Multitape Turing Machines Research Articles

R70-1 A Note on Computing Time for the Recognition of Context- Free Languages by a Single-Tape Turing Machine

Although considerable progress has been made in the understanding of the more "algebraic" properties of context-free languages, the quantitative aspects of the recognition and parsing of context- free languages are still far from well understood. It was shown by Kasami and Younger that for every context-free language there exists a multitape Turing machine which recognizes this language and uses no more than n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> operations to process input words of length n. For one-tape Turing machines Hartmanis showed, using the Younger algorithm, that every context-free language can be recognized in n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">5</sup> operations. This result has now been improved in the paper under review. It is shown, by a careful implementation of the Torii–Kasami–Ozaki algorithm on a one-tape Turing machine, that every context-free language can be recognized in n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</sup> operations. Again, just like for the n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> operation multitape result, it is not known how good this time bound is. On the other hand, the authors have been able to show in this paper that for linear context-free languages n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> is the least-upper time bound for their recognition on one-tape Turing machines. This is done by showing that every linear language is n2-recognizable, and by recalling that there are linear languages whose recognition requires n2 operations on a one-tape machine.

Quasi-realtime languages

The quasi-realtime languages are seen to be the languages accepted by nondeterministic multitape Turing machines in real time. The family of quasi-realtime languages forms an abstract family of languages closed under intersection, linear erasing, and reversal. It is identical with the family of languages accepted by nondeterministic multitape Turing machines in linear time. Every quasi-realtime language can be accepted in real time by a nondeterministic one stack, one pushdown store machine, and can be expressed as the length-preserving homomorphic image of the intersection of three context-free languages.

Time and tape complexity of pushdown automaton languages

An algorithm is presented which will determine whether any string w in Σ * , of length n , is contained in a language L ⊆ Σ * defined by a two-way nondeterministic pushdown automation. This algorithm requires time n 3 when implemented on a random access computer. It requires n 4 time and n 2 tape when implemented on a multitape Turing machine. If the pushdown automaton is deterministic, the algorithm requires n 2 time on a random access computer and n 2 log n time on a multitape Turing machine.

Classes of automata and transitive closure

A study is made of the classes of predicates accepted by three types of multitape Turing machine. In order of decreasing acceptance powers, these are the general Turing machine, the linear-bounded automaton, and the two-way multitape nonwriting automaton. Each class is shown to consist of all and only those predicates which can be defined by a corresponding class of predicate calculus formulas based on catenation, and involving as logical operators conjunction, disjunction, and a type of transitive closure on predicates of 2 n variables.

Real-Time Definable Languages

The closure properties of the class of languages defined by real-time, online, multi-tape Turing machines are proved. The results obtained are, for the most part, negative and, as one would expect, asymmetric. It is shown that the results remain valid for a broad class of real-time devices. Finally, the position of the class of real-time definable languages in the “classical” linguistic hierarchy is established.

Turing machines with a schedule to keep

It is shown that if a sequence of digits α = α 1 α 2 α 3 ··· is generated by a multitape Turing machine so that the n th digit α n is produced no later than time T ( n ), where T is a real-time-countable function, then there exists a multitape Turing machine which generates α and produces α n at exactly time T ( n ). This result makes it possible to strengthen some theorems of Hartmanis and Stearns and to answer a question of Yamada. Other applications yield a more general approach to countability and a new technique for dealing with time-limited computational complexity.

A note on computing time for recognition of languages generated by linear grammars

It is shown that (1) there exists a language L 0 which is generated by a linear grammar and is not T(n) -recognizable by any on-line multitape Turing machine if lim n→∞ T (n)/(n/log n) 2 = 0, and (2) any language generated by a linear grammar is n 2 -recognizable by an on-line single-tape Turing machine in the sense of Hartmanis and Stearns.

Two-Tape Simulation of Multitape Turing Machines

It has long been known that increasing the number of tapes used by a Turing machine does not provide the ability to compute any new functions. On the other hand, the use of extra tapes does make it possible to speed up the computation of certain functions. It is known that a square factor is sometimes required for a one-tape machine to behave as a two-tape machine and that a square factor is always sufficient. The purpose of this paper is to show that, if a given function requires computation time T for a k -tape realization, then it requires at most computation time T log T for a two-tape realization. The proof of this fact is constructive; given any k -tape machine, it is shown how to design an equivalent two-tape machine that operates within the stated time bounds. In addition to being interesting in its own right, the trade-off relation between number of tapes and speed of computation can be used in a diagonalization argument to show that if T ( n ) and U ( n ) are two time functions such that inf T ( n ) log T ( n ) ÷ U ( n ) = 0 then there exists a function that can be computed within the time bound U ( n ) but not within the time bound T ( n ).

Turing machines with restricted memory access

Multitape Turing machines which can use their storage tapes only as counters or as pushdown stores are investigated. The memory access restrictions are produced by regarding the machines as small computers (as in the formalism of Wang) and by restricting the instruction repertoires. Relationships are given linking machines which only accept or reject inputs and machines which emit output sequences as a function of their input. It is shown that without restrictions on computing time or amount of tape used that only six distinct classes of sets of strings (languages) are produced by the above memory access restrictions.