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Ordinal machines and admissible recursion theory

Peter Koepke, Benjamin Seyfferth
2009 Annals of Pure and Applied Logic  
We show that this provides a simple machine model adequate for classical admissible recursion theory as developed by G. Sacks and his school.  ...  For α an admissible ordinal, the basic notions of αrecursive or α-recursively enumerable are equivalent to being computable or computably enumerable by an α-machine, respectively.  ...  This work led to the concept of admissible ordinals and to α-recursion theory where recursions are carried out on (the elements of) an admissible ordinal α.  ... 
doi:10.1016/j.apal.2009.01.005 fatcat:2lrx5efocnflheozbiu7vmbr5u

A representation of recursively enumerable sets through Horn formulas in higher recursion theory

Juan A. Nido Valencia, Julio E. Solís Daun, Luis M. Villegas Silva
2016 Periodica Mathematica Hungarica  
We extend a classical result in ordinary recursion theory to higher recursion theory, namely that every recursively enumerable set can be represented in any model A by some Horn theory, where A can be  ...  any model of a higher recursion theory, like primitive set recursion, αrecursion, or β -recursion.  ...  Key words and phrases. Horn Theory, primitive recursive set functions, recursively enumerable set, α-recursion theory, primitive recursively closed ordinals, admissible recursion, β -recursion.  ... 
doi:10.1007/s10998-016-0148-x fatcat:i7wfhvaplvf7jd6ugcq5az5mbe

Ordinal Computability [chapter]

Peter Koepke
2009 Lecture Notes in Computer Science  
Ordinal computability uses ordinals instead of natural numbers in abstract machines like register or Turing machines.  ...  We give an overview of the computational strengths of α-β-machines, where α and β bound the time axis and the space axis of some machine model.  ...  from recursion theory, admissibility theory, descriptive set theory, and set theoretic constructibility theory.  ... 
doi:10.1007/978-3-642-03073-4_29 fatcat:e6aq26lgqvej7a6mhtlt3ni7je

Page 8 of Mathematical Reviews Vol. 52, Issue 1 [page]

1976 Mathematical Reviews  
His main theorem is: Let « be an admissible ordinal, C a regular a-r.e. set, and D a non-a-recursive «-r.e. set; then there are regular a-r.e. sets A and B such that AU B=C, AN B=g, A, BS, C, and such  ...  Previous generalizations of theorems from ordinary recursion theory to «-recursion theory had been limited to those theorems which, in ordinary recursion theory, were finite injury priority arguments in  ... 

Discrete Transfinite Computation Models [chapter]

Philip D. Welch
2011 Computability in Context  
The notion of admissible ordinal stands out, not least because of the development of α-recursion theory in the 1960's and 70's.  ...  Computation on ordinals, and ordinal length machines In the 70's the theory of α-recursion coming out of the meta-recursion of the 60's reached its highest stage of development.  ...  ([Welch (2005) ]) (i) Π 1 3 CA 0 , ACA 0 + AQI and Π 1 2 CA 0 are in descending order of strength in that each theory proves the the existence of β-models of the next.  ... 
doi:10.1142/9781848162778_0012 fatcat:u4uggbtvjzf7rlzd3gdgv62xym

Characteristics of discrete transfinite time Turing machine models: Halting times, stabilization times, and Normal Form theorems

P.D. Welch
2009 Theoretical Computer Science  
We provide, inter alia, a Normal form Theorem, and a characterisation of which ordinals start gaps in halting times of such machines.  ...  We give an acount of the basic determinants of the courses of computation of the Infinite Time Turing Machine model of Hamkins and Kidder, a model of computation which allows for transfinitely many steps  ...  Acknowledgements We should like to thank various people who have contributed questions, queries and discussion of the notions involved and struggled with the incomprehensibility of some of our previous  ... 
doi:10.1016/j.tcs.2008.09.050 fatcat:xixekgzcrbhb5eirbnoirhusoe

Page 4400 of Mathematical Reviews Vol. , Issue 83k [page]

1983 Mathematical Reviews  
Their dependence on varied techniques (from Barwise compactness and projecta-master code theory to ordinal recursion theory) indicates areas for further application. E. R.  ...  This point is now obvious, of course, after the many successful generalisations such as admissibility theory (including a-reeursion theory), various axiomatic recursion theories and also the Kleene theory  ... 

Bounding lemmata for non-deterministic halting times of transfinite Turing machines

Philip D. Welch
2008 Theoretical Computer Science  
Later we shall use some results and notions from admissibility theory (for which the reader may consult [1]) and from generalised recursion theory (see [7] ) and descriptive set theory (see [4] ).  ...  In particular we observe that there is a Uniform Bounding Lemma which states that if any total algorithm halts before the first ordinal admissible in the input x, then there is a recursive ordinal γ by  ...  Theorem 2 (Deolalikar, Hamkins, & Schindler; [2, Here ω CK 1 is the supremum of all recursive ordinals, and ω x 1 will be used to denote the supremum of all ordinals recursive in x (in both cases this  ... 
doi:10.1016/j.tcs.2007.12.014 fatcat:4jxkgsxtgjfepdsgnxz4tzfx5y

Page 1107 of Mathematical Reviews Vol. 43, Issue 5 [page]

1972 Mathematical Reviews  
The author investigates what first order statements true of the lattice of recursively enumerable (r.e.) sets remain true for recursion theory on admissible ordinals.  ...  For any admissible ordinal « let R, and Q, denote the lattices of a-r.e. sets and of a-r.e. subsets of w respectively.  ... 

Discrete Transfinite Computation [article]

Philip Welch
2014 arXiv   pre-print
We outline the connections between such models and the older theories of recursion in higher types, generalized recursion theory, and recursion on ordinals such as α-recursion.  ...  Variants of such machines are considered that have longer tapes than the standard model, or that work on ordinals rather than numbers.  ...  Some of the other generalizations of recursion theory, say to meta-recursion theory, as advocated by Kreisel and elucidated by Sacks and his school, and which later became ordinal α-recursion theory, we  ... 
arXiv:1409.5052v2 fatcat:vo4q5htmwfb2xl3zdfivecstha

Discrete Transfinite Computation [chapter]

P. D. Welch
2015 Turing's Revolution  
We outline the connections between such models and the older theories of recursion in higher types, generalized recursion theory, and recursion on ordinals such as α-recursion.  ...  Variants of such machines are considered that have longer tapes than the standard model, or that work on ordinals rather than numbers.  ...  Some of the other generalizations of recursion theory, say to meta-recursion theory, as advocated by Kreisel and elucidated by Sacks and his school, and which later became ordinal α-recursion theory, we  ... 
doi:10.1007/978-3-319-22156-4_6 fatcat:pghp6fyk2jdm5nylnyqibdj2ji

Page 3949 of Mathematical Reviews Vol. , Issue 88h [page]

1988 Mathematical Reviews  
It was shown in Part I [“Minimal degrees and 1-generic de- grees in higher recursion theory”, submitted] that for all admissible ordinals a, no 1-generic degree recursive in 0’ bounds a minimal degree.  ...  The class of such ordinals includes all constructible cardinals which are not Z, admissible, such as w.” 88h:03065 03D65 03F60 18B25 18B30 54B30 Mulry, Philip S. (1-COLG) Adjointness in recursion.  ... 

Page 3 of Mathematical Reviews Vol. , Issue 88a [page]

1988 Mathematical Reviews  
The goal of this extensive article is to begin the extension of this theory to the study of recursiveness for functionals and predicates of finite types over an arbitrary admissible set A.  ...  Summary: “We give a new characterization of the hyperarithmetic sets: a set X of integers is recursive in e, if and only if there is a Turing machine which computes X and ‘halts’ in less than or equal  ... 

Page 4646 of Mathematical Reviews Vol. , Issue 80M [page]

1980 Mathematical Reviews  
The author presents a (formal) theory Ty of operations, classes and ordinals, for which set theory provides one interpretation, and admissible sets another.  ...  and an analogue theory on admissible sets.  ... 

Page 3557 of Mathematical Reviews Vol. , Issue 83i [page]

1983 Mathematical Reviews  
models: RAM, storage modification machines, Turing machines, URM, polynomial time and real time equivalence of machine types, algorithms and recursive functions, recursive and recursively enumerable sets  ...  A new proof of Sacks’s theorem: “Every countable admissible ordinal >w is of the form w) for some TCw”, proceeds as follows.  ... 
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