work of Alan

arXiv:1304.6363v1
fatcat:nh5lsf23pvedleymkatyjksivi
*Turing**and*Alonzo Church in the 1930s. ... This article takes a*Turing*centenary look at how the interface between the computable world*and*the incomputable formed a central theme in Turing's work - from the early establishment of the standard ... On the one hand, we have numerical*analysis**and*scientific computation; on the other hand, we have the tradition of computation*theory*arising from logic*and*computer science. ...
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