Skip to main content
SpringerLink
Log in

Formal proof of integer adders using all-prefix-sums operation

  • Research Paper
  • Published:
Science China Information Sciences Aims and scope Submit manuscript

Abstract

Addition of two binary numbers is a fundamental operation in electronic circuits. Several integer adder architectures have been proposed. Their formal properties are well known, but the proofs are either incomplete or difficult to find. In this paper, we present a formal proof for the correctness of prefix adders. Both sequential and parallel algorithms are formalized and proved. In contrast to previous proofs using higher order functions and rewriting systems, our work is based on first order recursive equations, which are familiar to the computer arithmetic community and are therefore understandable by people working on VLSI circuit design. This study sets up a basis for further work on formal proofs of computer arithmetic algorithms.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Koren I. Computer Arithmetic Algorithms. Natick: A K Peters, 2002

    MATH  Google Scholar 

  2. Ladner R, Fischer M. Parallel prefix computation. J ACM, 1980, 27: 831–838

    Article  MathSciNet  MATH  Google Scholar 

  3. Lakshmivarahan S, Dhall S K. Parallel Computing Using the Prefix Problem. USA: Oxford University Press, 1994

    Google Scholar 

  4. Blelloch G E. Scans as primitive parallel operations. IEEE Trans Comput, 1989, 38: 1526–1538

    Article  Google Scholar 

  5. Blelloch G E. Prefix sums and their applications. In: Synthesis of Parallel Algorithms. San Francisco: Morgan Kaufmann Publisher, 1993

    Google Scholar 

  6. Kogge P M, Stone H S. A parallel algorithm for the efficient solution of a general class of recurrence equations. IEEE Trans Comput, 1973, C-22: 786–793

    Article  MathSciNet  Google Scholar 

  7. Brent R P, Kung H T. A regular layout for parallel adders. IEEE Trans Comput, 1982, C-31: 260–264

    Article  MathSciNet  Google Scholar 

  8. O’donnell J T, Runger G. Derivation of a logarithmic time carry lookahead addition circuit. J Funct Program, 2004, 14: 697–713

    Article  MATH  Google Scholar 

  9. Sheeran M. Hardware design and functional programming: A perfect match. J Univ Comput Sci, 2005, 11: 1135–1158

    Google Scholar 

  10. Kapur D, Subramaniam M. Mechanical verification of adder circuits using rewrite rulelaboratory. Form Method Syst Des, 1998, 13: 127–158

    Article  Google Scholar 

  11. Chen G, Liu F. Proofs of correctness and properties of integer adder circuits. IEEE Trans Comput, 2010, 59: 134–136

    Article  MathSciNet  Google Scholar 

  12. Howie J M. Fundamentals of Semigroup Theory. USA: Oxford University Press, 1996

    Google Scholar 

  13. Knowles S. A family of adders. In: Proceedings of the 15th IEEE Symposium on Computer Arithmetic, Vail, USA, 2001. 277–281

Download references

Author information

Authors and Affiliations

  1. School of Computer Science, National University of Defense Technology, Changsha, 410073, China

    Feng Liu & QingPing Tan

  2. Department of Electrical and Computer Engineering, Concordia University, Montreal, H3G 1M8, Canada

    Otmane Ait Mohamed

Authors
  1. Feng Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. QingPing Tan
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Otmane Ait Mohamed
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Feng Liu.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Liu, F., Tan, Q. & Mohamed, O.A. Formal proof of integer adders using all-prefix-sums operation. Sci. China Inf. Sci. 55, 1949–1960 (2012). https://doi.org/10.1007/s11432-011-4331-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11432-011-4331-7

Keywords

Search

Navigation