Abstract
We investigate the concept of semicomputability of relations on abstract structures. We consider three possible definitions of this concept, which all reduce to the classical notion of recursive enumerability over the natural numbers. By working in the algebra of the reals, with and without order, we find examples of sets which distinguish between these three notions. We also find interesting examples of sets of real and complex numbers which are semicomputable but not computable.
Research supported by SERC Research Grant GR/F 59070, by MRC Research Grant SPG 9017859, and by an academic travel grant from the British Council.
Research supported by a grant from the Science & Engineering Research Board of McMaster University, by a grant from the Natural Sciences and Engineering Research Council of Canada, and by an academic travel grant from the British Council.
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References
E. Becker (1986), On the real spectrum of a ring and its application to semialgebraic geometry, Bull. Amer. Math. Soc. (N.S.) 15, 19–60.
L. Blum, M. Shub and S. Smale (1989), On a theory of computation and complexity over the real numbers: NP-completeness, recursive functions and universal machines, Bull. Amer. Math. Soc. (N.S.) 21, 1–46.
T. Bröcker and L.C. Lander (1975), Differentiable Germs and Catastrophes, London Math. Soc.
E. Engeler (1968), Formal Languages: Automata and Structures, Markham Publ. Co.
E. Engeler (1975), On the solvability of algorithmic problems, Logic Colloquium '73 (H.E. Rose and J.C. Shepherdson, eds.), North-Holland, pp. 231–251.
H. Friedman (1971), Algorithmic procedures, generalised Turing algorithms and elememtary recursion theories, Logic Colloquium '69 (R.O. Gandy and C.M.E. Yates, eds.), North-Holland, pp. 361–389.
G.T. Herman and S.D. Isard (1970), Computability over arbitrary fields, J. London Math. Soc. (2) 2, 73–79.
S.C. Kleene (1952), Introduction to Metamathematics, North-Holland.
A. Kreczmar (1977), Programmability in fields, Fundamenta Informaticae 1, 195–230.
G. Kreisel and J.L. Krivine (1971), Elements of Mathematical Logic, North-Holland.
I.R. Shafarevich (1977), Basic Algebraic Geometry (transi. K.A. Hirsch), Springer-Verlag.
J. Shepherdson (1985), Algorithmic procedures, generalized Turing algorithms, and elementary recursion theory, Harvey Friedman's Research on the Foundations of Mathematics (L.A. Harrington, M.D. Morley, A. Ščedrov and S.G. Simpson, eds.), North-Holland, pp. 309–315.
A.S. Troelstra and D. van Dalen (1988), Constructivism in Mathematics: An Introduction, Vol. I, North-Holland.
J.V. Tucker (1980),. Computing in algebraic systems, Recursion Theory, Its Generalisations and Applications (F.R. Drake and S.S. Wainer, eds.), LMS/Cambridge University Press.
J. V. Tucker, S.S. Wainer and J. I. Zucker (1990), Provable computable functions on abstract data types, Proceedings of the 17th International Colloquium on Automata, Languages and Programming, Warwick University, England, July 1990 (M.S. Paterson, ed.), Lecture Notes in Computer Science 443, Springer-Verlag, pp. 745–760.
J.V. Tucker and J.I. Zucker (1988), Program Correctness over Abstract Data Types, with Error-State Semantics, CWI Monograph Series, North-Holland and the Centre for Mathematics and Computer Science (CWI), Amsterdam.
J.V. Tucker and J.I. Zucker (1989), Horn programs and semicomputable relations on abstract structures, Proceedings of the 16th International Colloquium on Automata, Languages and Programming, Stresa, Italy, July 1989, Lecture Notes in Computer Science 372, Springer-Verlag, pp. 745–760.
J.V. Tucker and J.I. Zucker (1991a), Provable computable selection functions on abstract structures, Leeds Proof Theory 1990 (P. Aczel, H. Simmons and S.S. Wainer, eds.), Cambridge University Press (to appear).
J.V. Tucker and J.I. Zucker (1991b), Deterministic and nondeterministic computation, and Horn programs, on abstract data structures, J. Logic Programming (to appear).
J.V. Tucker and J.I. Zucker (1991c), Specifications, relations and selection functions on abstract data types, Department of Computer Science & Systems, McMaster University, Technical Report 91-14.
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© 1992 Springer-Verlag Berlin Heidelberg
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Tucker, J.V., Zucker, J.I. (1992). Examples of semicomputable sets of real and complex numbers. In: Myers, J.P., O'Donnell, M.J. (eds) Constructivity in Computer Science. Constructivity in CS 1991. Lecture Notes in Computer Science, vol 613. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0021091
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DOI: https://doi.org/10.1007/BFb0021091
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