Any sequential machine M represents a function fM from input sequences to output symbols. A function f is representable if some finite-state sequential machine represents it. The function fM is called an n-th order approximation to a given function f if fM is equal to f for all input sequences of length less than or equal to n. It is proved that, for an arbitrary nonrepresentable function f, there are infinitely many n such that any sequential machine representing an nth order approximation to f has more than n/2 + 1 states. An analogous result is obtained for two-way sequential machines and, using these and related results, lower bounds are obtained for two-way sequential machines and, using these and related results, lower bounds are obtained on the amount of work tape required online and offline Turing machines that compute nonrepresentable functions. © 1967, ACM. All rights reserved.