Abstract
In the theory of computation on topological algebras there is a considerable gap between so-called abstract and concrete models of computation. In concrete models, unlike abstract models, the computations depend on the representation of the algebra. First, we show that with abstract models, one needs algebras with <i>partial operations</i>, and computable functions that are both <i>continuous</i> and <i>many-valued</i>. This many-valuedness is needed even to compute single-valued functions, and so <i>abstract models must be nondeterministic even to compute deterministic problems</i>. As an abstract model, we choose the "while"-array programming language, extended with a nondeterministic "countable choice" assignment, called the <i><b>WhileCC*</b></i> model. Using this, we introduce the concept of <i>approximable many-valued computation</i> on metric algebras. For our concrete model, we choose metric algebras with <i>effective representations</i>. We prove:(1) for any metric algebra <i>A</i> with an effective representation α, <i><b>WhileCC*</b></i> approximability implies computability in α, and (2) also the converse, under certain reasonable conditions on <i>A</i>. From (1) and (2) we derive an equivalence theorem between abstract and concrete computation on metric partial algebras. We give examples of algebras where this equivalence holds.
- Aberth, O. 1980. Computable Analysis. MIT Press, Cambridge, MA.Google Scholar
- Aberth, O. 2001. Computable Calculus. Addison-Wesley, Reading, MA.Google Scholar
- Apt, K. and Olderog, E.-R. 1991. Verification of Sequential and Concurrent Programs. Springer-Verlag, Berlin, Germany. Google Scholar
- Blum, L., Cucker, F., Shub, M., and Smale, S. 1998. Complexity and Real Computation. Springer-Verlag, Berlin, Germany. Google Scholar
- Blum, L., Shub, M., and Smale, S. 1989. On a theory of computation and complexity over the real numbers: NP-completeness, recursive functions and universal machines. Bull. Amer. Math. Soc. 21, 1--46.Google Scholar
- Brattka, V. 1996. Recursive characterisation of computable real-valued functions and relations. Theoret. Comput. Sci. 162, 45--77. Google Scholar
- Brattka, V. 1999. Recursive and computable operations over topological structures. Ph.D. dissertation, FernUniversität Hagen, Fachbereich Informatik, Hagen, Germany. Informatik Berichte 255, FernUniversität Hagen, July 1999.Google Scholar
- Ceitin, G. 1959. Algebraic operators in constructive complete separable metric spaces. Dok. Akad. Nauk SSSR 128, 49--52.Google Scholar
- de Bakker, J. 1980. Mathematical Theory of Program Correctness. Prentice Hall, Englewood Cliffs, NJ. Google Scholar
- Dijkstra, E. 1976. A Discipline of Programming. Prentice Hall, Englewood Cliffs, NJ. Google Scholar
- Edalat, A. 1995. Dynamical systems, measures, and fractals via domain theory. Inform. Computat. 120, 32--48. Google Scholar
- Edalat, A. 1997. Domains for computation in mathematics, physics and exact real arithmetic. Bull. Symbol. Log. 3, 401--452.Google Scholar
- Engelking, R. 1989. General Topology. Heldermann Verlag, Berlin, Lemge, Germany.Google Scholar
- Goguen, J., Thatcher, J., and Wagner, E. 1978. An initial approach to the specification, correctness and implementation of abstract data types. In Current Trends in Programming Methodology, vol. 4: Data Structuring, R. Yeh, Ed. Prentice Hall, Englewood Cliffs, NJ, 80--149.Google Scholar
- Grzegorczyk, A. 1955. Computable functions. Fundamenta Mathematicae 42, 168--202.Google Scholar
- Grzegorczyk, A. 1957. On the defintions of computable real continuous functions. Fundamenta Mathematicae 44, 61--71.Google Scholar
- Heath, M. 1997. Scientific Computing: An Introductory Survey. McGraw-Hill, New York, NY. Google Scholar
- Lacombe, D. 1955. Extension de la notion de fonction récursive aux fonctions d'une ou plusieurs variables réelles, I, II, III. C.R. Acad. Sci. Paris. 240, 2470--2480; 241, 13--14, 151--153.Google Scholar
- Meseguer, J. and Goguen, J. 1985. Initiality, induction and computability. In Algebraic Methods in Semantics, M. Nivat and J. Reynolds, Eds. Cambridge University Press, Cambridge, U.K., 459--541. Google Scholar
- Moschovakis, Y. 1964. Recursive metric spaces. Fundamenta Mathematicae 55, 215--238.Google Scholar
- Odifreddi, P. 1999. Classical Recursion Theory, 21st ed. North Holland, Amsterdam, The Netherlands.Google Scholar
- Pour-El, M. and Richards, J. 1989. Computability in Analysis and Physics. Springer-Verlag, Berlin, Germany.Google Scholar
- Royden, H. 1963. Real Analysis. Macmillan, London, U.K.Google Scholar
- Spreen, D. 1998. On effective topological spaces. J. Symbol. Log. 63, 185--221.Google Scholar
- Spreen, D. 2001. Representations versus numberings: On the relationship of two computability notions. Theoret. Comput. Sci. 263, 473--499. Google Scholar
- Stephenson, K. 1996. An algebraic approach to syntax, semantics and computation. Ph.D. dissertation, Department of Computer Science, University of Wales, Swansea, U.K.Google Scholar
- Stewart, K. 1998. Concrete and abstract models of computation over metric algebras. Ph.D. dissertation, Department of Computer Science, University of Wales, Swansea, U.K.Google Scholar
- Stoltenberg-Hansen, V. and Tucker, J. 1988. Complete local rings as domains. J. Symbol. Log. 53, 603--624.Google Scholar
- Stoltenberg-Hansen, V. and Tucker, J. 1995. Effective algebras. In Handbook of Logic in Computer Science, S. Abramsky, D. Gabbay, and T. Maibaum, Eds. Vol. 4. Oxford University Press, Oxford, U.K., 357--526. Google Scholar
- Stoltenberg-Hansen, V. and Tucker, J. 1999. Concrete models of computation for topological algebras. Theoret. Comput. Sci. 219, 347--378. Google Scholar
- Taylor, A. and Lay, D. 1980. Introduction to Functional Analysis. John Wiley & Sons, New York, NY. Google Scholar
- Tucker, J. and Zucker, J. 1988. Program Correctness over Abstract Data Types, with Error-State Semantics. CWI Monographs, vol. 6. North Holland, Amsterdam, The Netherlands. Google Scholar
- Tucker, J. and Zucker, J. 1992a. Examples of semicomputable sets of real and complex numbers. In Constructivity in Computer Science: Summer Symposium, San Antonio, Texas, June 1991, J. Myers, Jr. and M. O'Donnell, Eds. Lecture Notes in Computer Science, vol. 613. Springer-Verlag, Berlin, Germany, 179--198. Google Scholar
- Tucker, J. and Zucker, J. 1992b. Theory of computation over stream algebras, and its applications. In Mathematical Foundations of Computer Science 1992: 17th International Symposium, Prague, I. Havel and V. Koubek, Eds. Lecture Notes in Computer Science, vol. 629. Springer-Verlag, Berlin, Germany, 62--80. Google Scholar
- Tucker, J. and Zucker, J. 1994. Computable functions on stream algebras. In Proof and Computation: NATO Advanced Study Institute International Summer School at Marktoberdorf, 1993, H. Schwichtenberg, Ed. Springer-Verlag, Berlin, Germany, 341--382.Google Scholar
- Tucker, J. and Zucker, J. 1999. Computation by 'while' programs on topological partial algebras. Theoret. Comput. Sci. 219, 379--420. Google Scholar
- Tucker, J. and Zucker, J. 2000. Computable functions and semicomputable sets on many-sorted algebras. In Handbook of Logic in Computer Science, S. Abramsky, D. Gabbay, and T. Maibaum, Eds. Vol. 5. Oxford University Press, Oxford, U.K., 317--523. Google Scholar
- Tucker, J. and Zucker, J. 2002a. Abstract computability and algebraic specification. ACM Trans. Computat. Log. 3, 279--333. Google Scholar
- Tucker, J. and Zucker, J. 2002b. Computable total functions on metric algebras, universal algebraic specifications and dynamical systems. Tech. rep. CAS 02-04-JZ, Department of Computing & Software, McMaster University, Hamilton, Ont., Canada. (Accepted for publication in Journal of Logic and Algebraic Programming.)Google Scholar
- Weihrauch, K. 2000. Computable Analysis: An Introduction. Springer-Verlag, Berlin, Germany. Google Scholar
- Wirsing, M. 1991. Algebraic specification. In Handbook of Theoretical Computer Science, Vol. B: Formal Methods and Semantics, J. van Leeuwen, Ed. North Holland, Amsterdam, The Netherlands, 675--788. Google Scholar
Index Terms
- Abstract versus concrete computation on metric partial algebras
Recommendations
Partial algebras and complexity of satisfiability and universal theory for distributive lattices, boolean algebras and Heyting algebras
Characterizations are given for the classes of partial subalgebras of distributive lattices, boolean algebras and Heyting algebras. Thereby, complexity results are obtained for the satisfiability of quantifier-free first-order sentences in these ...
Computation by “while” programs on topological partial algebras
Special issue on computability and complexity in analysisThe uniqueness condition for the double pushout transformation of algebras
The double pushout approach to the algebraic graph transformation of hypergraphs was invented 30 years ago and it has been generalized since then to more general objects, like for instance relational systems or total and partial unary algebras. We have ...
Comments