Randomness and Recursive Enumerability

  • Authors:

  • Antonín Kucera;T. Slaman

  • Affiliations:

  • -;-

  • Venue:
  • SIAM Journal on Computing
  • Year:
  • 2002

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Abstract

One recursively enumerable real $\alpha$ dominates another one $\beta$ if there are nondecreasing recursive sequences of rational numbers $(a[n]:n\in\omega)$ approximating $\alpha$ and $(b[n]:n\in\omega)$ approximating $\beta$ and a positive constant C such that for all n, $C(\alpha-a[n])\geq(\beta-b[n])$. See [R. M. Solovay, Draft of a Paper (or Series of Papers) on Chaitin's Work, manuscript, IBM Thomas J. Watson Research Center, Yorktown Heights, NY, 1974, p. 215] and [G. J. Chaitin, IBM J. Res. Develop., 21 (1977), pp. 350--359]. We show that every recursively enumerable random real dominates all other recursively enumerable reals. We conclude that the recursively enumerable random reals are exactly the $\Omega$-numbers [G. J. Chaitin, IBM J. Res. Develop., 21 (1977), pp. 350--359]. Second, we show that the sets in a universal Martin-Löf test for randomness have random measure, and every recursively enumerable random number is the sum of the measures represented in a universal Martin-Löf test.