Complexity theory randomness

What is the complexity theory in physics?
The concept of complexity has its origins in quantum information science, an area developed within the framework of quantum mechanics.
The general idea behind complexity is to quantify how difficult it is to reach a certain quantum state starting from another one..
What is the concept of randomness?
Randomness describes a phenomenon in which the outcome of a single repetition is uncertain, but there is nonetheless a regular distribution of relative frequencies in a large number of repetitions.
The main focus here is on randomness and its relationship to variation and expectation..
What is the law of randomness?
These rules state that even though a single random event might be completely unpredictable, a collection of independent random events is extremely predictable — and the larger the number of events, the more predictable they become..
What is the theory of randomness?
A random sequence of events, symbols or steps often has no order and does not follow an intelligible pattern or combination.
Individual random events are, by definition, unpredictable, but if the probability distribution is known, the frequency of different outcomes over repeated events (or "trials") is predictable..
Why is it important to have randomness in the process?
Randomness has very important applications in many areas of mathematics.
In statistics, the selection of a random sample is important to ensure that a study is conducted without bias.
A simple random sample is obtained by numbering every member of the population of interest, and assigning each member a numerical label..
A random sequence of events, symbols or steps often has no order and does not follow an intelligible pattern or combination.
Individual random events are, by definition, unpredictable, but if the probability distribution is known, the frequency of different outcomes over repeated events (or "trials") is predictable.
The same principle applies to many other “random” events in life, including dice and roulette wheels.
They are not really random, we simply don't have the tools to do the mathematical calculations accurately enough to predict the outcome.
But true randomness does exists – at the very foundations of matter.

The Komogorov complexity of a finite binary string is the length of its shortest describing program. Intuitively, it is a measure of the amount of information the string contains. A string is random if it cannot be compressed, i.e., its shortest program is as long as the string itself.

From the point of view of computational complexity theory, there's no difference between a random sequence and a sequence which is hard to predict.

In this paper we present a complexity theoretic definition of randomness derived from Kolmogorov complexity. The Komogorov complexity of a finite binary string

A randomness extractor, often simply called an extractor, is a function, which being applied to output from a weak entropy source, together with a short, uniformly random seed, generates a highly random output that appears independent from the source and uniformly distributed.
Examples of weakly random sources include radioactive decay or thermal noise; the only restriction on possible sources is that there is no way they can be fully controlled, calculated or predicted, and that a lower bound on their entropy rate can be established.
For a given source, a randomness extractor can even be considered to be a true random number generator (TRNG); but there is no single extractor that has been proven to produce truly random output from any type of weakly random source.

The concept of a random sequence is essential in probability theory and statistics.
The concept generally relies on the notion of a sequence of random variables and many statistical discussions begin with the words let X1,...,Xn be independent random variables....
Yet as D.
H.
Lehmer stated in 1951: A random sequence is a vague notion... in which each term is unpredictable to the uninitiated and whose digits pass a certain number of tests traditional with statisticians.

Randomized polynomial time class of computational complexity theory

In computational complexity theory, randomized polynomial time (RP) is the complexity class of problems for which a probabilistic Turing machine exists with these properties:

Complexity theory randomness

What is the complexity theory in physics?

What is the concept of randomness?

What is the law of randomness?

What is the theory of randomness?

Why is it important to have randomness in the process?