… As a second example, by considering the power set of the … a randomness space, we introduce
a new notion of a random … idea and provide rigorous definitions of randomness spaces in …

… whenever we have to code "+" or some other algebraic operation allowed in a recursion we
use the … First we define the notion of diophantine equation: D 1.9 (Diophantine equation) A(n) …

… After this we turn to other notions of randomness such as … randomness, where the notion of
what constitutes a random set … notions of randomness, various approaches to the definition of …

… the definitions of the notion of a random sequence using the three main ideas described that
have dominated algorithmic randomness. … of other forms of related randomness notions that …

… In this paper, we define new notions of randomness based on the difference hierarchy. We
… Surprisingly, we can find yet another relationship with Martin-L�f randomness, this time from …

… The purposes of this paper are (i) to define a notion of randomness for sequences of … 0}
each occur once, whereas for all 16 other {0, 5} random sequences, at least one of these length-2 …

… ce test concept defining Martin-L�f randomness is replaced … weaker randomness notions,
like permutation randomness … one set to be more random than another? We also consider …

… notion of complexity serves not only to define randomness … Godel replaced the concept of
truth with that of provability and … theory of random numbers can be made plain through another …

… We show that their notion of injective randomness does not coincide with Martin-L�f
randomness. We start in the next subsection with the definitions of the different types of martingales …

… that we can state the definition (of "Benioff-random") in the … mathematical notions of randomness
and the physical notion, … so many different definitions of mathematical randomness; this …