Abstract
A recursive functiong is augmenting ifg(n) ≥n for alln. Ag-constructor for a setA is a Turing machine that on inputn outputs some element ofA betweenn andg(n), inclusive, if such an element exists. Ag-generator forA outputs all elements ofA betweenn andg(n). Ag-detector merely determines whether there exist any elements in the given range. Total constructors (generators, detectors) always halt, and indicate whether or not any elements exist in the set in the given range; partial machines need not halt in the event that no such elements exist.
We characterize the classes of sets having each of these types of machines. We also establish that the problems of determining, for a given recursively enumerable setA and augmenting functiong, whetherA has a total or partialg-generator or a totalg-constructor or detector are all ∑3-complete; and we determine, for fixedg, the relationships among the classes of sets having these different types of machines.
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Part of this work was done while the author was an AT&T Bell Labs Scholar at the Computer Science Department of the University of Rochester.
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Sanchis, L.A. On the effective generation of set elements within specified ranges. Math. Systems Theory 26, 327–341 (1993). https://doi.org/10.1007/BF01189853
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DOI: https://doi.org/10.1007/BF01189853