Phase Transition between Unidirectionality and Bidirectionality
The notion of weak truth-table reducibility plays an important role in recursion theory. In this paper, we introduce an elaboration of this notion, where a computable bound on the use function is explicitly specified. This elaboration enables us to deal with the notion of asymptotic behavior in a manner like in computational complexity theory, while staying in computability theory. We apply the elaboration to sets which appear in the statistical mechanical interpretation of algorithmic information theory. We demonstrate the power of the elaboration by revealing a critical phenomenon, i.e., a phase transition, in the statistical mechanical interpretation, which cannot be captured by the original notion of weak truth-table reducibility.
KeywordsBinary String Recursive Function Order Function Recursion Theory Consecutive Zero
Unable to display preview. Download preview PDF.
- 13.Reimann, J., Stephan, F.: On hierarchies of randomness tests. In: Proceedings of the 9th Asian Logic Conference, August 16-19, World Scientific Publishing, Novosibirsk (2005)Google Scholar
- 14.Solovay, R.M.: Draft of a paper (or series of papers) on Chaitin’s work... done for the most part during the period of September–December (1974); unpublished manuscript. IBM Thomas J. Watson Research Center, p. 215. Yorktown Heights, New York (May 1975)Google Scholar
- 16.Tadaki, K.: A statistical mechanical interpretation of algorithmic information theory. In: Local Proceedings of Computability in Europe 2008 (CiE 2008), June 15-20, pp. 425–434. University of Athens, Greece (2008); An Extended Version Available at arXiv:0801.4194v1Google Scholar
- 19.Tadaki, K.: A computational complexity-theoretic elaboration of weak truth-table reducibility. Research Report of CDMTCS 406 (July 2011)Google Scholar