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# Recursively enumerable sets modulo iterated jumps and extensions of Arslanov's completeness criterion

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Let We be the eth recursively enumerable (r.e.) set in a standard enumeration. The fixed point form of Kleene's recursion theorem asserts that for every recursive function f there exists e which is a fixed point of f in the sense that We = Wf(e). In this paper our main concern is to study the degrees of functions with no fixed points. We consider both fixed points in the strict sense above and fixed points modulo various equivalence relations on recursively enumerable sets.

Our starting point for the investigation of the degrees of functions without (strict) fixed points is the following result due to M. M. Arslanov [A1, Theorem 1] and known as the Arslanov completeness criterion. Proofs of this result may also be found in [So1, Theorem 1.3] and [So2, Chapter 12], and we will give a game version of the proof in §5 of this paper.

Theorem 1.1 (Arslanov). Let A be an r.e. set. Then A is complete (i.e. A has degree0′) iff there is a function f recursive in A with no fixed point.

## References

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[A1]Arslanov, M. M., On some generalizations of a fixed point theorem, Izvestiya Vysshikh Uchebnykh Zavedeniǐ Matematika 1981, no. 5 (228), pp. 916; English translation, Soviet Mathematics (Iz. VUZ), vol. 25 (1981), no. 5, pp. 1–10.
[A2]Arslanov, M. M., On a hierarchy of the degrees of unsolvability, Probabilistic Methods and Cybernetics, vyp. 18, Kazanskiǐ Gosudarstvennyǐ Universitet, Kazan', 1982, pp. 1017. (Russian)
[A3]Arslanov, M. M., Families of recursively enumerable sets and their degrees of unsolvability, Izvestiya Vysshikh Uchebnykh Zavedeniǐ Matematika 1985, no. 4 (275), pp. 1319; English translation, Soviet Mathematics (Iz. VUZ), vol. 29 (1985), no. 4, pp. 13-21.
[A4]Arslanov, M. M., Recursively enumerable sets and degrees of unsolvability, Izdatel'stvo Kazanskogo Gosudarstvennogo Universiteta, Kazan', 1986. (Russian)
[A5]Arslanov, M. M., Completeness in the arithmetical hierarchy and fixed points, Algebra i Logika (to appear): English translation, Algebra and Logic (to appear).
[ANS]Arslanov, M. M., Nadirov, R. F., and Solov'ev, V. D., Completeness criteria for recursively enumerable sets and some general theorems on fixed points, Izvestiya Vysshikh Uchebnykh Zavedeniǐ Matematika 1977, No. 4 (179), pp. 37; English translation, Soviet Mathematics (Iz. VUZ), vol. 21 (1977), pp. 1–4.
[D]Demuth, O., A notion of semigenericity, Commentationes Mathematicae Universitatis Carolinae, vol. 28 (1987), pp. 7174.
[EHK]Epstein, R. L., Haas, R., and Kramer, R., Hierarchies of sets and degrees below0′, Logic Year 1979–80 (Lerman, M.et al, editors), Lecture Notes in Mathematics, vol. 859, Springer-Verlag, Berlin, 1981, pp. 3248.
[J]Jockusch, C. G., Degrees of functions with no fixed points, Proceedings of the eighth international congress for logic, methodology and philosophy of science (to appear).
[JSh]Jockusch, C. G. and Shore, R. A., Pseudo jump operators. II: Transfinite iterations, hierarchies, and minimal covers, this Journal, vol. 49 (1984), pp. 12051236.
[JSo]Jockusch, C. G. and Soare, R. I., classes and degrees of theories, Transactions of the American Mathematical Society, vol. 173 (1972), pp. 3356.
[K1]Kučera, A., Measure, Π classes, and complete extensions of PA, Recursion theory week (Ebbinghaus, H.-D.et al, editors), Lecture Notes in Mathematics, vol. 1141, Springer-Verlag, Berlin, 1985, pp. 245259.
[K2]Kučera, A., An alternative, priority free solution to Post's problem. Mathematical foundations of computer science 1986 (Gruska, J.et al, editors), Lecture Notes in Computer Science, vol. 233, Springer-Verlag, Berlin, 1986, pp. 493500.
[K3]Kučera, A., On the role of 0′ in recursion theory, Logic Colloquium '86 (Drake, F. R. and Truss, J. K., editors), North-Holland, Amsterdam, 1988, pp. 133141.
[K4]Kučera, A., On the use of diagonally nonrecursive functions, Logic Colloquium '87 (to appear).
[La]Lachlan, A. H., On some games which are relevant to the theory of recursively enumerable sets, Annals of Mathematics, ser. 2, vol. 91 (1970), pp. 291310.
[Le]Lempp, S., Topics in recursively enumerable sets and degrees, Ph.D. thesis, University of Chicago, Chicago, Illinois, 1986.
[R]Rogers, H. Jr., Theory of recursive functions and effective computability, McGraw-Hill, New York, 1967.
[RND]Reingold, E., Nievergelt, J., and Deo, N., Combinatorial algorithms: theory and practice, Prentice-Hall, Englewood Cliffs, New Jersey, 1977.
[Sa]Sacks, G. E., On a theorem of Lachlan and Martin, Proceedings of the American Mathematical Society, vol. 18 (1967), pp. 140141.
[Sehl]Schwarz, S., Index sets of recursively enumerable sets, quotient lattices, and recursive linear orderings, Ph.D. thesis, University of Chicago, Chicago, Illinois, 1982.
[Sch2]Schwarz, S., Index sets related to the high-low hierarchy, Israel Journal of Mathematics (to appear).
[Sco]Scott, D., Algebras of sets binumerable in complex extensions of arithmetic, Recursive function theory (Dekker, J. C. E., editor), Proceedings of Symposia in Pure Mathematics, vol. 5, American Mathematical Society, Providence, Rhode Island, 1962, pp. 117121.
[So1]Soare, R. I., Fundamental methods for constructing recursively enumerable degrees, Recursion theory: its generalisations and applications (Proceedings of Logic Colloquium '79; Drake, F. and Wainer, S. S., editors), London Mathematical Society Lecture Note Series, vol. 45, Cambridge University Press, Cambridge, 1980, pp. 151.
[So2]Soare, R. I., Recursively enumerable sets and degrees: a study of computable functions and computably generated sets, Springer-Verlag, Berlin, 1987.
[ST]Scott, D. and Tennenbaum, S., On the degrees of complete extensions of arithmetic, Notices of the American Mathematical Society, vol. 7 (1960), pp. 242243 (abstract #568-3).

# Recursively enumerable sets modulo iterated jumps and extensions of Arslanov's completeness criterion

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