Theory of Computational Complexity, 2nd Edition

It is interesting to point out that for these complexity classes, an increase of a constant factor on the time or the space bounds does not change the complexity classes. The reason is that a TM with a larger alphabet set and a larger set of states can simulate, in one move, a constant number of moves of a TM with a smaller alphabet set and a smaller number of states. We only show these properties for deterministic complexity classes.

Proposition 1.12

(Tape Compression Theorem) Assume that c > 0. Then, .

Proof

Let M be a TM that runs in space bound s(n). We will construct a new TM M^′ that simulates M within space bound , using the same number of tapes. Assume that M uses the alphabet Γ of size k, has r states , and uses t tapes. Let m be a positive integer such that . Divide the squares in each work tape of M into groups; each group contains exactly m squares. The machine M^′ uses an alphabet of size k^m so that each square of M^′ can simulate a group of squares of M. In addition, M^′ has states, each state represented by , for some , , , . Each state encodes the information ...