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Chaitin's constant
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== Super Omega == The first {{mvar|n}} bits of [[Gregory Chaitin]]'s constant {{math|Ω}} are random or incompressible in the sense that they cannot be computed by a halting algorithm with fewer than {{math|''n'' − O(1)}} bits. However, consider the short but never halting algorithm which systematically lists and runs all possible programs; whenever one of them halts its probability gets added to the output (initialized by zero). After finite time the first {{mvar|n}} bits of the output will never change any more (it does not matter that this time itself is not computable by a halting program). So there is a short non-halting algorithm whose output converges (after finite time) onto the first {{mvar|n}} bits of {{math|Ω}}. In other words, the [[enumerable]] first {{mvar|n}} bits of {{math|Ω}} are highly compressible in the sense that they are [[limit-computable]] by a very short algorithm; they are not [[random]] with respect to the set of enumerating algorithms. [[Jürgen Schmidhuber]] constructed a limit-computable "Super {{math|Ω}}" which in a sense is much more random than the original limit-computable {{math|Ω}}, as one cannot significantly compress the Super {{math|Ω}} by any enumerating non-halting algorithm.<ref>{{cite journal |last=Schmidhuber |first=Jürgen |author-link=Jürgen Schmidhuber |year=2002 |title=Hierarchies of generalized Kolmogorov complexities and nonenumerable universal measures computable in the limit |journal=International Journal of Foundations of Computer Science |<!--Preprint title: Algorithmic Theories of Everything-->arxiv=quant-ph/0011122 |volume=13 |pages=587–612 |doi=10.1142/S0129054102001291 |number=4}}</ref> For an alternative "Super {{math|Ω}}", the [[universality probability]] of a [[prefix-free code|prefix-free]] [[universal Turing machine]] (UTM){{Snd}} namely, the probability that it remains universal even when every input of it (as a [[binary string]]) is prefixed by a random binary string{{Snd}} can be seen as the non-halting probability of a machine with oracle the third iteration of the [[halting problem]] (i.e., {{math|O{{sup|(3)}}}} using [[Turing jump]] notation).<ref>{{cite journal |last1=Barmpalias |first1=G. |first2=Dowe |last2=D. L. |title=Universality probability of a prefix-free machine |journal=Philosophical Transactions of the Royal Society A |volume=370 |issue=1 | pages=3488–3511 <!--Theme Issue 'The foundations of computation, physics and mentality: the Turing legacy'--> |editor1-first=Barry |editor1-last=Cooper |editor2-first=Samson |editor2-last=Abramsky |date=2012 |doi=10.1098/rsta.2011.0319|pmid=22711870 |bibcode=2012RSPTA.370.3488B |doi-access=free }}</ref>
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