Abstract
The elementary algebraic specifications form a small subset of the range of techniques available for algebraic specifications and are based on equational specifications with hidden functions and sorts and initial algebra semantics. General methods exist to show that all semicomputable and computable algebras can be characterised up to isomorphism by such specifications. Here we consider these specification methods for specific computable rational number arithmetics. In particular, we give an elementary equational specification of the 0-totalised rational function field ℚ0(X) with its degree operator as an auxiliary function.
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References
Adamek, J., Hebert, M., Rosicky, J.: On abstract data types presented by multiequations. Theoretical Computer Science 275, 427–462 (2002)
Bergstra, J.A., Tucker, J.V.: The completeness of the algebraic specification methods for data types. Information and Control 54, 186–200 (1982)
Bergstra, J.A., Tucker, J.V.: Initial and final algebra semantics for data type specifications: two characterisation theorems. SIAM Journal on Computing 12, 366–387 (1983)
Bergstra, J.A., Tucker, J.V.: Algebraic specifications of computable and semicomputable data types. Theoretical Computer Science 50, 137–181 (1987)
Bergstra, J.A., Tucker, J.V.: Equational specifications, complete term rewriting systems, and computable and semicomputable algebras. Journal of ACM 42, 1194–1230 (1995)
Bergstra, J.A., Tucker, J.V.: The data type variety of stack algebras. Annals of Pure and Applied Logic 73, 11–36 (1995)
Bergstra, J.A., Tucker, J.V.: The rational numbers as an abstract data type (submitted)
Bergstra, J.A., Tucker, J.V.: Elementary algebraic specifications of the rational complex numbers (submitted)
Calkin, N., Wilf, H.S.: Recounting the rationals. American Mathematical Monthly 107, 360–363 (2000)
Contejean, E., Marche, C., Rabehasaina, L.: Rewrite systems for natural, integral, and rational arithmetic. In: Comon, H. (ed.) RTA 1997. LNCS, vol. 1232, pp. 98–112. Springer, Heidelberg (1997)
Goguen, J.A.: Memories of ADJ. Bulletin of the European Association for Theoretical Computer Science 36, 96–102 (1989)
Goguen, J.A.: Tossing algebraic flowers down the great divide. In: Calude, C.S. (ed.) People and ideas in theoretical computer science, pp. 93–129. Springer, Singapore (1999)
Meseguer, J., Goguen, J.A.: Initiality, induction, and computability. In: Nivat, M. (ed.) Algebraic methods in semantics, pp. 459–541. Cambridge University Press, Cambridge (1986)
Meseguer, J., Goguen, J.A.: Remarks on remarks on many-sorted algebras with possibly emtpy carrier sets. Bulletin of the EATCS 30, 66–73 (1986)
Goguen, J.A., Diaconescu, R.: An Oxford Survey of Order Sorted Algebra. Mathematical Structures in Computer Science 4, 363–392 (1994)
Goguen, J.A., Thatcher, J.W., Wagner, E.G., Wright, J.B.: Initial algebra semantics and continuous algebras. Journal of ACM 24, 68–95 (1977)
Goguen, J.A., Thatcher, J.W., Wagner, E.G.: An initial algebra approach to the specification, correctness and implementation of abstract data types. In: Yeh, R.T. (ed.) Current trends in programming methodology. IV. Data structuring, pp. 80–149. Prentice-Hall, Engelwood Cliffs (1978)
Hodges, W.: Model Theory. Cambridge University Press, Cambridge (1993)
Kamin, S.: Some definitions for algebraic data type specifications. SIGLAN Notices 14(3), 28 (1979)
Khoussainov, B.: Randomness, computability, and algebraic specifications. Annals of Pure and Applied Logic, 1–15 (1998)
Khoussainov, B.: On algebraic specifications of abstract data types. In: Baaz, M., Makowsky, J.A. (eds.) CSL 2003. LNCS, vol. 2803, pp. 299–313. Springer, Heidelberg (2003)
Marongiu, G., Tulipani, S.: On a conjecture of Bergstra and Tucker. Theoretical Computer Science 67 (1989)
Meinke, K., Tucker, J.V.: Universal algebra. In: Abramsky, S., Gabbay, D., Maibaum, T. (eds.) Handbook of Logic in Computer Science, Mathematical Structures, vol. I, pp. 189–411. Oxford University Press, Oxford (1992)
Meseguer, J., Goguen, J.A.: Initiality, induction and computability. In: Nivat, M., Reynolds, J. (eds.) Algebraic methods in semantics, pp. 459–541. Cambridge University Press, Cambridge (1985)
Moss, L.: Simple equational specifications of rational arithmetic. Discrete Mathematics and Theoretical Computer Science 4, 291–300 (2001)
Moss, L., Meseguer, J., Goguen, J.A.: Final algebras, cosemicomputable algebras, and degrees of unsolvability. Theoretical Computer Science 100, 267–302 (1992)
Stoltenberg-Hansen, V., Tucker, J.V.: Effective algebras. In: Abramsky, S., Gabbay, D., Maibaum, T. (eds.) Handbook of Logic in Computer Science, Semantic Modelling, vol. IV, pp. 357–526. Oxford University Press, Oxford (1995)
Stoltenberg-Hansen, V., Tucker, J.V.: Computable rings and fields. In: Griffor, E. (ed.) Handbook of Computability Theory, pp. 363–447. Elsevier, Amsterdam (1999)
Terese: Term Rewriting Systems. Cambridge Tracts in Theoretical Computer Science, vol. 55. Cambridge University Press, Cambridge (2003)
Wagner, E.: Algebraic specifications: some old history and new thoughts. Nordic Journal of Computing 9, 373–404 (2002)
Wechler, W.: Universal algebra for computer scientists. EATCS Monographs in Computer Science. Springer, Heidelberg (1992)
Wirsing, M.: Algebraic specifications. In: van Leeuwen, J. (ed.) Handbook of Theoretical Computer Science, Formal models and semantics, vol. B, pp. 675–788. North-Holland, Amsterdam (1990)
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Bergstra, J.A. (2006). Elementary Algebraic Specifications of the Rational Function Field. 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_5
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DOI: https://doi.org/10.1007/11780342_5
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