Abstract
A real number is called left c.e. (right c.e.) if it is the limit of an increasing (decreasing) computable sequence of rational numbers. In particular, if a left c.e. real has a c.e. binary expansion, then it is called strongly c.e. While the strongly c.e. reals have nice computational properties, the class of strongly c.e. reals does not have good mathematical properties. In this paper, we show that, for any non-computable strongly c.e. real x, there are strongly c.e. reals \(y_1\) and \(y_2\) such that the difference \(x-y_1\) is neither left c.e., nor right c.e., and the sum \(x\,+\,{y}_2\) is not strongly c.e. Thus, the class of strongly c.e. reals is not closed under addition and subtraction in an extremely strong sense.
This research was done when the first author visited Arcadia University in the fall 2018. We appreciate very much the support of the Department of Mathematics and Computer Science, Arcadia University.
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Notes
- 1.
One of the referees pointed out an alternative proof of Theorem 1 which follows more easily from results in the literature. It is based on the observation that the difference of two left computable reals which are Solovay incomparable is not semi-computable. So, since Downey, Hirschfeldt and LaForte [9] have shown that Solovay reducibility coincides with computable Lipschitz reducibility on the strongly c.e. reals, it suffices to show that for any noncomputable c.e. set A there is a c.e. set B which is cl-incomparable with A. But, since Barmpalias [3] has shown that there are no maximal c.e. cl-degrees, this can be deduced from Sacks’ Splitting Theorem by duplicating the argument for cl-reducibility which we have given for ibT-reducibility in the remark following the proof of Lemma 2.
References
Ambos-Spies, K., Ding, D., Fan, Y., Merkle, W.: Maximal pairs of computably enumerable sets in the computably Lipschitz degrees. Theory Comput. Syst. 52(1), 2–27 (2013)
Ambos-Spies, K., Weihrauch, K., Zheng, X.: Weakly computable real numbers. J. Complex. 16(4), 676–690 (2000)
Barmpalias, G.: Computably enumerable sets in the solovay and the strong weak truth table degrees. In: Cooper, S.B., Löwe, B., Torenvliet, L. (eds.) CiE 2005. LNCS, vol. 3526, pp. 8–17. Springer, Heidelberg (2005). https://doi.org/10.1007/11494645_2
Barmpalias, G., Lewis-Pye, A.: A note on the differences of computably enumerable reals. In: Day, A., Fellows, M., Greenberg, N., Khoussainov, B., Melnikov, A., Rosamond, F. (eds.) Computability and Complexity. LNCS, vol. 10010, pp. 623–632. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-50062-1_37
Barmpalias, G., Lewis-Pye, A.: Differences of halting probabilities. J. Comput. Syst. Sci. 89, 349–360 (2017)
Calude, C.S., Hertling, P.H., Khoussainov, B., Wang, Y.: Recursively enumerable reals and Chaitin \(\varOmega \) numbers. Theor. Comput. Sci. 255, 125–149 (2001)
Downey, R., Terwijn, S.A.: Computably enumerable reals and uniformly presentable ideals. MLQ Math. Log. Q. 48(suppl. 1), 29–40 (2002)
Downey, R.G., Hirschfeldt, D.R.: Algorithmic Randomness and Complexity Theory and Application of Computability. Springer, Heidelberg (2010). https://doi.org/10.1007/978-0-387-68441-3
Downey, R.G., Hirschfeldt, D.R., LaForte, L.G.: Randomness and reducibility. J. Comput. Syst. Sci. 68(1), 96–114 (2004)
Downey, R.G., LaForte, G.L.: Presentations of computably enumerable reals. Theor. Comput. Sci. 284(2), 539–555 (2002)
Miller, J.S.: On work of Barmpalias and Lewis-Pye: a derivation on the D.C.E. reals. In: Day, A., Fellows, M., Greenberg, N., Khoussainov, B., Melnikov, A., Rosamond, F. (eds.) Computability and Complexity. LNCS, vol. 10010, pp. 644–659. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-50062-1_39
Robinson, R.M.: Review of Peter, R. Rekursive Funktionen. J. Symbol. Log. 16, 280–282 (1951)
Schröder, M.: Admissible representations in computable analysis. In: Beckmann, A., Berger, U., Löwe, B., Tucker, J.V. (eds.) CiE 2006. LNCS, vol. 3988, pp. 471–480. Springer, Heidelberg (2006). https://doi.org/10.1007/11780342_48
Sacks, G.E.: Degrees of Unsolvability. Princeton University Press, Princeton (1963)
Robert Irving Soare: Recursion theory and Dedekind cuts. Trans. Am. Math. Soc. 140, 271–294 (1969)
Turing, A.M.: On computable numbers, with an application to the “Entscheidungsproblem”. Proc. Lond. Math. Soc. 42(2), 230–265 (1936)
Weihrauch, K.: An introduction. In: Weihrauch, K. (ed.) Computable Analysis. Springer, Heidelberg (2000). https://doi.org/10.1007/978-3-642-56999-9_1
Guohua, W.: Regular reals. Math. Log. Q. 51(2), 111–119 (2005)
Zheng, X.: On the turing degrees of weakly computable real numbers. J. Log. Comput. 13(2), 159–172 (2003)
Zheng, X., Rettinger, R.: Weak computability and representation of reals. Math. Log. Q. 50(4/5), 431–442 (2004)
Zheng, X., Rettinger, R.: Computability of real numbers. In: Brattka, V., Hertling, P. (eds.) Handbook on Computability and Complexity in Analysis, Theory and Applications of Computability. Springer-Verlag (to appear)
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Ambos-Spies, K., Zheng, X. (2019). On the Differences and Sums of Strongly Computably Enumerable Real Numbers. In: Manea, F., Martin, B., Paulusma, D., Primiero, G. (eds) Computing with Foresight and Industry. CiE 2019. Lecture Notes in Computer Science(), vol 11558. Springer, Cham. https://doi.org/10.1007/978-3-030-22996-2_27
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