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Complexity and Relative Randomness for 1-Random Sets
In this chapter, we examine basic questions about the function K, as well as related questions about the orderings ⩽C and ⩽K, and other measures of...
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Preliminaries
Most of our notation, terminology, and conventions will be introduced as we go along, but we set down here a few basic ones that are used throughout...
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Computability Theory
In this chapter we will develop a significant amount of computability theory. Much of this technical material will not be needed until much later in...
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Kolmogorov Complexity of Finite Strings
This chapter is a brief introduction to Kolmogorov complexity and the theory of algorithmic randomness for finite strings. We will concentrate on...
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Complexity of Computably Enumerable Sets
In this section, we look at the initial segment complexity of c.e. sets, including a fascinating gap phenomenon uncovered by Kummer [226]. We begin...
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Algorithmic Randomness and Complexity
Intuitively, a sequence such as 101010101010101010… does not seem random, whereas 101101011101010100…, obtained using coin tosses, does. How can we...
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Other Notions of Algorithmic Randomness
In [348, 349], Schnorr analyzed his characterization of 1-randomness in terms of c.e. martingales (Theorem 6.3.4). He argued that this result...
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Lowness and Triviality for Other Randomness Notions
We now turn to lowness notions for other notions of randomness. We begin with Schnorr randomness. Since there is no universal Schnorr test, it is not...
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Measures of Relative Randomness
Measures of relative complexity such as Turing reducibility, wtt-reducibility, and so on have proven to be central tools in computability theory. The...
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Randomness-Theoretic Weakness
In this chapter, we introduce an important class of “randomnesstheoretically weak” sets, the K-trivial sets. As we will see, this class has several...
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Ω as an Operator
We have already seen that Chaitin’s Ω is a natural example of a 1-random real. We have also seen that, in algorithmic randomness, prefix-free...
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Strong Jump Traceability
As we have seen, the K-trivial sets form a class of “extremely low” sets, properly contained within the class of superlow sets and closed under join....
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Algorithmic Randomness and Turing Reducibility
In this chapter, we look at the distribution of 1-random and n-random degrees among the Turing (and other) degrees. Among the major results we...
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Relating Complexities
In this chapter, we look at some of the fundamental relationships between flavors of Kolmogorov complexity. The first four sections explore...
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Effective Reals
In this chapter we study the left computably enumerable reals, and discuss some other classes of effectively approximable reals. Left computably...
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Algorithmic Dimension
Not all classes of measure 0 are created equal, and the classical theory of dimension provides a method for classifying them. Likewise, some...
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Martin-Löf Randomness
In this chapter, we will introduce three cornerstone approaches to the definition of algorithmic randomness for infinite sequences. (i) The...
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Rough Sets and Knowledge Technology 4th International Conference, RSKT 2009, Gold Coast, Australia, July 14-16, 2009, Proceedings
This book constitutes the refereed proceedings of the Fourth International Conference on Rough Sets and Knowledge Technology, RSKT 2009, held in Gold...
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On Construction of Partial Association Rules
This paper is devoted to the study of approximate algorithms for minimization of partial association rule length. It is shown that under some natural...
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Estimation of Mutual Information: A Survey
A common problem found in statistics, signal processing, data analysis and image processing research is the estimation of mutual information, which...