Abstract
We provide several machine-independent characterizations of deterministic complexity classes in the model of computation proposed by L. Blum, M. Shub and S. Smale. We provide a characterization of partial recursive functions over any arbitrary structure. We show that polynomial time computable functions over any arbitrary structure can be characterized in term of safe recursive functions. We show that polynomial parallel time decision problems over any arbitrary structure can be characterized in terms of safe recursive functions with substitutions.
This author has been partially supported by City University of Hong Kong SRG grant 7001290.
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References
S. Bellantoni and S. Cook. A new recursion-theoretic characterization of the poly-time functions. Computational Complexity, 2:97–110, 1992.
Lenore Blum, Felipe Cucker, Michael Shub, and Steve Smale. Complexity and Real Computation. Springer Verlag, 1998.
Olivier Bournez, Jean-Yves Marion, and Paulin de Naurois. Safe recursion over an arbitrary structure: Deterministic polynomial time. Technical report, LORIA, 2002.
Lenore Blum, Mike Shub, and Steve Smale. On a theory of computation and complexity over the real numbers: Np-completeness, recursive functions, and universal machines. Bulletin of the American Mathematical Society, 21:1–46, 1989.
P. Clote. Computational models and function algebras. In D. Leivant, editor, LCC’94, volume 960, pages 98–130, 1995.
A. Cobham. The intrinsic computational difficulty of functions. In Y. Bar-Hillel, editor, Proceedings of the International Conference on Logic, Methodology, and Philosophy of Science, pages 24–30. North-Holland, Amsterdam, 1962.
S. A. Cook. Computability and complexity of higher-type functions. In Y. Moschovakis, editor, Logic from Computer Science, pages 51–72. Springer-Verlag, New York, 1992.
F. Cucker, M. Shub, and S. Smale. Separation of complexity classes in Koiran’s weak model. Theoretical Computer Science, 133(1):3–14, 11 October1994.
F. Cucker. Pℝ≠ NCℝ. Journal of Complexity, 8:230–238, 1992.
Heinz-Dieter Ebbinghaus and Jörg Flum. Finite Model Theory. Perspectives in Mathematical Logic. Springer-Verlag, Berlin, 1995.
R. Fagin. Generalized first order spectra and polynomial time recognizable sets. In R. Karp, editor, Complexity of Computation, pages 43–73. SIAMAMS, 1974.
J. B. Goode. Accessible telephone directories. The Journal of Symbolic Logic, 59(1):92–105, March 1994.
Y. Gurevich. Algebras of feasible functions. In Twenty Fourth Symposium on Foundations of Computer Science, pages 210–214. IEEE Computer Society Press, 1983.
M. Hofmann. Type systems for polynomial-time computation, 1999. Habilitation.
N. Immerman. Descriptive Complexity. Springer, 1999.
N. Jones. The expressive power of higher order types or, life without cons. 2000.
D. Leivant. Intrinsic theories and computational complexity. In LCC’94, number 960, pages 177–194, 1995.
D. Leivant and J-Y Marion. Lambda calculus characterizations of polytime. fi, 19(1,2):167,184, September 1993.
Daniel Leivant and Jean-Yves Marion. Ramified recurrence and computational complexity II: substitution and poly-space. In L. Pacholski and J. Tiuryn, editors, Computer Science Logic, 8th Workshop, CSL’94, volume 933 of Lecture Notes in Computer Science, pages 369–380, Kazimierz, Poland, 1995. Springer.
K. Meer. A note on a P ≠NP result for a restricted class of real machines. Journal of Complexity, 8:451–453, 1992.
Christian Michaux. Une remarque `a propos des machines sur _ introduites par Blum, Shub et Smale. In C. R. Acad. Sc. de Paris, volume 309 of 1, pages 435–437. 1989.
J-Y Marion and J-Y Moyen. Efficient first order functional program interpreter with time bound certifications. In LPAR, volume 1955, pages 25–42. Springer, Nov 2000.
Bruno Poizat. Les petits cailloux. aléas, 1995.
B. Kapron R. Irwin and J. Royer. On characterizations of the basic feasible functionals. 11:117–153, 2001.
V. Sazonov. Polynomial computability and recursivity in finite domains. Elektronische Informationsverarbeitung und Kybernetik, 7:319–323, 1980.
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Bournez, O., Cucker, F., de Naurois, P.J., Marion, JY. (2003). Computability over an Arbitrary Structure. Sequential and Parallel Polynomial Time. In: Gordon, A.D. (eds) Foundations of Software Science and Computation Structures. FoSSaCS 2003. Lecture Notes in Computer Science, vol 2620. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-36576-1_12
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