Theory of one-tape linear-time Turing machines

Abstract

A theory of one-tape (one-head) linear-time Turing machines is essentially different from its polynomial-time counterpart since these machines are closely related to finite state automata. This paper discusses structural-complexity issues of one-tape Turing machines of various types (deterministic, nondeterministic, reversible, alternating, probabilistic, counting, and quantum Turing machines) that halt in linear time, where the running time of a machine is defined as the length of any longest computation path. We explore structural properties of one-tape linear-time Turing machines and clarify how the machines' resources affect their computational patterns and power.is short, then this machine recognizes only a regular language. Using the non-regularity measure of Dwork and Stockmeyer [13], the second claim asserts that any language accepted by a machine with short crossing sequences has constantly-bounded non-regularity. Extending Hennie's argument, Kobayashi [25] later showed that any language recognized by one-tape o(n log n)-time deterministic Turing machines should be regular as well. This time bound o(n log n) is actually optimal since certain one-tape O(n log n)-time deterministic Turing machines can recognize non-regular languages.Unlike polynomial-time computation, one-tape linear-time nondeterministic computation is sensitive to the definition of the machine's running time. Such sensitivity is also observed in average-case complexity theory [44]. By taking his weak definition that defines the running time of a nondeterministic Turing machine to be the length of a "shortest" accepting path, Michel [30] demonstrated that one-tape nondeterministic Turing machines running in linear time (in the sense of his weak definition) solve even NP-complete problems. Clearly, his weak definition gives an enormous power to one-tape nondeterministic machines and therefore it does not seem to offer any interesting features of time-bounded nondeterminism. On the contrary, the strong definition (in Michel's term) requires the running time to be the length of any "longest" (both accepting and rejecting) computation path. This strong definition provides us with a reasonable basis to study the effect of linear-time bounded computations. We therefore adopt his strong definition of running time and, throughout this paper, all one-tape time-bounded Turing machines are assumed to accommodate this strong definition. By expanding Kobayashi's result, we prove that one-tape o(n log n)-time nondeterministic Turing machines recognize only regular languages.The model of alternating Turing machines of Chandra, Kozen, and Stockmeyer [6] naturally expand the model of nondeterministic machines. The number of alternations of such an alternating Turing machine seems to enhance the computational power of the machine; however, our strong definition of running time makes it possible for us to prove that a constant number of alternations do not give any additional computational power to one-tape linear-time alternating Turing machines; namely...

Verifying whether one-tape Turing machines run in linear time

“…They also proved that a one-tape deterministic Turing machine which runs in time o(n log n), produces only crossing sequences of bounded length and accepts a regular language. Later Tadaki, Yamakami and Lin [10] proved the same for one-tape non-deterministic Turing machines. Properties of crossing sequences generated by one-tape non-deterministic Turing machines were also analyzed by Pighizzini [8].…”

Nonuniform Families of Polynomial-Size Quantum Finite Automata and Quantum Logarithmic-Space Computation with Polynomial-Size Advice

“…Our proof of ptime-2BQ ⊆ 2P essentially follows an argument used in [30,Lemma 8.1], in which, for any linear-time one-tape QTM with Q-amplitudes, its acceptance probability on input x is calculated by two appropriate functions computed by the acceptance probabilities of liner-time one-tape PTMs. A major deviation from [30] is that we allow arbitrary real transition probabilities in [0, 1] for 2pfa's. This simplifies our construction of the desired 2pfa.…”

Complexity Bounds of Constant-Space Quantum Computation

“…It is not difficult to verify that co-SL R coincides with the family of all languages L recognized by 1pfa's M with "non-strict cut points" (which requires p M,acc (x) ≥ η instead of p M,acc (x) > η) for certain constants η ∈ [0, 1]. It is known in [32] that SL Q and SL = Q are characterized in terms of one-tape linear-time Turing machines (namely, 1-PLIN and 1-C = LIN). Despite our past efforts, we still do not know whether SL R is closed under complementation, whether SL = R is included in SL R , and whether SL = R contains any non-recursive language (see, e.g., [18] for references therein).…”