By Andrzej Szepietowski (eds.)
ISBN-10: 3540486690
ISBN-13: 9783540486695
ISBN-10: 3540583556
ISBN-13: 9783540583554
This complete monograph investigates the computational strength of Turing machines with sublogarithmic area. The stories are dedicated to the Turing laptop version brought by means of Stearns, Hartmanis, and Lewis (1965) with a two-way read-only enter tape and a separate two-way read-write paintings tape. The ebook offers the major effects on area complexity, additionally as regards the periods of languages appropriate, lower than the point of view of a sublogarithmic variety of cells used in the course of computation. It originates from classes given via the writer on the Technical college of Gdansk and Gdansk college in 1991 and 1992. It used to be finalized in 1994 whilst the writer visited Paderborn collage and contains the newest contributions to the field.
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Additional info for Turing Machines with Sublogarithmic Space
Example text
Let x, u, v E ~*, and u =--k v. Then the following two statements are equivalent: Lower Bounds for Two-way Turing Machines 29 9 there is a k space-bounded accepting COml~utation of M o n z u 9 there is a k space-bounded accepting computation of M on zv. Proof. Let us consider some k space-bounded accepting computation of M on ~U: where ~0 is the initial configuration and ~,,, is accepting. We shall show that there is also a k space-bounded accepting computation on z v . -. ~,n M can cross the boundary between z and u several times.
Let M be a middle L(n) space-bounded one-way deterministic or nondeterministic Turing machine. Then either Theorem Lower Bounds for the Middle Mode of Space Complexity 35 > c l o g n for some constant c > 0 and infinitely many n, or 9 M accepts a regular language and space used by M is bounded by a constant. * L(n) Proof. Suppose that there is no constant upper bound on space used by M. an is the shortest accepted input causing M to use exactly k space. Consider two computations of M on w : one that uses exactly k space on w and the other that accepts w.
Hence, there are at most 2kdk different S~v(r ) and S~lv(r) = 0 if r > 2kd k. Hence, the number of different S~Iu is bounded by (2kd~) 2kdk -- 22k~d2k. Thus < 2 2k~d2k " So we can derive that k > cloglog n for some constant c > O. Hence, L(n) > e log log n for infinitely many n. 2. Lower Bounds for One-way Turing Machines In this section we present lower bounds for accepting non-regular languages by one-way Turing machines. In this case the lower bound is equal to log n for strongly space-bounded deterministic, nondeterministic, and alternating, and weakly space-bounded deterministic Turing machines; and is equal to log log n for weakly space-bounded nondeterministic and alternating Turing machines (Stearns et.
Turing Machines with Sublogarithmic Space by Andrzej Szepietowski (eds.)
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