Work Tape Research Articles

A note on some simultaneous relations among time, space, and reversal for single work tape nondeterministic turing machines

Simultaneous resource bounded complexity classes for nondeterministic single worktape off-line Turing machines are considered such as time-space bounded classes, denoted by NTISP 1 ( T, S ), reversal-space bounded classes, denoted by NRESP 1 ( R, S ), and time-reversal bounded classes, denoted by NTIRE 1 ( T, R ). It is shown that NRESP 1 ( R ( n ), S ( n )) contains NTISP 1 ( S ( n ), R ( n )) and is contained in NTISP 1 ( R ( n ) S ( n ) n 2 log n , R ( n ) log n ). The following corollaries follow: (1) the affirmative solution to the nondeterministic single worktape version of the NC = ? SC problem, NTIRE 1 (poly, polylog) = NTISP 1 (poly, polylog), and (2) a reversal-space trade-off, NRESP 1 (polylog, poly) = NRESP 1 (poly, polylog).

A tradeoff theorem for space and reversal

In this note, the following result is proved: Let M be an off-line Turing machine, n the input length, S the work space and R the total number of reversals of the work tape heads. Then RS= o( n) implies RS= O(1).

Folding of the plane and the design of systolic arrays

Systolic arrays were first introduced by Kung (see, e.g., [2] and [3]) as devices composed of processors of a few different types, which are regularly and locally connected. These processors are activated in a synchronous way by a unique clock which is the only global communication between them. This paper is a continuation of the work presented in [I] where ‘folding’ has been proposed as a technique for the design of systolic arrays. Here we first study the power of folding as a general geometric transformation. Then, as an application we show that two congruent sequences on a ‘regular’ grid can be identified by a limited number of foldings (cf. Theorem 3.5). This result can be used in the design of systolic arrays. Because of the importance of this motivation we start with an example of a systolic array which has been already briefly described in [ 11. Consider Kung and Leiserson’s hex-connected processor array for matrix multiplication [3, pp. 276-2801 modified in such a way that it applies to dense matrices. Fig. 1 illustrates the case where the dimension n of the matrix is equal to 3: the left-to-right and the right-to-left flows correspond to the two matrices A and B to be multiplied and the bottom-to-top flow to the product C = A x B. Each node represents an inner product step processor, i.e., a processor computing one step of a scalar product: s +s + ab (see Fig. 2). Assume we want to use this array to compute the different powers of a matrix A, which basically amounts to computing A, A X A = A2,. . . , Ak X Ak = A2k ).... One solution is to iteratively feed the different outputs of step k coming out in the upper part of the array, to both left and right inputs of step k + 1, i.e., to connect each yi to (Y~ and pi. However, in doing this we would create non-local connections and break the regularity of the layout. Instead, we can first fold the array (as one would fold a sheet of paper) along the axis 1, the righthand side coming on top of the left-hand side. The new array computes the same functions as the original one, very much in the same way as a Turing machine with a one way infinite working tape can simulate a Turing machine with a two-way infinite working tape, by folding the latter. Then, we can fold the new array along the axis 2 (the left-hand side on top of the right-hand side) and again along the axis 3 (the right-hand side on top of the left-hand side). The processors cxi, pi, yi will eventually occupy the same place and we must connect them to each other thus introducing only local connections. As a result the regularity of the initial lay-out is preserved. The price to pay for it is that the new array consists of up to 8 = 23 more

An improved simulation result for ink-bounded turing machines

A (one tape, deterministic) Turing Machine is f( n) ink bounded if the machine changes a symbol of its work tape at most f( n) times while processing an input of length n. The result of our paper is the construction of an “ink efficient” universal machine which, for any f( n) ink bounded machine M and input x, can simulate the processing of M on x or detect that M is looping infinitely on input x. The universal machine requires O( f( n) 1+ ϵ ink for this simulation, where e is an arbitrarily small positive number. As a corollary we establish that for ink-constructable g: For any ϵ > 0, if inf n→∞ ( f( n) 1+ ge / g( n)) = 0 then there is a language in INK( g(n)) not in INK( f(n)).

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.

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.

Some Bounds on the Storage Requirements of Sequential Machines and Turing Machines

Any sequential machine M represents a function f M from input sequences to output symbols. A function f is representable if some finite-state sequential machine represents it. The function f M is called an n-th order approximation to a given function f if f M is equal to f for all input sequences of length less than or equal to n . It is proved that, for an arbitrary nonrepresentable function f , there are infinitely many n such that any sequential machine representing an n th order approximation to f has more than n /2 + 1 states. An analogous result is obtained for two-way sequential machines and, using these and related results, lower bounds are obtained for two-way sequential machines and, using these and related results, lower bounds are obtained on the amount of work tape required online and offline Turing machines that compute nonrepresentable functions.

Design and characteristics of a variable-length record sort using new fixed-length record sorting techniques

This paper describes the application of several new techniques for sorting fixed-length records to the problem of variable-length record sorting. The techniques have been implemented on a Sylvania 9400 computer system with 32,000 fixed-length words of memory. Specifically, the techniques sequence variable-length records of unrestricted size, produce long initial strings of data, merge strings of data at the power of T - 1, where T is the number of work tapes in a system, and do not restrict the volume of input data.