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The Church Problem for Countable Ordinals

2009
*
Logical Methods in Computer Science
*

We consider a natural generalization of

doi:10.2168/lmcs-5(2:5)2009
fatcat:nlikkkg23zgfhhvavn2a2bur6q
*the**Church**problem*to*countable**ordinals*and investigate games of arbitrary*countable*length. ... We prove that determinacy and decidability parts of*the*Bu}chi and Landweber theorem hold*for*all*countable**ordinals*and that its full extension holds*for*all*ordinals*< \omega\^\omega. ... Acknowledgments I am very grateful to Amit Shomrat*for*his insightful comments. ...##
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Synthesis of Finite-state and Definable Winning Strategies

2009
*
Foundations of Software Technology and Theoretical Computer Science
*

We consider two natural generalizations of

doi:10.4230/lipics.fsttcs.2009.2332
dblp:conf/fsttcs/Rabinovich09
fatcat:awc7tm6ulrgqzk7p3zrq4vhx5y
*the**Church**problem*to*countable**ordinals*:*the*first deals with finite-state strategies;*the*second deals with MLO-definable strategies. ... We investigate games of arbitrary*countable*length and prove*the*computability of these generalizations of Church's*problem*. ... Acknowledgments I am very grateful to Amit Shomrat*for*his insightful comments. ...##
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Page 2763 of Mathematical Reviews Vol. , Issue 85g
[page]

1985
*
Mathematical Reviews
*

*The*results generalize those on

*the*elementary theory ET|co] of all

*countable*

*ordinals*[cf. J. E. ...

*For*example, if T is finite, monadic, and

*Church*- Rosser, then

*the*following questions are decidable: (I) Is Mr a group? ...

##
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Orders, reduction graphs and spectra

1999
*
Theoretical Computer Science
*

*For*well-ordered spectra, we show that they can represent all

*countable*successor

*ordinals*. ... Then, considering constructive

*ordinals*, we address

*the*well-known

*problem*of knowing which

*ordinals*are lambda representable (that is

*for*which

*ordinal*CL there exists a lambda term T such that a is isomorphic ... We thank Stefano Sorgi and Ivan0 Salvo

*for*drawing

*the*illustrations. ...

##
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On Turing's legacy in mathematical logic and the foundations of mathematics

2013
*
Arbor: Ciencia, Pensamiento y Cultura
*

*The*

*Church*-Kleene system O of

*ordinal*notations:

*The*number 1 is a notation

*for*

*the*

*ordinal*0, 2 a is a notation

*for*

*the*successor of |a| and 3 . 5 e is a notation

*for*

*the*supremum of

*the*sequence |an| ... But

*for*T α to be effectively given,

*the*

*ordinal*α must be computable. This leads to

*the*notion of

*ordinal*notations, i.e., a computable way to assign

*countable*

*ordinals*to natural numbers. ...

##
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On the questions P ?= NP ∩ co-NP and NP ?= co-NP for infinite time Turing machines
[article]

2003
*
arXiv
*
pre-print

Schindler recently addressed two versions of

arXiv:math/0305445v1
fatcat:y4gc2yrzk5dcbn4mft6kf3mbsa
*the*question P ?= NP*for*Turing machines running in transfinite*ordinal*time. These versions differ in their definition of input length. ...*The*corresponding complexity classes are labelled P, NP and P^+, NP^+. Schindler showed that P ≠ NP and P^+ ≠ NP^+. ... INTRODUCTION*The*fundamental open*problems*in complexity theory*for*Turing ma- versions of*the*classes P and NP in a Turing machine running in transfinite*ordinal*time. ...##
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Page 5959 of Mathematical Reviews Vol. , Issue 97J
[page]

1997
*
Mathematical Reviews
*

Summary: “We extend

*the*analysis of*the*decision*problem**for*modules over a group ring Z[G] to*the*case in which G is a cyclic group of square-free order. ...*The*paper includes a short analysis of*the*decision*problem**for**the*theories of (finitely generated) modules over Z{¢,,,], where m is a positive integer and ¢,,, is a primitive mth root of 1.” ...##
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Page 892 of Mathematical Reviews Vol. , Issue 84c
[page]

1984
*
Mathematical Reviews
*

It is shown that

*the*word*problem**for**Church*- Rosser systems is decidable in linear time.” ... This is an expository paper, presenting decidability results*for**problems*concerning deterministic read-only transformers and*for*related*problems*concerning context-free grammars, finite automata over ...##
###
Page 368 of American Mathematical Society. Transactions of the American Mathematical Society Vol. 100, Issue 2
[page]

1961
*
American Mathematical Society. Transactions of the American Mathematical Society
*

Their existence supports

*the*view that*the*“constructively accessible”*ordinals*covered by C may not give as natural a stopping place as*the*“con- structively*countable*”*ordinals*covered by Sy}. ...*Church*,*The*constructive second number class, Bull. Amer. Math. Soc. vol. 44 (1938) pp. 224-232. [CK] A.*Church*and S. C. Kleene, Formal definitions in*the*theory of*ordinal*numbers, Fund. ...##
###
Cantor's Proof in the Full Definable Universe

2010
*
The Australasian Journal of Logic
*

all objects that can be defined not just in one formal language but by means of

doi:10.26686/ajl.v9i0.1818
fatcat:wk24fuy55nhadofovyuk5lo2ui
*the*full expressive power of natural language: this universe seems to be*countable*on one account and uncountable on another ... Cantor's proof that*the*powerset of*the*set of all natural numbers is uncountable yields a version of Richard's paradox when restricted to*the*full definable universe, that is, to*the*universe containing ... it from below, then, under*the**Church*-Turing thesis, only*the**Church*-Kleene constructive*ordinals*would exist in*the*definable universe and*the*first*Church*-Kleene non-constructive*ordinal*ω CK 1 would ...##
###
Ordinal Recursion Theory
[article]

1996
*
arXiv
*
pre-print

In this article, intended

arXiv:math/9609203v1
fatcat:qbrxz3s5mjet7i4ya3igsgob2q
*for**the*Handbook of Recursion Theory, we survey recursion theory on*the**ordinal*numbers, with sections devoted to α-recursion theory, β-recursion theory and*the*study of*the*admissibility ... This view is supported by*the*successful solution of Post's*problem**for*all admissible*ordinals*(Theorem 2 below). We give here some examples of admissible*ordinals*: (a) α = ω,*the*classical case. ... But this contradicts*the*well-ordering of*ordinals*. This proves*the*claim. It follows from*the*claim that κ 2 and hence α * 2 is*countable*. We conclude that α is*countable*. ...##
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Decreasing Diagrams with Two Labels Are Complete for Confluence of Countable Systems

2018
*
International Conference on Rewriting Techniques and Applications
*

*The*decreasing diagrams technique is one of

*the*strongest and most versatile methods

*for*proving confluence of abstract reduction systems, it is complete

*for*

*countable*systems, and it has many well-known ... Surprisingly, we find that two labels suffice

*for*proving confluence

*for*every abstract rewrite system having

*the*cofinality property, thus in particular

*for*every confluent,

*countable*system. ... Finally, we are thankful to

*the*reviewers

*for*many useful suggestions. ...

##
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Topological Complexity of omega-Powers : Extended Abstract
[article]

2008
*
arXiv
*
pre-print

This is an extended abstract presenting new results on

arXiv:0809.1812v1
fatcat:v64qfyhasvhwdoe72lc3ubtp6e
*the*topological complexity of omega-powers (which are included in a paper "Classical and effective descriptive complexities of omega-powers" available ... from arXiv:0708.4176) and reflecting also some open questions which were discussed during*the*Dagstuhl seminar on "Topological and Game-Theoretic Aspects of Infinite Computations" 29.06.08 - 04.07.08. ...*the**Church*-kleene*ordinal*. ...##
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Indefinite Extensibility in Natural Language

2013
*
The Monist
*

We also point out that object-object mapping theories of semantics,

doi:10.5840/monist201396211
fatcat:4c6oowi6hbfvvg2oc7yfpezebi
*the*usual account*for**the*possibility of non intentional semantics, doesn't seem able to account*for**the*indefinitely extensible productivity ...*The*Monist's call*for*papers*for*this issue ended: "if formalism is true, then it must be possible in principle to mechanize meaning in a conscious thinking and language-using machine; if intentionalism ...*the*least non*Church*-Kleene-constructive*ordinal* 1 CK (*Church*1936;*Church*and Kleene 1936; Kleene 1938) . ...##
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Page 2852 of Mathematical Reviews Vol. , Issue 87f
[page]

1987
*
Mathematical Reviews
*

(b)

*For*any p, if © is a function on*the**ordinal*numbers then @ is ES,-recursive if and only if it is uniformly ©; (p) in every admissible set.” ... Some open*problems*indicate interesting directions of further research. {*For**the*entire collection see MR 87b:03004.} Detlef G. ...
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