Multitape Turing Machines Research Articles

The recognition of deterministic CFLs in small time and space

Let S(n) be a nice space bound such that log2 n ⩽ S(n) ⩽ n. Then every DCFL is recognized by a multitape Turing machine simultaneously in time O(n2/S(n)) and space O(S(n)), and this time bound is optimal. If the machine is allowed a random access input, then the time bound can be improved so that the time-space product is O(n1 + ɛ).

Space-bounded simulation of multitape turing machines

A new proof of a theorem of Hopcroft, Paul, and Valiant is presented: Every deterministic multitape Turing machine of time complexityT(n) can be simulated by a deterministic Turing machine of space complexityT(n)/logT(n). The proof includes an overlap argument.

A simplified proof of the real-time recognizability of palindromes on turing machines

A comparatively short proof is given of the recognizability of palindromes in real time on multitape Turing machines. It is based on the same idea as the original proof by the author, and on Z. Galil's idea for simplifying the proof by using the Fischer-Paterson algorithm for finding all symmetric suffixes in linear time.

On alternation II

Every deterministic t(n)-time bounded multitape Turing machine can be simulated by an alternating t(n) loglog t(n)/log t(n)-time bounded Turing machine. If the depth of every directed acyclic graph with n edges can be reduced to log n by removing only o(n) edges, then in linear time nondeterministic multitape Turing machines can recognize mor languages than deterministic multitape Turing machines. For some graphs reduction of the depth to log n requires the removal of Ω(n/loglog n) edges. A graph theoretic condition is given, which implies that obliviousness reduces the power of multitape Turing machines.

On alternation

Every alternating t(n)-time bounded multitape Turing machine can be simulated by an alternating t(n)-time bounded 1-tape Turing machine. Every nondeterministic t(n)-time bounded 1-tape Turing machine can be simulated by an alternating (n + (t(n)) 1/2)-timebounded 1-tape Turing machine. For wellbehaved functions t(n) every nondeterministic t(n)-time bounded 1-tape Turing machine can be simulated by a deterministic ((nlogn)1/2 + (t(n))1/2)-tape bounded off-line Turing machine. These results improve or extend results by Chandra-Stockmeyer, Lipton-Tarjan and Paterson.

Saving Space in Fast String-Matching

The string-matching problem is to find all instances (as contiguous substrings) of a “pattern” character string x in a longer “text" string ”. The naive algorithm, trying the pattern from scratch starting at each successive text position, requires only a fixed number of auxiliary storage locations but time proportional to $|x|\cdot|Y|$ (worst case). On the other hand, the fast algorithm of Knuth, Morris, and Pratt requires only time proportional to $|Y|$ but extra space proportional to $|x|$ (every case). Algorithms described in this paper reduce the extra space used by the Knuth–Morris–Pratt algorithm down to $O(\log |x|)$, and the time for the naive algorithm down to $O(|x|^\varepsilon |y|)$ for any fixed $\varepsilon > 0$. Also described are implementations on two-way multihead finite automata and multitape Turing machines.

Space-bounded complexity classes and iterated deterministic substitution

We investigate the effect on the space complexity when a language family K is extended by means of iterated λ-free deterministic substitution to the family η(K). If each language in K is accepted by a one-way nondeterministic multi-tape Turing machine within space S(n) for some monotonic space bound S(n) ⩾ log n, then η(K) is included in NSPACE(S(n)). Thus for each monotonic space bound S(n) ⩾ n, the inclusion K ⊆ NSPACE(S(n)) implies that η(K) is also included in NSPACE(S(n)). An implication similar to the latter one also holds for DSPACE(S(n)). Consequently, some well-known space-bounded complexity classes such as the families of (non)deterministic context-sensitive languages, of two-way (non)deterministic nonerasing stack automaton languages, and PLPACE are AFL's closed under intersection and iterated λ-free deterministic substitution. On the other hand no complexity class which includes DSPACE(log n) is closed under controlled iterated λ-free (non) deterministic substitution.

Polynomial Algorithms for Deterministic Pushdown Automata

An algorithm is presented for converting a deterministic pushdown automaton (dpda) of size n into an equivalent dpda that always halts. The dpda produced is of size $O(n)$. The algorithm operates in linear time on a random access machine (but may require the allocation of $O(n^2 )$ storage), and in time $O(n^2 )$ on a multi-tape Turing machine. Related results on polynomial time algorithms for dpda equivalence problems and for two-way pushdown automata language recognition problems are discussed.

Palindrome recognition in real time by a multitape turing machine

A real-time Turing machine algorithm that finds the smallest nontrivial initial palindrome in the input string is constructed. A small modification of this algorithm yields a real-time Turing machine algorithm which finds all initial palindromes in the input string.

One way finite visit automata

A one-way preset Turing machine with base L is a nondeterministic on-line Turing machine with one working tape preset to a member of L. FINITEREVERSAL( L ) (FINITEVISIT ( L )) is the class of languages accepted by one-way preset Turing machines with bases in L which are limited to a finite number of reversals (visits). For any full semiAFL L , FINITEREVERSAL ( L ) is the closure of L under homomorphic replication or, equivalently, the closure of L under iteration of controls on linear context-free grammars while FINITEVISIT ( L ) is the result of applying controls from L to absolutely parallel grammars or, equivalently, the closure of L under deterministic two-way finite state transductions. If L is a full AFL with L ≠ FINITEVISIT( L ), then FINITEREVERSAL( L ) ≠ FINITEVISIT( L ). In particular, for one-way checking automata, k + 1 reversals are more powerful than k reversals, k + 1 visits are more powerful than k visits, k visits and k + 1 reversals are incomparable and there are languages definable within 3 visits but no finite number of reversals. Finite visit one-way checking automaton languages can be accepted by nondeterministic multitape Turing machines in space log 2 n. Results on finite visit checking automata provide another proof that not all context-free languages can be accepted by one-way nonerasing stack automata.

Real-time recognition of substring repetition and reversal

Real-time multitape Turing machine algorithms are presented for recognizing the languages {wxyxz∈Σ*:|w|=r|x,|z| =t|x|} and {wxyxRz∈Σ*:|w|=r|x,|z| =t|x|} for fixedr, s, andt and for string-matching with “forced mismatches.”

Computational Complexity of Multitape Turing Machines and Random Access Machines

In recent years there has been an increasing interest in analyzing the computational complexity of programs. The multitape Turing machine has become the standard model used for evaluating time and storage complexity, even though such machines are not much like any existing computers. Some authors, however, implement their algorithms not on Turing machines but on random access machines. In 1972 Cook introduced a formal model of a random access machine. This model is closer to real computer, for real computers calculate the address of desired storage cell within a short time before fetching its content.

On Time Versus Space

It is shown that every deterministic multitape Turing machine of time complexity t ( n ) can be simulated by a deterministic Turing machine of tape complexity t ( n )/log t ( n ). Consequently, for tape constructable t ( n ), the class of languages recognizable by multitape Turing machines of time complexity t ( n ) is strictly contained in the class of languages recognized by Turing machines of tape complexity t ( n ). In particular the context-sensitive languages cannot be recognized in linear time by deterministic multitape Turing machines.

The complexity of the membership problem for some extensions of context-free languagest†

The time and tape complexity of some families of languages defined in the literature by altering methods of generation by context-free grammars is considered. Specifically; it is shown that the following families of languages can be recognized by deterministic multitape Turing machines either in polynomial time or within (log n)2 tape: 1) the context independent developmental (EOL) languages; 2) the simple matrix languages; 3) the languages generated by derivation restricted state grammars.: 4) the languages generated by linear context-free grammars with certain non-regular control sets; 5) the languages generated by certain classes of vector grammars. In fact, these languages are of the same tape complexity as context-free languages. Other results indicate the complexity of EDOL languages and the effects on complexity of applying the homomorphic replication operator to regular and context-free languages.

On Families of Languages Defined by Time-Bounded Random Access Machines

There are essentially two results described in this paper. First, it is shown that for any random access machine (RAM) time-constructable function $T(n) \geqq n$, there are languages L such that L can be recognized in time $O(T(n))$ by a RAM (using the unit cost measure), but L cannot be recognized by any deterministic multitape Turing machine in time $O(T(n))$. Secondly, a family of random access stored program machines (RASP’S) are considered. For these RASP’S it is shown that there is an arbitrarily complex (infinitely often) partial recursive function $f(n)$. which has only 0–1 values (whenever defined) such that $f(n)$ can be computed in time $F(n)$ by some RASP, but cannot be computed in time $(1 - \varepsilon )F(n)$, for any $\varepsilon > 0$, by any RASP in this family.

Hierarchies of turing machines with restricted tape alphabet size

It is shown that for any real constants b > a ≥0, multitape Turing machines operating in space L 1 ( n )=[ bn r ] can accept more sets than those operating in space L 2 ( n )=[ an r ] provided the number of work tapes and tape alphabet size are held fixed. It is also shown that Turing machines with k +1 work tapes are more powerful than those with k work tapes if the tape alphabet size and the amount of work space are held constant.

Theorems on factorization and primality testing

1. Introduction. This paper is concerned with the problem of obtaining theoretical estimates for the number of arithmetical operations required to factorize a large integer n or test it for primality. One way of making these problems precise uses a multi-tape Turing machine (e.g. (1), although we require a version with an input tape). At the start of the calculation n is written in radix notation on one of the tapes, and the machine is to stop after writing out the factors in radix notation or after writing one of two symbols denoting ‘prime’ or ‘composite’. There are, of course, other definitions which could be used; but the differences between these are unimportant for our purpose.

A hierarchy for nondeterministic time complexity

We prove the following theorem in this paper: For any real numbers r 1 , r 2 , 1≤ r 1 < r 2 , there is a set A of strings which has nondeterministic time complexity n r 2, but not nondeterministic time complexity n r 1. The computing devices are nondeterminsitic multitape Turing machines.

Pushdown automata with counters

The computational power of 2-way pushodown acceptors with m (≥1) additional counters (called m C-PDA) is investigated. It is shown that given any multitape Turing machine M that accepts within time T(n) , where n is the input length, one can find a 3C-PDA M ′ such that M and M ′ accept the same set of inputs and if M accepts an input w of length n within time T(n) , then M ′ accepts w storing no more than count T(n) in any of the counters. For any such Turing machine, one can also find a 2C-PDA which accepts the same set within count T(n) 2 . A number of other results relating the power of 1C-PDA and those of other types of acceptors are also obtained.

Note on tape reversal complexity of languages

The number of tape reversals required for the recognition of a set of inputs by a 1-tape Turing machine (TM) has been proposed before as a measure of complexi It is shown that if a set is recognizable by a multi-tape TM with no more than R(n) tape reversals, where n is the input length, then the set is recog It is also shown that if a set is recognizable by a multi-tape TM within time T(n) , then the set is recognizable by a 1-tape off-line TM with no more than Upper bounds on the number of reversals necessary for the recognition of a number of well-known classes of languages are also obtained.