Bounded Randomness

Algorithmic Information Theory Using Kolmogorov Complexity

“…Information can therefore fail to be compressible due to one of two reasons: It can have high entropy and contain true chaos, or it can contain highly complex information in an orderly fashion. This behavior has recently been identified and made precise by researchers and many recent projects have explored this area [1,3]. The evidence from studies suggests that high computational strength is accompanied by low randomness content.…”

“…To develop tools to overcome these problems in practice, a fundamental theoretical question to consider is the issue of how the degree of incompressibility varies with computational power of the information being tested [1][2]. For instance suppose we are given the outcome of a series of random coin toss.…”

Hierarchy of Computably Enumerable Degrees II

“…In the paper [11], Brodhead, Downey and Ng showed that totally ω-c.a. degrees also capture a notion of randomness related to Martin-Löf randomness.…”

“…We pause the enumeration of U n at a stage s at which A s P U n,s . Theorem 3.6 (Brodhead, Downey, Ng [11]). A c.e.…”

Sub-computable Boundedness Randomness

“…Uniform relative Kurtz randomness is studied by Kihara and Miyabe [18], who proved van Lambalgen's theorem holds for their definition. Another special type of bounded randomness was recently studied by Brodhead, Downey and Ng [8].As shown by Wang [34], Kurtz random reals may not be stochastic in the sense of Church [13]. For example, it need not be the case that the number of occurrences of 0's in a Kurtz random sequence X tends to 1/2 in the limit.…”