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Trivial Reals
"... Solovay showed that there are noncomputable reals ff such that H(ff _ n) 6 H(1n) + O(1), where H is prefixfree Kolmogorov complexity. Such Htrivial reals are interesting due to the connection between algorithmic complexity and effective randomness. We give a new, easier construction of an Htrivi ..."
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Solovay showed that there are noncomputable reals ff such that H(ff _ n) 6 H(1n) + O(1), where H is prefixfree Kolmogorov complexity. Such Htrivial reals are interesting due to the connection between algorithmic complexity and effective randomness. We give a new, easier construction of an Htrivial real. We also analyze various computabilitytheoretic properties of the Htrivial reals, showing for example that no Htrivial real can compute the halting problem. Therefore, our construction of an Htrivial computably enumerable set is an easy, injuryfree construction of an incomplete computably enumerable set. Finally, we relate the Htrivials to other classes of "highly nonrandom " reals that have been previously studied.
EXTENSIONS OF EMBEDDINGS BELOW COMPUTABLY ENUMERABLE DEGREES
, 2010
"... Abstract. Toward establishing the decidability of the two quantifier theory of the ∆0 2 Turing degrees with join, we study extensions of embeddings of uppersemilattices into the initial segments of Turing degrees determined by computably enumerable sets, in particular the degree of the halting set ..."
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Abstract. Toward establishing the decidability of the two quantifier theory of the ∆0 2 Turing degrees with join, we study extensions of embeddings of uppersemilattices into the initial segments of Turing degrees determined by computably enumerable sets, in particular the degree of the halting set 0 ′. We obtain a good deal of sufficient and necessary conditions. Contents
DISCONTINUOUS PHENOMENA
"... ABSTRACT. We discuss the relationship between discontinuity and definability in the Turing degrees, with particular reference to degree invariant solutions to Post’s Problem. Proofs of new results concerning definability in lower cones are outlined. 1. ..."
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ABSTRACT. We discuss the relationship between discontinuity and definability in the Turing degrees, with particular reference to degree invariant solutions to Post’s Problem. Proofs of new results concerning definability in lower cones are outlined. 1.
Π 0 1 CLASSES AND STRONG DEGREE SPECTRA OF RELATIONS
"... Abstract. We study the weak truthtable and truthtable degrees of the images of subsets of computable structures under isomorphisms between computable structures. In particular, we show that there is a low c.e. set that is not weak truthtable reducible to any initial segment of any scattered compu ..."
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Abstract. We study the weak truthtable and truthtable degrees of the images of subsets of computable structures under isomorphisms between computable structures. In particular, we show that there is a low c.e. set that is not weak truthtable reducible to any initial segment of any scattered computable linear order. Countable Π 0 1 subsets of 2 ω and Kolmogorov complexity play a major role in the proof.
Resolute sequences in . . .
, 2012
"... We study infinite sequences whose initial segment complexity is invariant under effective insertions of blocks of zeros inbetween their digits. Surprisingly, such resolute sequences may have nontrivial initial segment complexity. In fact, we show that they occur in many well known classes from comp ..."
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We study infinite sequences whose initial segment complexity is invariant under effective insertions of blocks of zeros inbetween their digits. Surprisingly, such resolute sequences may have nontrivial initial segment complexity. In fact, we show that they occur in many well known classes from computability theory, e.g. in every jump class and every high degree. Moreover there are degrees which consist entirely of resolute sequences, while there are degrees which do not contain any. Finally we establish connections with the contiguous c.e. degrees, the ultracompressible sequences, the anticomplex sequences thus demonstrating that this class is an interesting superclass of the sequences with trivial initial segment complexity.
STRONG DEGREE SPECTRA OF RELATIONS
, 2008
"... For my husband Andrew, and children Prudence, Jancis, and Rutherford. One of the main areas of study in computable model theory is examining how certain aspects of a computable structure may change under an isomorphism to another computable structure. Let A be a computable structure, and let R be an ..."
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For my husband Andrew, and children Prudence, Jancis, and Rutherford. One of the main areas of study in computable model theory is examining how certain aspects of a computable structure may change under an isomorphism to another computable structure. Let A be a computable structure, and let R be an additional relation on the domain of A, so it is not named in the language of A. Harizanov defined the Turing degree spectrum of R on A to be the set of all Turing degrees of the images of R under all isomorphisms from A onto computable structures. Similarly, we define this notion for strong degrees such as weak truthtable degrees and truthtable degrees. We show that the conditions necessary for the Turing degree spectrum to contain all Turing degrees, found by Harizanov, are also enough to have the truthtable degree spectrum to contain all truthtable degrees. We further study the degreetheoretic complexity of initial segments of computable linear orderings. In particular, let L be a computable linear ordering of order type ω+ω ∗. Harizanov showed that the Turing degree spectrum of the ωpart of L is all of the limit computable
RESOLUTE SEQUENCES IN INITIAL SEGMENT COMPLEXITY
"... Abstract. We study infinite sequences whose initial segment complexity is invariant under effective insertions of blocks of zeros inbetween their digits. Surprisingly, such resolute sequences may have nontrivial initial segment complexity. In fact, we show that they occur in many well known classes ..."
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Abstract. We study infinite sequences whose initial segment complexity is invariant under effective insertions of blocks of zeros inbetween their digits. Surprisingly, such resolute sequences may have nontrivial initial segment complexity. In fact, we show that they occur in many well known classes from computability theory, e.g. in every jump class and every high degree. Moreover there are degrees which consist entirely of resolute sequences, while there are degrees which do not contain any. Finally we establish connections with the contiguous c.e. degrees, the ultracompressible sequences, the anticomplex sequences thus demonstrating that this class is an interesting superclass of the sequences with trivial initial segment complexity. 1.