Some Bounds on the Storage Requirements of Sequential Machines and Turing Machines

Abstract

Any sequential machine M represents a function f _M from input sequences to output symbols. A function f is representable if some finite-state sequential machine represents it. The function f _M is called an n-th order approximation to a given function f if f _M 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 n th 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.