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The Church Problem for Countable Ordinals

Alexander Rabinovich, Wolfgang Thomas
2009 Logical Methods in Computer Science  
We consider a natural generalization of 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.  ... 
doi:10.2168/lmcs-5(2:5)2009 fatcat:nlikkkg23zgfhhvavn2a2bur6q

Synthesis of Finite-state and Definable Winning Strategies

Alexander Rabinovich, Marc Herbstritt
2009 Foundations of Software Technology and Theoretical Computer Science  
We consider two natural generalizations of 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.  ... 
doi:10.4230/lipics.fsttcs.2009.2332 dblp:conf/fsttcs/Rabinovich09 fatcat:awc7tm6ulrgqzk7p3zrq4vhx5y

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?  ... 

Orders, reduction graphs and spectra

B. Intrigila, M.Venturini Zilli
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.  ... 
doi:10.1016/s0304-3975(98)00141-8 fatcat:c2mpiprum5futesfm63au34aai

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

Joan Bagaria
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.  ... 
doi:10.3989/arbor.2013.764n6002 fatcat:bludoedrnrf57jq4iocd24bzau

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

Vinay Deolalikar
2003 arXiv   pre-print
Schindler recently addressed two versions of 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.  ... 
arXiv:math/0305445v1 fatcat:y4gc2yrzk5dcbn4mft6kf3mbsa

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.”  ... 

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 countableordinals 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

Laureano Luna, William Taylor
2010 The Australasian Journal of Logic  
all objects that can be defined not just in one formal language but by means of 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  ... 
doi:10.26686/ajl.v9i0.1818 fatcat:wk24fuy55nhadofovyuk5lo2ui

Ordinal Recursion Theory [article]

Chi Tat Chong, Sy D. Friedman
1996 arXiv   pre-print
In this article, intended 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.  ... 
arXiv:math/9609203v1 fatcat:qbrxz3s5mjet7i4ya3igsgob2q

Decreasing Diagrams with Two Labels Are Complete for Confluence of Countable Systems

Jörg Endrullis, Jan Willem Klop, Roy Overbeek, Michael Wagner
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.  ... 
doi:10.4230/lipics.fscd.2018.14 dblp:conf/rta/EndrullisKO18 fatcat:v6cugaleincodeescqeyye5awa

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

Olivier Finkel
2008 arXiv   pre-print
This is an extended abstract presenting new results on 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.  ... 
arXiv:0809.1812v1 fatcat:v64qfyhasvhwdoe72lc3ubtp6e

Indefinite Extensibility in Natural Language

Laureano Luna, Sherwood J. B. Sugden
2013 The Monist  
We also point out that object-object mapping theories of semantics, 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) .  ... 
doi:10.5840/monist201396211 fatcat:4c6oowi6hbfvvg2oc7yfpezebi

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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