5 research outputs found
Nondeterministic one-tape off-line Turing machines and their time complexity
In this paper we consider the time and the crossing sequence complexities of
one-tape off-line Turing machines. We show that the running time of each
nondeterministic machine accepting a nonregular language must grow at least as
n\log n, in the case all accepting computations are considered (accept
measure). We also prove that the maximal length of the crossing sequences used
in accepting computations must grow at least as \log n. On the other hand, it
is known that if the time is measured considering, for each accepted string,
only the faster accepting computation (weak measure), then there exist
nonregular languages accepted in linear time. We prove that under this measure,
each accepting computation should exhibit a crossing sequence of length at
least \log\log n. We also present efficient implementations of algorithms
accepting some unary nonregular languages.Comment: 18 pages. The paper will appear on the Journal of Automata, Languages
and Combinatoric
Verifying Time Complexity of Deterministic Turing Machines
We show that, for all reasonable functions , we can
algorithmically verify whether a given one-tape Turing machine runs in time at
most . This is a tight bound on the order of growth for the function
because we prove that, for and , there
exists no algorithm that would verify whether a given one-tape Turing machine
runs in time at most .
We give results also for the case of multi-tape Turing machines. We show that
we can verify whether a given multi-tape Turing machine runs in time at most
iff for some .
We prove a very general undecidability result stating that, for any class of
functions that contains arbitrary large constants, we cannot
verify whether a given Turing machine runs in time for some
. In particular, we cannot verify whether a Turing machine
runs in constant, polynomial or exponential time.Comment: 18 pages, 1 figur
Space Complexity of Stack Automata Models
This paper examines several measures of space complexity on variants of stack automata: non-erasing stack automata and checking stack automata. These measures capture the minimum stack size required to accept any word in a language (weak measure), the maximum stack size used in any accepting computation on any accepted word (accept measure), and the maximum stack size used in any computation (strong measure). We give a detailed characterization of the accept and strong space complexity measures for checking stack automata. Exactly one of three cases can occur: the complexity is either bounded by a constant, behaves (up to small technicalities explained in the paper) like a linear function, or it grows arbitrarily larger than the length of the input word. However, this result does not hold for non-erasing stack automata; we provide an example when the space complexity grows with the square root of the input length. Furthermore, an investigation is done regarding the best complexity of any machine accepting a given language, and on decidability of space complexity properties