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The marginal utility of the Turing machine computational resources running time and storage space are studied. A technique is developed which, unlike diagonalization, applies equally well to nondeterministic and deterministic automata. For f, g time or space bounding functions with f(n + 1) small compared to g(n), it is shown that, in terms of word length n, there are languages which are accepted by Turing machines operating with time or space g(n) but which are accepted by no Turing machine operating within time or space f(n). The proof involves use of the recursion theorem together with "padding" or "translational" techniques of formal language theory. Relations between worktape alphabet size, number of worktape heads, number of input heads, and Turing machine storage space are established. Within every common subexponential space bound, it is shown that enlarging the worktape alphabet always increases computing power. A hierarchy of two-way multihead finite automata is obtained even in the nondeterministic case. Results that are only slightly weaker are obtained for Turing machines that accept only languages over a one-letter alphabet. {PB 236-777.AS}