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On the complexity of random strings (Extended Abstract)
 IN STACS 96
, 1996
"... We show that the set R of Kolmogorov random strings is truthtable complete. This improves the previously known Turing completeness of R and shows how the halting problem can be encoded into the distribution of random strings rather than using the time complexity of nonrandom strings. As an applic ..."
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Cited by 8 (1 self)
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We show that the set R of Kolmogorov random strings is truthtable complete. This improves the previously known Turing completeness of R and shows how the halting problem can be encoded into the distribution of random strings rather than using the time complexity of nonrandom strings. As an application we obtain that Post's simple set is truthtable complete in every Kolmogorov numbering. We also show that the truthtable completeness of R cannot be generalized to sizecomplexity with respect to arbitrary acceptable numberings. In addition we note that R is not frequency computable.
Reflections on Quantum Computing
, 2000
"... In this rather speculative note three problems pertaining to the power and limits of quantum computing are posed and partially answered: a) when are quantum speedups possible?, b) is fixedpoint computing a better model for quantum computing?, c) can quantum computing trespass the Turing barrier? 1 ..."
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In this rather speculative note three problems pertaining to the power and limits of quantum computing are posed and partially answered: a) when are quantum speedups possible?, b) is fixedpoint computing a better model for quantum computing?, c) can quantum computing trespass the Turing barrier? 1 When are quantum speedups possible? This section discusses the possibility that speedups in quantum computing can be achieved only for problems which have a few or even unique solutions [12]. For instance, this includes the computational complexity class UP [15]. Typical examples are Shor's quantum algorithm for prime factoring [18] and Grover's database search algorithm [13] for a single item satisfying a given condition in an unsorted database (see also Gruska [14]). In quantum complexity, one popular class of problems is BQP,whichisthe set of decision problems that can be solved in polynomial time (on a quantum computer) so that the correct answer is obtained with probability at l...
Chaitin Omega Numbers and Strong Reducibilities
, 1997
"... We prove that any Chaitin # number (i.e., the halting probability of a universal selfdelimiting Turing machine) is wttcomplete, but not ttcomplete. In this way we obtain a whole class of natural examples of wttcomplete but not ttcomplete r.e. sets. The proof is direct and elementary. 1 Introdu ..."
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We prove that any Chaitin # number (i.e., the halting probability of a universal selfdelimiting Turing machine) is wttcomplete, but not ttcomplete. In this way we obtain a whole class of natural examples of wttcomplete but not ttcomplete r.e. sets. The proof is direct and elementary. 1 Introduction Kucera [8] has used Arslanov's completeness criterion 1 to show that all random sets of r.e. Tdegree are in fact Tcomplete. Hence, every Chaitin # number is Tcomplete. In this paper we will strengthen this result by proving that every Chaitin # number is weak truthtable complete. However, no Chaitin # number can be ttcomplete as, because of a result stated by Bennett [1] (see Juedes, Lathrop, and Lutz [9] for a proof), there is no random sequence x such that K # tt x. 2 Notice that in this way we obtain a whole class of natural examples of wttcomplete but not ttcomplete r.e. sets (a fairly complicated construction of such a set was given by Lachlan [10]). # The first has...
Chaitin Ω numbers and halting problems
, 2009
"... ... 1975] introduced Ω number as a concrete example of random real. The real Ω is defined as the probability that an optimal computer halts, where the optimal computer is a universal decoding algorithm used to define the notion of programsize complexity. Chaitin showed Ω to be random by discovering ..."
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... 1975] introduced Ω number as a concrete example of random real. The real Ω is defined as the probability that an optimal computer halts, where the optimal computer is a universal decoding algorithm used to define the notion of programsize complexity. Chaitin showed Ω to be random by discovering the property that the first n bits of the basetwo expansion of Ω solve the halting problem of the optimal computer for all binary inputs of length at most n. In the present paper we investigate this property from various aspects. We consider the relative computational power between the basetwo expansion of Ω and the halting problem by imposing the restriction to finite size on both the problems. It is known that the basetwo expansion of Ω and the halting problem are Turing equivalent. We thus consider an elaboration of the Turing equivalence in a certain manner.