Notre Dame Journal of Formal Logic

Program Size Complexity for Possibly Infinite Computations

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


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.

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

First available in Project Euclid: 31 January 2005

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Zentralblatt MATH identifier

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

program size complexity Kolmogorov complexity infinite computations


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.

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