Computing A Glimpse of Randomness

Computing A Glimpse of Randomness - arXiv:nlin.CD/0112022...

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Unformatted text preview: arXiv:nlin.CD/0112022 v1 17 Dec 2001 Computing A Glimpse of Randomness Cristian S. Calude, Michael J. Dinneen, Chi-Kou Shu Department of Computer Science, University of Auckland, Private Bag 92019, Auckland, New Zealand E-mails: { cristian,mjd,cshu004 } @cs.auckland.ac.nz Abstract A Chaitin Omega number is the halting probability of a universal Chaitin (self- delimiting Turing) machine. Every Omega number is both computably enumerable (the limit of a computable, increasing, converging sequence of rationals) and ran- dom (its binary expansion is an algorithmic random sequence). In particular, every Omega number is strongly non-computable. The aim of this paper is to describe a procedure, which combines Java programming and mathematical proofs, for com- puting the exact values of the first 63 bits of a Chaitin Omega: 000000100000010000100000100001110111001100100111100010010011100. Full description of programs and proofs will be given elsewhere. 1 Introduction Any attempt to compute the uncomputable or to decide the undecidable is without doubt challenging, but hardly new (see, for example, Marxen and Buntrock [24], Stewart [32], Casti [10]). This paper describes a hybrid procedure (which combines Java program- ming and mathematical proofs) for computing the exact values of the first 63 bits of a concrete Chaitin Omega number , Ω U , the halting probability of the universal Chaitin (self-delimiting Turing) machine U , see [15]. Note that any Omega number is not only uncomputable, but random, making the computing task even more demanding. Computing lower bounds for Ω U is not difficult: we just generate more and more halting programs. Are the bits produced by such a procedure exact? Hardly . If the first bit of the approximation happens to be 1, then sure, it is exact. However, if the provisional bit given by an approximation is 0, then, due to possible overflows, nothing prevents the first bit of Ω U to be either 0 or 1. This situation extends to other bits as well. Only an initial run of 1’s may give exact values for some bits of Ω U . The paper is structured as follows. Section 2 introduces the basic notation. Com- putably enumerable (c.e.) reals, random reals and c.e. random reals are presented in Section 3. Various theoretical difficulties preventing the exact computation of any bits of an Omega number are discussed in Section 4. The register machine model of Chaitin [15] is discussed in Section 5. In section 6 we summarize our computational results con- cerning the halting programs of up to 84 bits long for U . They give a lower bound for Ω U which is proved to provide the exact values of the first 63 digits of Ω U in Section 7. Chaitin [13] has pointed out that the self-delimiting Turing machine constructed in the preliminary version of this paper [9] is universal in the sense of Turing (i.e., it is capable to simulate any self-delimiting Turing machine), but it is not universal in the sense of algorithmic information theory because the “price” of simulation is not bounded...
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This note was uploaded on 09/23/2010 for the course MATH 1121 taught by Professor Dr.mcgrawhill during the Spring '10 term at SUNY Buffalo.

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Computing A Glimpse of Randomness - arXiv:nlin.CD/0112022...

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