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By Suel Memon

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A very similar partial speed-up will be used in the next section to characterize time constructible functions. 1 If A is accepted by a Turing machine M within space s(n), and c is a real number with 0 < c < 1, then A is accepted by a Turing machine Mr within space c·s(n). Let r be an integer such that r·c 2 2, and assume that M has k tapes. We construct a Turing machine Mr with k tapes which performs the simulation. Each symbol of the tape alphabet of Mr is an r-tuple of symbols of the alphabet of M.

For the case of oracle Turing machines, running time is defined exactly as in the case of non-oracle machines. However, there is some controversy on the appropriate conventions for measuring the amount of space used by oracle Turing machines: should the oracle tape be counted as a work tape? We take the position that the oracle tape is to be bounded by any space bound we consider, unless otherwise indicated. We should issue the warning, anyway, that in many cases (such as, for instance, if space bounds are sublinear) we will impose on the oracle tape a space bound different from the work tape space bound.

This process is known as a godelization of Turing machines, and the number corresponding to each machine is its Godel number. By convention, numbers that do not correspond to a string correctly encoding a machine are considered Gooel numbers of some arbitrarily fixed trivial machine. In this way we can list in order all the Turing machines, and refer to the nth machine, meaning the Turing machine with Gooel number n. Configurations and computations can be described as well by words over some alphabet, and we will identify configurations and computations with the words describing them.