Algebraic Model of an Arithmetic Unit for TTE-Computable Normalized Rational Numbers

  • Gregorio de Miguel Casado
  • Juan Manuel García Chamizo
  • María Teresa Signes Pont
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4497)


A formal specification of an arithmetic unit for computable normalized rational numbers is proposed. This specification, developed under the scope of the paradigm known as algebraic models of processors, exploits the connection between the signed digit representation for rational numbers in Type-2 Theory of Effectivity and online arithmetic in Computer Arithmetic. The proposal aims for specification formalization and calculation reliability together with implementation feasibility.


formal methods for VLSI design Type-2 Theory of Effectivity online arithmetic 


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  1. 1.
    Andrews, D., Sass, R., Anderson, E., Agron, J., Peck, W., Stevens, J., Baijot, F., Komp, E.: The Case for High Level Programming Models for Reconfigurable Computers. In: Proc. of the 2006 International Conference on Engineering of Reconfigurable Systems & Algorithms, pp. 21–32 (2006)Google Scholar
  2. 2.
    Avizienis, A.: Signed-digit number representations for fast parallel arithmetic. IRE Trans. Electronic Computers 10, 389–400 (1961)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Blanck, J.: Exact real arithmetic systems: Results of competition, Computability and Complexity in Analysis. In: Blank, J., Brattka, V., Hertling, P. (eds.) CCA 2000. LNCS, vol. 2064, pp. 390–394. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  4. 4.
    Blanck, J.: Exact real arithmetic using centred intervals and bounded error terms. Journal of Logic and Algebraic Programming 66, 207–240 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Borkar, S.: Getting Gigascale Chips: Challenges and Opportunities in Continuing Moore’s Law. ACM Queue 1(7), 26–33 (2003)CrossRefGoogle Scholar
  6. 6.
    de Miguel Casado, G., García Chamizo, J.M.: The Role of Algebraic Models and Type-2 Theory of Effectivity in Special Purpose Processor Design. In: Beckmann, A., Berger, U., Löwe, B., Tucker, J.V. (eds.) CiE 2006. LNCS, vol. 3988, pp. 137–146. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  7. 7.
    Ercegovac, M.D., Lang, T.: Digital Arithmetic. M. Kaufmann, Seattle (2004)Google Scholar
  8. 8.
    Fox, A.C.J., Harman, N.A.: Algebraic Models of Correctness for Abstract Pipelines. The Journal of Algebraic and Logic Programming 57(1-2), 71–107 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Gowland, P., Lester, D.: A Survey of Exact Arithmetic Implementations. In: Blank, J., Brattka, V., Hertling, P. (eds.) CCA 2000. LNCS, vol. 2064, pp. 30–47. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  10. 10.
    Harman, N.A. and Tucker, J.V.: Algebraic models of microprocessors: the verification of a simple computer. In: V Stravridou (ed), Mathematics of Dependable Systems II. Oxford : Clarendon Press; New York, Oxford University Press, (1997) 135–170Google Scholar
  11. 11.
    Harman, N.A.: Models of Timing Abstraction in Simultaneous Multithreaded and Multi-Core Processors, Logical Approaches to Computational Barriers, Report Series, Vol. CSR 7-2006, (2006) 129–139Google Scholar
  12. 12.
    Hayes, B.: A Lucid Interval. American Scientist 91, 484–488 (2003)CrossRefGoogle Scholar
  13. 13.
    ISSCC Roundtable: Embedded Memories for the Future, IEEE Design and Test of Computers, vol. 20, pp. 66–81 (2003)Google Scholar
  14. 14.
    Lynch, T., Schulte, M.: A High Radix On-line Arithmetic for Credible and Accurate Computing. Journal of UCS 1, 439–453 (1995)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Post, D.E., Votta, L.G.: Computational science demands a new paradigm. Physics today 58(1), 35–41 (2005)CrossRefGoogle Scholar
  16. 16.
    Schröder, M.: Admissible Representations in Computable Analysis. In: Calamoneri, T., Finocchi, I., Italiano, G.F. (eds.) CIAC 2006. LNCS, vol. 3998, pp. 471–480. Springer, Heidelberg (2006)Google Scholar
  17. 17.
    Weihrauch, K.: Computable Analysis. Springer, Heidelberg (2000)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Gregorio de Miguel Casado
    • 1
  • Juan Manuel García Chamizo
    • 1
  • María Teresa Signes Pont
    • 1
  1. 1.SPA-Lab,University of Alicante, 03690, San Vicente del Raspeig, AlicanteSpain

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