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- W171088929 abstract "The thesis is concerned the question of characterizing those computably enumerable (c.e.) classes of computably enumerable sets which have repetition (an injective enumeration). This problem can be traced back to 1958, when Friedberg proved that the class of all computably enumerable sets can be injectively enumerated. We go beyond the scope of the literature by extending the study to the problem of characterizing the c.e. classes which are c.e. bounded number of repetitions and repetitions.An investigation of the question restricted to classes of cofinite sets leads to satisfying answer in special case but demonstrates the difficulties of the general problem.The property of class of being c.e. at most repetitions is shown to behave more naturally than injective enumerability. We prove an extension theorem characterization as corollary. We also show that the corresponding statement does not hold for the property of being c.e. bounded repetitions.By fixing an ${cal C}$ of all computably enumerable classes it is possible to define index sets corresponding to properties of such classes. The index sets related to problems of enumerability various constraints on the number of repetitions are shown to be complete at their natural level in the Arithmetical Hierarchy. For example: ${e : {cal rm has a computable enumeration without repetition}$ is $Sigmasb6$-complete, and ${e : {cal C}sp{(e)} rm has a computable enumeration with finite repetitions}$ is $Sigmasb6$-complete." @default.
- W171088929 created "2016-06-24" @default.
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- W171088929 date "1995-01-01" @default.
- W171088929 modified "2023-09-24" @default.
- W171088929 title "Computable enumeration and the problem of repetition" @default.
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