The reduced nearest neighbor rule (Corresp.)

Fig. 3 shows PG,.,,,,, (m) for various numbers of observations N and various sizes of memory m. The Chernoff bound on P,*(a) was used for N 2 32. Quite naturally one does better with more memory. The P~,sym(m) curve for any given value of m follows the P,*(co) line for low values of N, diverges from it for larger values of N, and approaches a nonzero limit P,*(m) as N + co. This behavior is easily explained. Any given machine can “remember” all of the observations for low values of N. Here infinite memory offers no advantages. For larger values of N, a finite-state machine necessarily loses some information and thus does not do so well as one with infinite memory. As N -+ co, Pz sym(m) approaches Pm*(m), the infinite-time lower bound on the probability of error, since from [I] we know that for N = co the optimal machine is symmetric.