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On the Lower Bound to the Memory of Finite State Machines
September 1969 (vol. 18 no. 9)
pp. 856861
ASCII Text  x  
K. Vairavan, "On the Lower Bound to the Memory of Finite State Machines," IEEE Transactions on Computers, vol. 18, no. 9, pp. 856861, September, 1969.  
BibTex  x  
@article{ 10.1109/TC.1969.222782, author = {K. Vairavan}, title = {On the Lower Bound to the Memory of Finite State Machines}, journal ={IEEE Transactions on Computers}, volume = {18}, number = {9}, issn = {00189340}, year = {1969}, pages = {856861}, doi = {http://doi.ieeecomputersociety.org/10.1109/TC.1969.222782}, publisher = {IEEE Computer Society}, address = {Los Alamitos, CA, USA}, }  
RefWorks Procite/RefMan/Endnote  x  
TY  JOUR JO  IEEE Transactions on Computers TI  On the Lower Bound to the Memory of Finite State Machines IS  9 SN  00189340 SP856 EP861 EPD  856861 A1  K. Vairavan, PY  1969 KW  Finite memory KW  finite state machines KW  lower bound on memory KW  minimal canonical realizations. VL  18 JA  IEEE Transactions on Computers ER   
A finite state machine (FSM) is said to have finite memory ? if ? is the least integer such that yk = f(Xk , Xk 1,... Xk? , Yk1 , ... ?k? ) where yk and Xk represent the output and input at time k. If no such ? exists, then by convention the memory is said to be infinite. It has been observed [1] that if the memory of a pnary input, qnary output nstate minimal nondegenerate FSM is finite, then the memory ? is bounded as follows: Recently, considerable attention has been devoted to the study of the upper bound on ? [2][5]. In this paper we examine the lower bound on ?. We show that the lower bound is tight for all positive integers n and certain values of p and q. We also show that when n = 4^{k}, k a positive integer, there exist binary input, binary output minimal FSMs with minimal memory. It will be seen that this is equivalent to showing that for all positive integers ? there exist binary input, binary output minimal FSMs with the maximum number of states n=2^{2?}. Finally, we enumerate the equivalence classes of these finite memory machines with memory ? and n = 2^{2?}states.
Index Terms:
Finite memory, finite state machines, lower bound on memory, minimal canonical realizations.
Citation:
K. Vairavan, "On the Lower Bound to the Memory of Finite State Machines," IEEE Transactions on Computers, vol. 18, no. 9, pp. 856861, Sept. 1969, doi:10.1109/TC.1969.222782
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