On Processes and Structures

We consider an approach to computability in admissible sets based on a general notion of computable process, with Σ-predicates and Σ-operators as special cases, inspired by ideas from the Ershov-Scott theory of approximation spaces. We present some results from different topics in generalized computability, including reducibilities on admissible sets and structures, general notion of a jump, and computable analysis (more exactly, computability over the reals), obtained with the help of this approach, and state some open questions.

[1]  Benedikt Löwe,et al.  New Computational Paradigms , 2005 .

[2]  Luigi Acerbi,et al.  Shifting and Lifting of Cellular Automata , 2007 .

[3]  A. Stukachev,et al.  A jump inversion theorem for the semilattices of sigma-degrees , 2010 .

[4]  B. Dahn.  Admissible sets and structures , 1978 .

[5]  Antonio Montalb An,et al.  NOTES ON THE JUMP OF A STRUCTURES , 2009 .

[6]  Antonio Montalbán,et al.  Notes on the Jump of a Structure , 2009, CiE.

[7]  Benedikt Löwe,et al.  Mathematical Theory and Computational Practice, 5th Conference on Computability in Europe, CiE 2009, Heidelberg, Germany, July 19-24, 2009. Proceedings , 2009, CiE.

[8]  A. S. Morozov,et al.  On the Relation of Σ-Reducibility Between Admissible Sets , 2004 .

[9]  Alexey Stukachev,et al.  On Mass Problems of Presentability , 2006, TAMC.

[10]  I︠U︡riĭ Leonidovich Ershov.  Definability and Computability , 1996 .

[11]  V. Baleva,et al.  The jump operation for structure degrees , 2006, Arch. Math. Log..

[12]  A. I. Stukachyov.  Σ-Admissible Families over Linear Orders , 2002 .

[13]  Manfred Broy,et al.  Formal Methods in Programming and Their Applications , 1993, Lecture Notes in Computer Science.

[14]  Alexey Stukachev.  Presentations of Structures in Admissible Sets , 2005, CiE.

[15]  Dana S. Scott,et al.  Outline of a Mathematical Theory of Computation , 1970 .

[16]  Yu. L. Ershov,et al.  The theory of A-spaces , 1973 .

[17]  Alexandra A. Soskova.  A Jump Inversion Theorem for the Degree Spectra , 2007, CiE.

[18]  V. Puzarenko.  A certain reducibility on admissible sets , 2009 .

[19]  A. Stukachev,et al.  Degrees of presentability of structures. I , 2007 .

[20]  Ivan N. Soskov,et al.  A Jump Inversion Theorem for the Degree Spectra , 2009, J. Log. Comput..