# Expected Acceptance Counts for Finite Automata with Almost Uniform Input

## 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.

## Keywords

Multiplication Algorithm Finite Automaton Probability Generate Function Sequential Machine Input String## Preview

Unable to display preview. Download preview PDF.

## References

- [B1]Booth, A. D.: A signed binary multiplication technique. Quart. J. Mech. Appl. Math.,
**4**(1951) 236–240zbMATHCrossRefMathSciNetGoogle Scholar - [B2]Booth, A. D.: Review of “A proof of the modified Booth’s algorithm for multiplication” by Louis P. Rubinfeld. Math. Rev.,
**53**#4610Google Scholar - [C]Chomsky, N., Miller, G. A.: Finite-state languages. Inform, and Control,
**1**(1958) 91–112CrossRefMathSciNetzbMATHGoogle Scholar - [F1]Freiman, C. V.: Statistical analysis of certain binary division algorithms. Proc. IRE,
**49**(1961) 91–103CrossRefMathSciNetGoogle Scholar - [F2]Frougny, C: On-the-fly algorithms and sequential machines. IEEE Trans, on Computers,
**49**(2000) 859–863CrossRefMathSciNetGoogle Scholar - [H]Hamming, R. W.: On the distribution of numbers. Bell System Tech. J.,
**49**(1970) 1609–1625MathSciNetzbMATHGoogle Scholar - [M]MacSorley, O. L.: High-speed arithmetic in binary computers. Proc. IRE,
**49**(1961) 67–91CrossRefMathSciNetGoogle Scholar - [N]Newcomb, S.: Note on the frequency of use of the different digits in natural numbers,. Amer. J. Math.,
**4**(1881) 39–40CrossRefMathSciNetGoogle Scholar - [S]Shannon, C. E.: A mathematical theory of communication. Bell System Tech. J.,
**27**(1948) 379–423, 623–655MathSciNetGoogle Scholar - [T]Tocher, K. D.: Techniques of multiplication and division for automatic binary computers. Quart. J. Mech. Appl. Math.,
**11**(1958) 364–384zbMATHCrossRefMathSciNetGoogle Scholar - [W1]Wallis, J.: Arithmetica infinitorum. Oxford, 1656Google Scholar
- [W2]Whittaker, E. T., Watson, G. N.: A course of modern analysis. Cambridge, 1963Google Scholar