Chaitin-Kolmogorov randomness (also called algorithmic randomness) defines a string (usually of bits) as being random if and only if it is shorter than any computer program that can produce that string. It is fairly easy to see that most strings of a given length are close to random in this sense. In computer programming and some branches of mathematics, strings are sequences of various simple objects. ... This article is about the unit of information. ... It has been suggested that random number be merged into this article or section. ... // A computer program or software program (usually abbreviated to a program) is a step-by-step list of instructions written for a particular computer architecture in a particular computer programming language. ...
Chaitin-Kolmogorov randomness distinguishes, at least in principle, between numbers that are generated by pseudo-random number generators and true random numbers. Chaitin's incompleteness theorem shows that though we know that most strings are random in the above sense, the fact that a specific string is random can never be proven, if the string's length is above a certain threshold. A pseudorandom number generator (PRNG) is an algorithm that generates a sequence of numbers, the elements of which are approximately independent of each other. ... In statistics, a random number is a single observation (outcome) of a specified random variable. ... Gregory J. Chaitin (born 1947) is an American contemporary mathematician and computer scientist. ...
Contrast with statistical randomness. A numeric sequence is said to be statistically random when it contains no recognizable patterns or regularities; sequences such as the results of a die roll, or the digits of Pi (as far as we can tell) exhibit statistical randomness. ...
Randomizedalgorithms (to solve a problem) are allowed to make errors and produce incorrect outputs on some sequences of random bits.
For decision problems, if the randomizedalgorithm also provides a witness along with a yes/no answer, then the correctness of the answer can be verified by evaluating the witness (like search problems).
The randomized reductions defined in Definition 6.2 are closed for randomized ap-time and they are transitive.
Randomizedalgorithms are particularly useful when faced with a malicious "adversary" or attacker who deliberately tries to feed a bad input to the algorithm (see competitive analysis).
The most basic randomizedcomplexity class is RP, which is the class of decision problems for which there is an efficient (polynomial time) randomizedalgorithm (or probabilistic Turing machine) which recognizes NO-instances with absolute certainty and recognizes YES-instances with a probability of at least 1/2.
Historically, the study of randomizedalgorithms was spurred by the discovery by Miller and Rabin in 1976 that the problem of determining the primality of a number can be solved efficiently by a randomizedalgorithm.
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