Bennett, Charles H. `Logical depth and physical complexity', in The Universal Turing Machine: a Half Century Survey, Oxford, 1988.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....looked at the connection between sophistication and logical depth for in nite strings. Koppel s paper uses a di erent de nition of logical depth imposing totality in the functions de ning logical depth and de ning the length of a time bound by the smallest program describing it. Bennett [Ben88] formally de ned the s signi cant logical depth of an object x as the time required by a standard universal Turing machine to generate x by a program that is no more than s bits longer than the shortest descriptions of x. Antunes et al. AFvM01] considered logical depth as one instantiation of ....

....a description of length C(x) Levin [Lev73] introduced a useful variant weighing program size and running time. De nition 7. For any strings x; y, the Levin complexity of x given y is Ct(xjy) min fjpj log t : U(p; y) halts in at most t steps and outputs xg: After some attempts, Bennett [Ben88] formally de ned the s signi cant logical depth of a string x as the time required by a standard universal Turing machine to generate x by a program that is no more than s bits longer than the shortest descriptions of x. A string x is called logically deep if it takes a lot of time to generate ....

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C. H. Bennett. Logical depth and physical complexity. In R. Herken, editor, The Universal Turing Machine: A Half-Century Survey, pages 227-257. Oxford University Press, 1988.

....Melkebeek [AFvM01] propose a notion of Computational Depth as a measure of nonrandom information in a string. Intuitively strings of high depth are low Kolmogorov complexity strings (and hence nonrandom) but a resource bounded machine cannot identify this fact. Indeed, Bennett s logical depth [Ben88] can be viewed as such a measure, but its de nition is rather technical. With simplicity in mind, Antunes et al. suggest that the di erence between two Kolmogorov complexity measures captures the intuitive Research done during an academic internship at NEC. This author is partially supported ....

C. H. Bennett. Logical depth and physical complexity. In R. Herken, editor, The Universal Turing Machine: A Half-Century Survey, pages 227-257. Oxford University Press, 1988.

....sequence in polynomial time (i.e. decide the nth bit of the sequence in time polynomial in the length of the binary representation of n) 11] 4. Even with oracle access to z, most recursive sequences cannot be computed in polynomial time. This appears to be folklore, known at least since [3]. Facts (i) and (ii) tell us that K contains far less information than z. In contrast, facts (iii) and (iv) tell us that K is computationally much more useful than z. That is, the information in K is more usefully organized than that in z. Bennett [3] introduced the notion of computational ....

....to be folklore, known at least since [3] Facts (i) and (ii) tell us that K contains far less information than z. In contrast, facts (iii) and (iv) tell us that K is computationally much more useful than z. That is, the information in K is more usefully organized than that in z. Bennett [3] introduced the notion of computational depth (also called logical depth ) in order to quantify the degree to which the information in an object has been organized. In particular, for infinite binary sequences, Bennett defined two levels of depth, strong depth and weak depth, and argued that ....

[Article contains additional citation context not shown here]

C. H. Bennett, Logical depth and physical complexity, In R. Herken, editor, The Universal Turing Machine: A Half-Century Survey, pp. 227--257. Oxford University Press, 1988.

.... inference helped to inspire less general yet practically more feasible principles of minimum description length [68, 46] as well as time bounded restrictions of Kolmogorov complexity, e.g. 28, 69, 39] as well as the concept of logical depth of x, the runtime of the shortest program of x [6]. Equation (14) makes predictions of the entire future, given the past. This seems to be the most general approach. Solomonoff [60] focuses just on the next bit in a sequence. Although this provokes surprisingly nontrivial problems associated with translating the bitwise approach to alphabets ....

C. H. Bennett. Logical depth and physical complexity. In The Universal Turing Machine: A Half Century Survey, volume 1, pages 227--258. Oxford University Press, Oxford and Kammerer & Unverzagt, Hamburg, 1988.

.... helped to inspire less general yet practically more feasible principles of minimum description length [95, 66, 44] as well as time bounded restrictions of Kolmogorov complexity, e.g. 42, 2, 96, 56] as well as the concept of logical depth of x, the runtime of the shortest program of x [8]. Equation (15) makes predictions of the entire future, given the past. This seems to be the most general approach. Solomonoff [83] focuses just on the next bit in a sequence. Although this provokes surprisingly nontrivial problems associated with translating the bitwise approach to alphabets ....

C. H. Bennett. Logical depth and physical complexity. In The Universal Turing Machine: A Half Century Survey, volume 1, pages 227--258. Oxford University Press, Oxford and Kammerer & Unverzagt, Hamburg, 1988.

....sets of r.e. T degree are in fact T complete. Hence, every Chaitin # number is T complete. In this paper we will strengthen this result by proving that every Chaitin # number is weak truth table complete. However, no Chaitin # number can be tt complete as, because of a result stated by Bennett [1] (see Juedes, Lathrop, and Lutz [9] for a proof) there is no random sequence x such that K # tt x. 2 Notice that in this way we obtain a whole class of natural examples of wtt complete but not tt complete r.e. sets (a fairly complicated construction of such a set was given by Lachlan [10] ....

C. H. Bennett. Logical depth and physical complexity, in R. Herken (ed.). The Universal Turing Machine. A Half-Century Survey, Oxford University Press, Oxford, 1988, 227--258.

....exhaustive search. The solution. The only solution to the problem is: make all w i equal to 1. The Kolmogorov complexity of this solution is small, since there is a short program that computes it. Its Levin complexity is small, too, since its logical depth (the runtime of its shortest program (Bennett, 1988)) is less than 400 time steps. The training data. To illustrate the generalization capability of search for solution candidates with low Levin complexity, only 3 training examples are used. They were randomly chosen from the 161; 700 possible inputs. The first training example is the binary ....

Bennett, C. H. (1988). Logical depth and physical complexity. In The Universal Turing Machine: A Half Century Survey, volume 1, pages 227--258. Oxford University Press, Oxford and Kammerer & Unverzagt, Hamburg.

....sequence in polynomial time (i.e. decide the nth bit of the sequence in time polynomial in the length of the binary representation of n) 11] 4. Even with oracle access to z, most recursive sequences cannot be computed in polynomial time. This appears to be folklore, known at least since [3]. Facts (i) and (ii) tell us that K contains far less information than z. In contrast, facts (iii) and (iv) tell us that K is computationally much more useful than z. That is, the information in K is more usefully organized than that in z. Bennett [3] introduced the notion of computational ....

....to be folklore, known at least since [3] Facts (i) and (ii) tell us that K contains far less information than z. In contrast, facts (iii) and (iv) tell us that K is computationally much more useful than z. That is, the information in K is more usefully organized than that in z. Bennett [3] introduced the notion of computational depth (also called logical depth ) in order to quantify the degree to which the information in an object has been organized. In particular, for infinite binary sequences, Bennett defined two levels of depth, strong depth and weak depth, and argued that ....

[Article contains additional citation context not shown here]

C. H. Bennett, Logical depth and physical complexity, In R. Herken, editor, The Universal Turing Machine: A Half-Century Survey, pp. 227--257. Oxford University Press, 1988.

....then it is # P btt reducible to a sparse problem. Arvind, Kobler, and Mundhenk [9, 10] have improved on this result in several respects. A problem that is random in the sense of Martin Lof [64] has extremely high information content. It is sometimes (e.g. in the context of computational depth [13]) of interest to know whether an algorithm can, from a random object, compute an object that could not have been computed from any random object with significantly less resources. In this connection, Book, Lutz, and Martin [17] proved that, if RAND is the set of all random decision problems, then ....

C. H. Bennett. Logical depth and physical complexity. In R. Herken, editor, The Universal Turing Machine: A Half-Century Survey, pages 227--257. Oxford University Press, London, 1988.

....it is relatively straightforward to interpret textual compression within any formalism, be it propositional logic, first order logic or context free grammar rules. Thirdly, this would bring the PAC learning model into line with other approaches developed from algorithmic complexity theory [40, 6, 2] and Bayesian statistics. According to Bayes Law P r(HjO) P r(H) P r(OjH) P r(O) If we treat bit encoded descriptions as though they were the outcome of repeatedly tossing an unbiased coin, the prior probability of a string of bits s of length l is simply 2 Gammal . Now by taking logs we ....

C. Bennett. Logical depth and physical complexity. In R. Herken, editor, The Universal Turing Machine A Half Century Survey, pages 227--257. Kammerer and Unverzagt, Hamburg, 1988.

....(infinite) However, a given quantity of information may be organized in various ways, rendering it more or less useful for various computational purposes. In order to quantify the degree to which the information in a computational, physical, or biological object has been organized, Bennett [4, 5] has extended algorithmic information theory by defining and investigating the computational depth of binary strings and binary sequences. Roughly speaking, the computational depth (called logical depth by Bennett [4, 5] of an object is the amount of time required for an algorithm to derive the ....

....a computational, physical, or biological object has been organized, Bennett [4, 5] has extended algorithmic information theory by defining and investigating the computational depth of binary strings and binary sequences. Roughly speaking, the computational depth (called logical depth by Bennett [4, 5]) of an object is the amount of time required for an algorithm to derive the object from its shortest description. Precise definitions appear in the sections to follow. Since this shortest description contains all the information in the object, the depth thus represents the amount of ....

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C. H. Bennett. Logical depth and physical complexity. In R. Herken, editor, The Universal Turing Machine: A Half-Century Survey, pages 227--257. Oxford University Press, 1988.

....space search could not be expected to converge within such a lattice. We must therefore look to some alternative model to guide and constrain the search through this more complex lattice. 3. 2 Algorithmic information theory Following the lead of Kolmogorov [8] various information theorists [4, 26, 2] have investigated the relationship between computation, randomness and message complexity. The basic intuition rests on the observation that although the strings 010100110111001100010110101100 and 010101010101010101010101010101 have approximately the same Shannon information content [25] the ....

C. Bennett. Logical depth and physical complexity. In R. Herken, editor, The Universal Turing Machine A Half Century Survey, pages 227--257. Kammerer and Unverzagt, Hamburg, 1988.

....in physics cannot measure intervals of time less than 10 26 seconds. 12 It advances one frame at a time. apparent paradox comes from the fact that human eyes cannot resolve the short time intervals between frames. For a detailed discussion we quote Mellor [31] Prigogine [36] Bennett [4] raised the following important question: Is self organisation an asymptotical qualitative phenomenon (like phase transitions) More precisely, are there physically reasonable models in which complexity appropriately defined increases without bound in time and space The answer is yes for ....

C. H. Bennett. Logical depth and physical complexity, in R. Herken (ed.). The Universal Turing Machine. A Half-Century Survey, Oxford University Press, Oxford, 1988, 227-258.

....have defined measures of complexity that attempt to capture specific types of organization or information. Typical proposals for such a measure are specific and designed to measure an object s complexity under a restricted model or domain. See [8, 10, 11, 24, 22] for example. However, Bennett [2, 3] has defined a complexity measure based on programs for universal Turing machines that does capture the desired complexity criteria and is universal for all objects that can be digitally encoded. Under Bennett s definition, the computational depth [2, 3] of a binary data string is roughly the ....

....11, 24, 22] for example. However, Bennett [2, 3] has defined a complexity measure based on programs for universal Turing machines that does capture the desired complexity criteria and is universal for all objects that can be digitally encoded. Under Bennett s definition, the computational depth [2, 3] of a binary data string is roughly the amount of time required to generate the string from a description of the string of nearly minimal length. A parameter s is used to define nearly minimal in section 3. A description of an object contains all the essential information required to ....

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C. H. Bennett. Logical depth and physical complexity. In R. Herken, editor, The Universal Turing Machine: A Half-Century Survey, pages 227--257. Oxford University Press, 1988.

.... that P = ALMOST GammaPm [Amb86] P = ALMOST GammaP btt = ALMOST GammaP log n GammaT [TB91] BPP = ALMOST GammaP T [BG81, Amb86] BPP = ALMOST GammaP tt [Amb86, TB91] AM = ALMOST GammaNP T [NW88] and PH = ALMOST GammaPH [NW88] 2 Note that BPP = P T (RAND) REC has already been proved in [Ben88]. The class RAND is considered to be the class of those languages having the greatest possible information content. It is well known that there is a constant c such that for all languages A and all n, the Kolmogorov complexity of the finite language An = fx 2 A j jxj ng is not greater than 2 ....

C. Bennett. Logical depth and physical complexity. In R. Herken (ed.), The Universal Turing Machine: A Half-Century Survey, pages 227--257. Oxford University Press, 1988.

....a similar proof one can actually show, using Theorem 5.1, that for any C 2 2 , there exists a C n random set in tt (C (n) even in R( Sigma C n ) but not in btt (C (n) It is known that the Martin Lof random set of Theorem 5. 2 (i) cannot have the same tt degree as K (Bennett [2], Juedes, Lathrop, and Lutz [4] Note that there is not an analogue of the result r.e. fA 2 2 : A is MartinL of randomg) 1 for the case of Delta n measure. In fact, it is easy to see that Delta n (fA 2 2 : A is Delta n randomg) 6= 1. Namely, for every Delta n martingale d ....

C. H. Bennett, Logical depth and physical complexity, in: R. Herken, ed., The Universal Turing Machine: A Half Century Survey, Oxford Univ. Press (1988) 227-257.

....exhaustive search. The solution. The only solution to the problem is: make all w i equal to 1. The Kolmogorov complexity of this solution is small, since there is a short program that computes it. Its Levin complexity is small, too, since its logical depth (the runtime of its shortest program (Bennett, 1988)) is less than 400 time steps. The difficulty. If the training set is very small (e.g. if there are just four or five training examples) then conventional perceptron algorithms will not solve this apparently simple problem. They will not achieve good generalization on unseen test data. One ....

Bennett, C. H. (1988). Logical depth and physical complexity. In The Universal Turing Machine: A Half Century Survey, volume 1, pages 227--258. Oxford University Press, Oxford and Kammerer & Unverzagt, Hamburg.

....Calude [4] gives a satisfactory description of the quantity of information of individual finite strings and infinite sequences. The same quantity of information may be organised in various ways; in order to quantify the degree of organisation of the information in a string or a sequence, Bennett [2], Juedes, Lathrop, and Lutz [13] and others, have considered the computational depth. Roughly speaking, the computational depth of an object is the amount of time required for an algorithm to derive the object from its shortest description. Bennett [2] showed that the characteristic sequence K ....

....in a string or a sequence, Bennett [2] Juedes, Lathrop, and Lutz [13] and others, have considered the computational depth. Roughly speaking, the computational depth of an object is the amount of time required for an algorithm to derive the object from its shortest description. Bennett [2] showed that the characteristic sequence K of the halting problem is strongly deep, while no random sequence is strongly deep. Investigating this matter further, Juedes, Lathrop, and Lutz [13] have considered the notion of usefulness of infinite sequences. A sequence x is useful if all ....

[Article contains additional citation context not shown here]

C. H. Bennett. Logical depth and physical complexity, in R. Herken (ed.). The Universal Turing Machine. A Half-Century Survey, Oxford University Press, Oxford, 1988, 227--258.

No context found.

Bennett, Charles H. `Logical depth and physical complexity', in The Universal Turing Machine: a Half Century Survey, Oxford, 1988.

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C. H. Bennett, Logical depth and physical complexity, In R. Herken, editor, The Universal Turing Machine: A Half-Century Survey, pp. 227--257. Oxford University Press, 1988.

No context found.

C. H. Bennett. Logical depth and physical complexity. In R. Herken, editor, The Universal Turing Machine: A Half-Century Survey, pages 227--257. Oxford University Press, 1988.

No context found.

C. H. Bennett. Logical depth and physical complexity. In R. Herken, editor, The Universal Turing Machine: A Half-Century Survey, pages 227-257. Oxford University Press, Oxford, 1988.

No context found.

C. H. Bennett. Logical Depth and Physical Complexity, pages 227--257. Oxford University Press, Oxford, UK, 1988.

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C. H. Bennett, Logical depth and physical complexity, in: R. Herken, ed., The Universal Turing Machine: A Half Century Survey, Oxford Univ. Press (1988) 227-257. 12

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C. H. Bennett. Logical depth and physical complexity. In The Universal Turing Machine: A Half Century Survey, volume 1, pages 227--258. Oxford University Press, Oxford and Kammerer & Unverzagt, Hamburg, 1988.

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