Skip to main content
SpringerLink
Log in

Expected Acceptance Counts for Finite Automata with Almost Uniform Input

  • Conference paper
  • First Online:
Book cover Algorithms and Computation (ISAAC 2002)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2518))

Included in the following conference series:

  • 988 Accesses

Abstract

If a sequence of independent unbiased random bits is fed into a finite automaton, it is straightforward to calculate the expected number of acceptances among the first n prefixes of the sequence. This paper deals with the situation in which the random bits are neither independent nor unbiased, but are nearly so. We show that, under suitable assumptions concerning the automaton, if the the difference between the entropy of the first n bits and n converges to a constant exponentially fast, then the change in the expected number of acceptances also converges to a constan texponentially fast. We illustrate this result with a variety of examples in which numbers folio wing the reciprocal distribution, whidi governs the significands of floating-point numbers, are recoded in the execution of various multiplication algorithms.

This research was supported by an NSERC Research Grant and a Canada Research Chair.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Booth, A. D.: A signed binary multiplication technique. Quart. J. Mech. Appl. Math., 4 (1951) 236–240

    Article  MATH  MathSciNet  Google Scholar 

  2. Booth, A. D.: Review of “A proof of the modified Booth’s algorithm for multiplication” by Louis P. Rubinfeld. Math. Rev., 53 #4610

    Google Scholar 

  3. Chomsky, N., Miller, G. A.: Finite-state languages. Inform, and Control, 1 (1958) 91–112

    Article  MathSciNet  MATH  Google Scholar 

  4. Freiman, C. V.: Statistical analysis of certain binary division algorithms. Proc. IRE, 49 (1961) 91–103

    Article  MathSciNet  Google Scholar 

  5. Frougny, C: On-the-fly algorithms and sequential machines. IEEE Trans, on Computers, 49 (2000) 859–863

    Article  MathSciNet  Google Scholar 

  6. Hamming, R. W.: On the distribution of numbers. Bell System Tech. J., 49 (1970) 1609–1625

    MathSciNet  MATH  Google Scholar 

  7. MacSorley, O. L.: High-speed arithmetic in binary computers. Proc. IRE, 49 (1961) 67–91

    Article  MathSciNet  Google Scholar 

  8. Newcomb, S.: Note on the frequency of use of the different digits in natural numbers,. Amer. J. Math., 4 (1881) 39–40

    Article  MathSciNet  Google Scholar 

  9. Shannon, C. E.: A mathematical theory of communication. Bell System Tech. J., 27 (1948) 379–423, 623–655

    MathSciNet  Google Scholar 

  10. Tocher, K. D.: Techniques of multiplication and division for automatic binary computers. Quart. J. Mech. Appl. Math., 11 (1958) 364–384

    Article  MATH  MathSciNet  Google Scholar 

  11. Wallis, J.: Arithmetica infinitorum. Oxford, 1656

    Google Scholar 

  12. Whittaker, E. T., Watson, G. N.: A course of modern analysis. Cambridge, 1963

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, University of British Columbia, 2366 Main Mall, Vancouver, V6T 1Z4, Canada

    Nie Holas Pippenger

Authors
  1. Nie Holas Pippenger
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. School of Computer Science, Carleton University, 1125 Colonel By Drive, Ottawa, Ontario, Canada, K1S 5B6

    Prosenjit Bose  & Pat Morin  & 

Rights and permissions

Reprints and permissions

Copyright information

© 2002 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Pippenger, N.H. (2002). Expected Acceptance Counts for Finite Automata with Almost Uniform Input. In: Bose, P., Morin, P. (eds) Algorithms and Computation. ISAAC 2002. Lecture Notes in Computer Science, vol 2518. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-36136-7_56

Download citation

  • DOI: https://doi.org/10.1007/3-540-36136-7_56

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-00142-3

  • Online ISBN: 978-3-540-36136-7

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation