Paul, W.J., Seifers, J.I., Simon, J., "An Information Theoretic Approach to time Bounds of On--Line Computation," J. Computers and System Sciences 23, pp. 108--126, 1981.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....id, c 1. 2 Related work concerning higher dimensional Turing machines can be found e.g. in [7] where under the different constraint of on line computations the tradeoff between time and dimensionality is investigated. Upper bounds for the reduction of the dimensions are dealt with e.g. in [9, 11, 12, 14]. Here, on one hand, we are going to present time hierarchies below linear time for any dimension. On the other hand, dimension hierarchies are presented for every time bound in the range in question. Thus, we obtain a two dimensional time dimension hierarchy. The basic notions and a preliminary ....

Paul, W., Seiferas, J. I., and Simon, J. An information-theoretic approach to time bounds for on-line computation. J. Comput. System Sci. 23 (1981), 108--126.

....move. Thus we say that the range of different memory access cost functions spans the difference between a TM and a RAM. We have not been able to obtain a good converse to (d) the combinatorial problem is that a TM with d dimensional tapes may change its head direction often, and the methods of [16] have more than the log t(n) overhead of the one dimensional case [6] For quasilinear time, i.e. time qlin = n(log n) O(1) the extra log n factors in the above simulations do not matter. Following Schnorr [20] we write DQL and NQL for the TM time classes DTIME[qlin] and NTIME[qlin] The ....

W. Paul, J. Seiferas, and J. Simon. An information-theoretic approach to time bounds for on-line computation. J. Comp. Sys. Sci., 23:108--126, 1981.

....is our justification for this concept of dimensionality. Lemma 7.5 says that it is no less restrictive than the older concept given by d dimensional Turing machines. For d 1 we suspect that it is noticeably more restrictive. The d dimensional tape reduction theorem of Paul, Seiferas, and Simon [58] gives t 0 (n) roughly equal to t(n) 1 1=d , and when ported to a BM, incurs memory access charges close to t(n) 1 2=d . Intuitively, the problem is that a d dimensional TM can change the direction of motion of its tape head(s) at any step, whereas this would be considered a break in ....

W. Paul, J. Seiferas, and J. Simon, An information-theoretic approach to time bounds for on-line computation, J. Comp. Sys. Sci., 23 (1981), pp. 108--126.

....the whole range of models in Section 1. The second inclusion in (a) follows because the Hennie Stearns simulation [HS66, HU79, WW86] is memory efficient under 1 . We suspect that for d 1 the converse simulation in (c) requires notably more than the O(log t) overhead of the d = 1 case (a) see [PSS81] for related matters. The intuitive reason is that a d TM may often change its head direction, but in going to a BM this is a break in pipelining and subject to a memory access charge. Let us abbreviate quasi linear time by qlin : n(log n) O(1) and following [Sch78] DTIME[qlin] to DQL, ....

W. Paul, J. Seiferas, and J. Simon. An information-theoretic approach to time bounds for on-line computation. J. Comp. Sys. Sci., 23:108--126, 1981.

....report appeared in the Proceedings of STOC 94, the Twenty Sixth Annual ACM Symposium on the Theory of Computing, pp. 668 675. Typeset by A M S T E X JIANG, SEIFERAS, AND VIT ANYI, HEADS VS. TAPES 2 indeed turned out to be the case, although the proofs have been disproportionately difficult [Ra63, He66, Gr77, Aa74, PSS81, Pa82, DGPR84, Ma85, LV88, LLV92, MSST93, PSSN90]. The case of deficiency in the number of heads allowed on each tape has turned out to be the most delicate, because it involves a surprise: A larger number of singlehead tapes can compensate for the absence of multihead tapes [MRF67, FMR72, LS81] For example, four single head tapes suffice for ....

.... gives us a tight bound on the number of single head tapes needed to recognize the particular language L in real time, since three do suffice [MRF67, FMR72] Thus L is another example of a language with number of tapes complexity 3, rather different from the one first given by Aanderaa [Aa74, PSS81]. For the latter, even a two head tape, even if enhanced by instantaneous head tohead jumps and allowed to operate probabilistically, was not enough [PSSN90] Historically, multihead tapes were introduced in Hartmanis and Stearns seminal paper [HS65] which outlined a linear time 1 simulation ....

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W. J. Paul, J. I. Seiferas, and J. Simon, An information-theoretic approach to time bounds for on-line computation, Journal of Computer and System Sciences 23, 2 (October, 1981), 108--126.

....is not restricted to trivialities. The proofs of the theorems in this paper may not be easy. However, the theorems are of the type that are used as a tool. Once derived, our theorems are easy to apply. 1. 1 Prelude The first application of Kolmogorov complexity in the theory of computation was in [19, 20]. By re doing proofs of known results, it was shown that static, descriptional (program size) complexity of a single random string can be used to obtain lower bounds on dynamic, computational (running time) complexity. None of the inventors of Kolmogorov complexity originally had these ....

.... N be a standard recursive, invertible, one one encoding of pairs of natural numbers in natural numbers. This idea can be iterated to obtain a pairing from triples of natural numbers with natural numbers hx; y; zi = hx; hy; zii, and so on. Any of the usual definitions of Kolmogorov complexity in [11, 20, 13] will do for the sequel. We are interested in the shortest effective description of a finite object x. To fix thoughts, consider the problem of describing a string x over 0 s and 1 s. Let T 1 ; T 2 ; be the standard enumeration of Turing machines. Since T i computes a partial recursive ....

W.J. Paul, J.I. Seiferas, and J. Simon. An information theoretic approach to time bounds for on-line computation. J. Comput. Syst. Sci., 23:108--126, 1981.

....version of this report appeared in the Proceedings of STOC 94, the Twenty Sixth Annual ACM Symposium on the Theory of Computing, pp. 668 675. Typeset by A M S T E X JIANG, SEIFERAS, AND VIT ANYI, HEADS VS. TAPES 2 be the case, although the proofs have been disproportionately difficult [Ra63, He66, Gr77, Aa74, PSS81, Pa82, DGPR84, Ma85, LV88, LLV92, MSST93, PSSN90]. The case of deficiency in the number of heads allowed on each tape has turned out to be the most delicate, because it involves a surprise: A larger number of singlehead tapes can compensate for the absence of multihead tapes [MRF67, FMR72, LS81] For example, four single head tapes suffice for ....

.... gives us a tight bound on the number of single head tapes needed to recognize the particular language L in real time, since three do suffice [MRF67, FMR72] Thus L is another example of a language with number of tapes complexity 3, rather different from the one first given by Aanderaa [Aa74, PSS81]. For the latter, even a two head tape, even if enhanced by instantaneous head tohead jumps and allowed to operate probabilistically, was not enough [PSSN90] Historically, multihead tapes were introduced in Hartmanis and Stearns seminal paper [HS65] which outlined a linear time 1 simulation ....

[Article contains additional citation context not shown here]

W. J. Paul, J. I. Seiferas, and J. Simon, An information-theoretic approach to time bounds for on-line computation, Journal of Computer and System Sciences 23, 2 (October, 1981), 108--126.

....It has been known for over twenty years that all multitape Turing machines can be simulated on line by 2 tape Turing machines in time O(n log n) HS2] and by 1 tape Turing machines in time O(n 2 ) HU] Since then, many other models of computation have been introduced and compared. See [Aa, DGPR, HS1, HS2, HU, LS, PSS, Pa, Vi2]. In addition to different storage mechanisms, real time, on line and off line machines have been studied. An on line simulation essentially simulates step by step each move of the simulated machine. In this paper, we consider off line machines, where an answer is given only once the whole input ....

....string x. By a simple counting argument, we know that there are strings x of each length such that K(x) jxj. These strings are called incompressible or K random. For completeness we recall the notions of Kolmogorov complexity of binary strings and those of self delimiting descriptions, see e.g. [PSS] or [LV] Fix an effective coding C of all Turing machines as binary strings, such that no code is a prefix of any other code. Denote the code of Turing machine M by C(M ) The Kolmogorov complexity with respect to C, of a binary string x, denoted as K C (x) is the length of the smallest binary ....

Paul, W.J., J.I. Seiferas, and J. Simon, "An information theoretic approach to time bounds for on-line computation, J. Computer and System Sciences, vol. 23, pp. 108-126, 1981.

....storage devices a fine distinction can be made by the capabilities in real time. Thus, 9] showed that two one head tape units are more powerful in real time than one such unit. Later [1] generalized this by demonstrating superiority of k 1 one head tape units over k such units, in real time. In [6] a new information theoretic argument was introduced to strip the proof in [1] down to its essentials, while [8] strengthened the result by exploiting the techniques further. Moreover, it was shown that considering multihead tape units resulted in an analogous real time hierarchy [6, 12] Thus, it ....

....real time. In [6] a new information theoretic argument was introduced to strip the proof in [1] down to its essentials, while [8] strengthened the result by exploiting the techniques further. Moreover, it was shown that considering multihead tape units resulted in an analogous real time hierarchy [6, 12]. Thus, it appeared, adding a head in general increases computing power. But does it also make a difference whether the heads reside on the same tape The advantage obtained by placing the heads on the same tape is that they can read each others writing. Nonetheless, showing that it is impossible ....

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W.J. Paul, J.I. Seiferas and J. Simon, An information theoretic approach to time bounds for on-line computation, J. Computer and System Sciences 23 (1981), 108 - 126.

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Paul, W.J., Seifers, J.I., Simon, J., "An Information Theoretic Approach to time Bounds of On--Line Computation," J. Computers and System Sciences 23, pp. 108--126, 1981.

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