Abstract
Most of the existing work in real number computation theory concentrates on complexity issues rather than computability aspects. Though some natural problems like deciding membership in the Mandelbrot set or in the set of rational numbers are known to be undecidable in the Blum-Shub-Smale (BSS) model of computation over the reals, there has not been much work on different degrees of undecidability. A typical question into this direction is the real version of Post’s classical problem: Are there some explicit undecidable problems below the real Halting Problem?
In this paper we study three different topics related to such questions: First an extension of a positive answer to Post’s problem to the linear setting. We then analyze how additional real constants increase the power of a BSS machine. And finally a real variant of the classical word problem for groups is presented which we establish reducible to and from (that is, complete for) the BSS Halting problem.
Part of this work was done while K. Meer visited the Forschungsinstitut für Diskrete Mathematik at the University of Bonn, Germany. The hospitality during the stay is gratefully acknowledged. He was partially supported by the IST Programme of the European Community, under the PASCAL Network of Excellence, IST-2002-506778 and by the Danish Natural Science Research Council SNF. This publication only reflects the authors’ views.
M. Ziegler is supported by DFG (project Zi1009/1-1) and by JSPS (ID PE 05501).
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References
Baker, A.: Transcendental Number Theory. Camb. Univ. Press, Cambridge (1975)
Blum, L., Cucker, F., Shub, M., Smale, S.: Complexity and Real Computation. Springer, Heidelberg (1998)
Ben-David, S., Meer, K., Michaux, C.: A note on non-complete problems in NP ℝ. Journal of Complexity 16(1), 324–332 (2000)
Boone, W.W.: The word problem. Proc. Nat. Acad. Sci. U.S.A 44, 265–269 (1958)
Blum, L., Shub, M., Smale, S.: On a Theory of Computation and Complexity over the Real Numbers: \(\mathcal{NP}\)-Completeness, Recursive Functions, and Universal Machines. Bulletin of the American Mathematical Society (AMS Bulletin) 21, 1–46 (1989)
Chapuis, O., Koiran, P.: Saturation and stability in the theory of computation over the reals. Annals of Pure and Applied Logic 99, 1–49 (1999)
Cucker, F., Grigoriev, D.Y.: On the power of real turing machines over binary inputs. SIAM Journal on Computing 26(1), 243–254 (1997)
Cucker, F., Koiran, P.: Computing over the Real with Addition and Order: Higher Complexity Classes. Journal of Complexity 11, 358–376 (1995)
Cucker, F.: The arithmetical hierarchy over the reals. Journal of Logic and Computation 2(3), 375–395 (1992)
Deimling, K.: Nonlinear Functional Analysis. Springer, Heidelberg (1985)
Derksen, H., Jeandel, E., Koiran, P.: Quantum automata and algebraic groups. J. Symbolic Computation 39, 357–371 (2005)
Fournier, H., Koiran, P.: Lower Bounds Are Not Easier over the Reals. In: Welzl, E., Montanari, U., Rolim, J.D.P. (eds.) ICALP 2000. LNCS, vol. 1853, pp. 832–843. Springer, Heidelberg (2000)
Friedberg, R.M.: Two recursively enumerable sets of incomparable degrees of unsolvability. Proc. Natl. Acad. Sci. 43, 236–238 (1957)
Koiran, P.: A weak version of the Blum-Shub-Smale model. In: Proceedings FOCS 1993, pp. 486–495 (1993)
Koiran, P.: Computing over the Reals with Addition and Order. Theoretical Computer Science 133, 35–48 (1994)
Koiran, P.: Elimination of Constants from Machines over Algebraically Closed Fields. Journal of Complexity 13, 65–82 (1997)
Lyndon, R.C., Schupp, P.E.: Combinatorial Group Theory. Springer, Heidelberg (1977)
Michaux, C.: Machines sur les réels et problèmes \(\mathcal{NP}\)-complets. Séminaire de Structures Algébriques Ordonnées, Prépublications de l’equipe de logique mathématique de Paris 7 (1990)
Michaux, C.: Ordered rings over which output sets are recursively enumerable. In: Proceedings of the AMS, vol. 111, pp. 569–575 (1991)
Muchnik, A.A.: Solution of Post’s reduction problem and of certain other problems in the theory of algorithms. Trudy Moskov Mat. Obsc. 7, 391–405 (1958)
Meer, K., Ziegler, M.: An explicit solution to Post’s problem over the reals. In: Liśkiewicz, M., Reischuk, R. (eds.) FCT 2005. LNCS, vol. 3623, pp. 467–478. Springer, Heidelberg (2005)
Meer, K., Ziegler, M.: On the word problem for a class of groups with infinite presentations (preprint 2006)
Novikov, P.S.: On the algorithmic unsolvability of the word problem in group theory. Trudy Mat. Inst. Steklov 44, 1–143 (1959)
Post, E.L.: Recursively enumerable sets of positive integers and their decision problems. Bull. Amer. Math. Soc. 50, 284–316 (1944)
Tucker, J.V.: Computability and the algebra of fields. J. Symbolic Logic 45, 103–120 (1980)
Tucker, J.V., Zucker, J.I.: Computable functions and semicomputable sets on many sorted algebras. In: Abramskz, S., Gabbay, D., Maibaum, T. (eds.) Handbook of Logic for Computer Science. Logic and Algebraic Methods, vol. V, pp. 317–523. Oxford University Press, Oxford (2000)
Weihrauch, K.: Computable Analysis. Springer, Heidelberg (2001)
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Meer, K., Ziegler, M. (2006). Uncomputability Below the Real Halting Problem. In: Beckmann, A., Berger, U., Löwe, B., Tucker, J.V. (eds) Logical Approaches to Computational Barriers. CiE 2006. Lecture Notes in Computer Science, vol 3988. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11780342_39
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DOI: https://doi.org/10.1007/11780342_39
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