Notre Dame Journal of Formal Logic

Program Size Complexity for Possibly Infinite Computations

Verónica Becher, Santiago Figueira, André Nies, and Silvana Picchi

Abstract

We define a program size complexity function $H^\infty$ as a variant of the prefix-free Kolmogorov complexity, based on Turing monotone machines performing possibly unending computations. We consider definitions of randomness and triviality for sequences in ${\{0,1\}}^\omega$ relative to the $H^\infty$ complexity. We prove that the classes of Martin-Löf random sequences and $H^\infty$-random sequences coincide and that the $H^\infty$-trivial sequences are exactly the recursive ones. We also study some properties of $H^\infty$ and compare it with other complexity functions. In particular, $H^\infty$ is different from $H^A$, the prefix-free complexity of monotone machines with oracle A.

Article information

Source
Notre Dame J. Formal Logic, Volume 46, Number 1 (2005), 51-64.

Dates
First available in Project Euclid: 31 January 2005

Permanent link to this document
https://projecteuclid.org/euclid.ndjfl/1107220673

Digital Object Identifier
doi:10.1305/ndjfl/1107220673

Mathematical Reviews number (MathSciNet)
MR2131546

Zentralblatt MATH identifier
1102.68036

Subjects
Primary: 68Q30: Algorithmic information theory (Kolmogorov complexity, etc.) [See also 03D32] 68Q05: Models of computation (Turing machines, etc.) [See also 03D10, 68Q12, 81P68]

Keywords
program size complexity Kolmogorov complexity infinite computations

Citation

Becher, Verónica; Figueira, Santiago; Nies, André; Picchi, Silvana. Program Size Complexity for Possibly Infinite Computations. Notre Dame J. Formal Logic 46 (2005), no. 1, 51--64. doi:10.1305/ndjfl/1107220673. https://projecteuclid.org/euclid.ndjfl/1107220673


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