### Table 1. An example of the permutation problem

2003

"... In PAGE 1: ... In this paper we focus speci cally on the case where the feature values have been permuted in a random manner. Table1 shows a simple example of this type of permutation problem. We would like to be able to learn the joint probability density of the original data on the left given only the permuted data and knowl- edge of the type of permutations that may have been Table 1.... In PAGE 1: ...pplied to the data (e.g., cyclic shifts). Two questions naturally arise: (a) how hard is this type of learning problem in general? and (b) what kinds of algorithms can we use to solve this problem in practice? In considering the rst problem, our intuition tells us that the \more di erent quot; the features in the original (unpermuted) table are then the \easier quot; the unscram- bling problem may be. For example, in Table1 , the distributions of each individual feature in the table on the left appear quite di erent from each other, so that one hopes that given enough data one could eventu- ally recover a model for the original data given only permuted data. In Section 2 we make this notion of learnability precise by introducing the notion of a Bayes-optimal permutation error rate.... In PAGE 4: ... E? C is proportional to the overlap of the individual fea- ture densities p (~xi) in the space S. For example, for the data on the left in Table1 we would expect the overlap of the 4 densities, as re ected by E? C, to be quite small. Furthermore, we would expect intuitively that the permutation error rate E? P should also be low in this case, and more generally that it should be re- lated to E? C in some manner.... ..."

Cited by 3

### Table2. Complexity classes of automata with logarithmically space-bounded tape and empty alternation

1994

"... In PAGE 5: ... In the following, for X 2 fLOG, PDA{TIME(pol), PDA, P, PSPACEg and a function g, where we again admit the cases that g is a constant or that g is unbounded, let EA log g X denote the set of all languages recognized by logspace Turing machines augmented with storage of type X, which make g(n) 1 empty alternations. The main results of this chapter are collected in Table2 , which is the \empty quot; analogue of Table 1. 3.... ..."

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### Table 1: Number of permutations original new

"... In PAGE 11: ....3.1 Algorithmic Complexity The overall complexity of the general transformation method is given by the total number of permutations that the function is evaluated for. It can be determined for both variants of the method according to Table1 , depending on the number of -cuts (m + 1) and the number of uncertain parameters n. In general, one can say that if the number of -cuts (m + 1) is large compared to the number of uncertain parameters, the new scheme o ers a signi cantly better complexity.... In PAGE 20: ...formation method: f2(x1; x2) = (x1 2)4 + (x2 2)2 + 0:2 1 : (19) If we would like to compute f2 using fuzzy numbers decomposed into 20 -levels, we would have to consider nperm = 20 X k=1 k2 = 2870 (20) permutations according to Table1 . Counting the total number of operations o required to execute one function evaluation in the two-dimensional space, we count o = 7.... ..."

### Table 3: Some of permutation and selection extendible parameter spaces for a design with two homologous factors.

2002

"... In PAGE 19: ... When this is applied to the other irreducible representations, a pattern starts to emerge: in general, the sum to zero constraint is not retained under selection. Table3 contains some of the most important linear parameter spaces thus obtained.2 Table 3: Some of permutation and selection extendible parameter spaces for a design with two homologous factors.... In PAGE 19: ... The differential effect model DE is such extendible model. In Table3 the differential effect model DE for a particular has been extended to a model that includes as a parameter. This is a param- eter space that is both extendible under permutations and selections.... In PAGE 19: ... For each the residual sums of squares RSS( ) of the DE is be min- imized for fi, before minimizing RSS( ) with respect to theta. In the ap- pendix we prove that the residual sums of squares RSS( ) of the differential 2Several permutation and slection extendible parameter spaces are not included in Table3 . Several, such as the model of functions that take on a constant c on the diagonal and twice that constant 2c on the off-diagonal, are not very useful for inference.... ..."

### TABLE 1. Summary of representation space changes

1997

### Table 1: Tentative comparison of artificial immune system with classifier systems

1999

"... In PAGE 6: ...1101011010110...110101 Figure 2: The Lifecycle of a Detector. 5 COMPARISON WITH CLASSIFIER SYSTEMS The AIS outlined in Section 4 resembles the architecture of a classifier system[11], although most of the details are different (see Table1 ). The mapping between classifier sys- tems and our AIS is not 1-1, however.... ..."

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### Table 1: Tentative comparison of artificial immune system with classifier systems

1999

"... In PAGE 6: ...1101011010110...110101 Figure 2: The Lifecycle of a Detector. 5 COMPARISON WITH CLASSIFIER SYSTEMS The AIS outlined in Section 4 resembles the architecture of a classifier system[11], although most of the details are different (see Table1 ). The mapping between classifier sys- tems and our AIS is not 1-1, however.... ..."

Cited by 1

### Table 1: Tentative comparison of artificial immune system with classifier systems

"... In PAGE 6: ...1101011010110...110101 Figure 2: The Lifecycle of a Detector. 5 COMPARISON WITH CLASSIFIER SYSTEMS The AIS outlined in Section 4 resembles the architecture of a classifier system[11], although most of the details are different (see Table1 ). The mapping between classifier sys- tems and our AIS is not 1-1, however.... ..."

### Table 1: Search Space Complexity.

1991

"... In PAGE 9: ... Figure 1 illustrates this three-pronged approach. Despite the deceptive appearance, the graph is drawn to scale based on the numbers in Table1 . The diagram shows the number of pieces on the board (vertically) versus the logarithm of the number of positions (base 10) with that many pieces on the board (horizontally).... ..."

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### Table 3: Permutations and their permutation vectors

1998

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