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Recursively Enumerable Reals and Chaitin Ω Numbers
"... A real is called recursively enumerable if it is the limit of a recursive, increasing, converging sequence of rationals. Following Solovay [23] and Chaitin [10] we say that an r.e. real dominates an r.e. real if from a good approximation of from below one can compute a good approximation of from b ..."
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Cited by 39 (3 self)
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A real is called recursively enumerable if it is the limit of a recursive, increasing, converging sequence of rationals. Following Solovay [23] and Chaitin [10] we say that an r.e. real dominates an r.e. real if from a good approximation of from below one can compute a good approximation of from below. We shall study this relation and characterize it in terms of relations between r.e. sets. Solovay's [23]like numbers are the maximal r.e. real numbers with respect to this order. They are random r.e. real numbers. The halting probability ofa universal selfdelimiting Turing machine (Chaitin's Ω number, [9]) is also a random r.e. real. Solovay showed that any Chaitin Ω number islike. In this paper we show that the converse implication is true as well: any Ωlike real in the unit interval is the halting probability of a universal selfdelimiting Turing machine.
Weakly Useful Sequences
, 2004
"... An infinite binary sequence x is defined to be (i) strongly useful if there is a computable time bound within which every decidable sequence is Turing reducible to x; and (ii) weakly useful if there is a computable time bound within which all the sequences in a nonmeasure 0 subset of the set of dec ..."
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Cited by 7 (2 self)
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An infinite binary sequence x is defined to be (i) strongly useful if there is a computable time bound within which every decidable sequence is Turing reducible to x; and (ii) weakly useful if there is a computable time bound within which all the sequences in a nonmeasure 0 subset of the set of decidable sequences are Turing reducible to x. Juedes,
On the dynamic qualitative behaviour of universal computation
 Complex Systems
, 2012
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unknown title
, 2009
"... Existence of biological uncertainty principle implies that we can never find ’THE ’ measure for biological complexity. ..."
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Existence of biological uncertainty principle implies that we can never find ’THE ’ measure for biological complexity.
Emergence in and Engineering of Complex MAS
"... As far as the laws of Mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality (Albert Einstein 1) We discuss the implications of emergence and complexity for the engineering of MAS. In particular, we argue that while formalisms may play a role ..."
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As far as the laws of Mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality (Albert Einstein 1) We discuss the implications of emergence and complexity for the engineering of MAS. In particular, we argue that while formalisms may play a role in specification and implementation of MAS, they do not provide predictive power in complex systems. Thus we must look to other tools for understanding these systems. Statistical methods may go some way to helping us, but they too have their limitations. Rather, we argue that we must revert to classic scientific method: observing the systems, positing theories and hypotheses, then testing and improving them. That the study of complex MAS has to be more of a natural (as opposed to formal) science. This has the consequence that there will be severe limitations on the extent to which one can “design ” complex MAS. We illustrate this case with two examples of MAS, both of which display system level dynamics that cannot directly be predicted from the behaviour of individuals. We call upon those in the ESOA community to explicitly reject those tenets that are only useful with simple MAS. 1.