论文信息 - Ordinal Computability

Ordinal Computability

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 *** -β -machines, where *** and β bound the time axis and the space axis of some machine model. The spectrum ranges from classical Turing computability to ***-***-computability which corresponds to Godel 's model of constructible sets. To illustrate some typical techniques we prove a new result on Infinite Time Register Machines (= ***-*** -register machines) which were introduced in [6]: a real number x *** *** 2 is computable by an Infinite Time Register Machine iff it is Turing computable from some finitely iterated hyperjump 0(n ).

Peter Koepke | P. Koepke

[1] Joel David Hamkins,et al. Infinite Time Turing Machines , 1998, Journal of Symbolic Logic.

[2] Benedikt Löwe,et al. Logical Approaches to Computational Barriers: CiE 2006 , 2007, J. Log. Comput..

[3] Peter A. Cholak. Review: Stephen G. Simpson, Subsystems of Second Order Arithmetic , 1999, Journal of Symbolic Logic.

[4] Peter Koepke,et al. Register computations on ordinals , 2008, Arch. Math. Log..

[5] Peter Koepke,et al. Turing computations on ordinals , 2005, Bull. Symb. Log..

[6] Stephen G. Simpson,et al. Subsystems of second order arithmetic , 1999, Perspectives in mathematical logic.

[7] Peter Koepke,et al. An Enhanced Theory of Infinite Time Register Machines , 2008, CiE.

[8] Joel David Hamkins,et al. Infinite Time Turing Machines , 2000 .

[9] H. E. Rose. COMPUTABILITY-AN INTRODUCTION TO RECURSIVE FUNCTION THEORY , 1981 .

[10] Peter Koepke,et al. Infinite Time Register Machines , 2006, CiE.

[11] John Longley. Interpreting Localized Computational Effects Using Operators of Higher Type , 2008, CiE.

[12] Benjamin Seyfferth,et al. Ordinal machines and admissible recursion theory , 2009, Ann. Pure Appl. Log..