Prefix-Like Complexities and Computability in the Limit

Computability in the limit represents the non-plus-ultra of constructive describability. It is well known that the limit computable functions on naturals are exactly those computable with the oracle for the halting problem. However, prefix (Kolmogorov) complexities defined with respect to these two models may differ. We introduce and compare several natural variations of prefix complexity definitions based on generalized Turing machines embodying the idea of limit computability, as well as complexities based on oracle machines, for both finite and infinite sequences.

[1]  Cristian S. Calude,et al.  Coins, Quantum Measurements, and Turing's Barrier , 2002, Quantum Inf. Process..

[2]  István Németi,et al.  Non-Turing Computations Via Malament–Hogarth Space-Times , 2001 .

[3]  John Case,et al.  On learning limiting programs , 1992, COLT '92.

[4]  H. Keisler,et al.  Handbook of mathematical logic , 1977 .

[5]  R. V. Freivald Functions Computable in the Limit by Probabilistic Machines , 1974, MFCS.

[6]  Jürgen Schmidhuber,et al.  Hierarchies of Generalized Kolmogorov Complexities and Nonenumerable Universal Measures Computable in the Limit , 2002, Int. J. Found. Comput. Sci..

[7]  Joseph R. Shoenfield,et al.  Degrees of unsolvability , 1959, North-Holland mathematics studies.

[8]  Robin Milner,et al.  On Observing Nondeterminism and Concurrency , 1980, ICALP.

[9]  Maribel Fernández,et al.  Curry-Style Types for Nominal Terms , 2006, TYPES.

[10]  André Nies,et al.  Program Size Complexity for Possibly Infinite Computations , 2005, Notre Dame J. Formal Log..

[11]  Jürgen Schmidhuber,et al.  Algorithmic Theories of Everything , 2000, ArXiv.

[12]  Stephen G. Simpson,et al.  Degrees of Unsolvability: A Survey of Results , 1977 .

[13]  Eugene Asarin,et al.  Noisy Turing Machines , 2005, ICALP.

[14]  Susumu Hayashi,et al.  Towards Limit Computable Mathematics , 2000, TYPES.

[15]  William I. Gasarch,et al.  Book Review: An introduction to Kolmogorov Complexity and its Applications Second Edition, 1997 by Ming Li and Paul Vitanyi (Springer (Graduate Text Series)) , 1997, SIGACT News.

[16]  J. Schmidhuber,et al.  Prefix-like Complexities of Finite and Infinite Sequences on Generalized Turing Machines. , 2005 .

[17]  Ming Li,et al.  An Introduction to Kolmogorov Complexity and Its Applications , 2019, Texts in Computer Science.

[18]  Jan Poland A coding theorem for Enumerable Output Machines , 2004, Inf. Process. Lett..

[19]  Nikolai K. Vereshchagin,et al.  Descriptive complexity of computable sequences , 2002, Theor. Comput. Sci..

[20]  Péter Gács,et al.  On the relation between descriptional complexity and algorithmic probability , 1981, 22nd Annual Symposium on Foundations of Computer Science (sfcs 1981).

[21]  Gregory J. Chaitin,et al.  A recent technical report , 1974, SIGA.

[22]  A. Kolmogorov Three approaches to the quantitative definition of information , 1968 .

[23]  Santiago Figueira,et al.  Kolmogorov Complexity for Possibly Infinite Computations , 2005, J. Log. Lang. Inf..