This Article 
 Bibliographic References 
 Add to: 
Some Characterizations of Functions Computable in On-Line Arithmetic
June 1994 (vol. 43 no. 6)
pp. 752-755

After a short introduction to on-line computing, we prove that the functions computable in on-line by a finite automaton are piecewise affine functions whose coefficients are rational numbers (i.e., the functions f(x)=ax+b, or f(x,y)=ax+by+c where a, b, and c are rational). A consequence of this study is that multiplication, division and elementary functions of operands of arbitrarily long length cannot be performed using bounded-size operators.

[1] A. Avizienis, "Signed-digit number representations for fast parallel arithmetic,"IRE Trans. Electron. Comput., vol. 10, pp. 389-400, 1961.
[2] M. D. Ercegovac and K. S. Trivedi, "On line algorithms for division and multiplication,"IEEE Trans. Comput., vol. C-26, no. 7, pp. 681-687, July 1977.
[3] M. D. Ercegovac and P. K. G. Tu, "A radix-4 on-line division algorithm," presented at8th Symp. Comput. Arithmetic, Como, Italy, May 1987.
[4] M. J. Irwin and R. M. Owens, "Digit pipelined arithmetic as illustrated by the paste-up system,"IEEE Comput. Mag., pp. 61-73, Apr. 1987.
[5] M. J. Irwin, "An arithmetic unit for Online computation," Ph.D. dissertation, Dept. of Comput. Sci., Univ. of Illinois, Champaign-Urbana, IL; Tech. Rep. UIUCDCS-R-77-873, 1977.
[6] J. G. Rusnak and K. S. Trivedi, "Higher radix on-line division," inProc. 4th IEEE Symp. Comput. Arithmetic, Oct. 1978, pp. 183-189.
[7] M. D. Ercegovac, "An on-line square rooting algorithm," inProc. 4th IEEE Symp. Comput. Arithmetic, Oct. 1978, pp. 183-189.
[8] R. M. Owens, "Digit online algorithms for pipelined architectures," Ph.D. dissertation, Dep. Comput. Sci., The Pennsylvania State Univ., 1980.
[9] M. D. Ercegovac, "On-line arithmetic: An overview,"SPIE, vol. 495,Real Time Signal ProcessingVII, 1984, pp. 86-93.
[10] R. M. Owens, "Techniques to reduce the inherent limitations of fully digit on-line arithmetic,"IEEE Trans. Comput., vol. C-32, no. 4, Apr. 1983.
[11] H. J. Sips and H. X. Lin, "A new model for on-line arithmetic with an application to the reciprocal calculation,"J. Parallel Distributed Comput., pp. 218-230, 1990.
[12] J. Duprat, Y. Herreros, and J. M. Muller, "Some results about on-line computation of functions," inProc. 9th Symp. Comput. Arithmetic, Santa Monica, CA, Sept. 1989, pp. 112-118.
[13] E. Wiedmer, "Computing with infinite objects,"Theorem Comput. Sci., vol. 10, pp. 133-155, 1980.
[14] J. Vuillemin, "Exact real computer arithmetic with continued fractions,"IEEE Trans. Comput., vol. 39, no. 8, pp. 1087-1105, Aug. 1990.

Index Terms:
digital arithmetic; finite automata; computability; online computing; finite automaton; piecewise affine functions; rational numbers; multiplication; division; elementary functions; operands; arbitrarily long length.
J.-M. Muller, "Some Characterizations of Functions Computable in On-Line Arithmetic," IEEE Transactions on Computers, vol. 43, no. 6, pp. 752-755, June 1994, doi:10.1109/12.286308
Usage of this product signifies your acceptance of the Terms of Use.