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Concept
Chaitin's constant
Summary
In the computer science subfield of algorithmic information theory, a Chaitin constant (Chaitin omega number) or halting probability is a real number that, informally speaking, represents the probability that a randomly constructed program will halt. These numbers are formed from a construction due to Gregory Chaitin.
Although there are infinitely many halting probabilities, one for each method of encoding programs, it is common to use the letter Ω to refer to them as if there were only one. Because Ω depends on the program encoding used, it is sometimes called Chaitin's construction when not referring to any specific encoding.
Each halting probability is a normal and transcendental real number that is not computable, which means that there is no algorithm to compute its digits. Each halting probability is Martin-Löf random, meaning there is not even any algorithm which can reliably guess its digits.
Let P be a programming language with only 5 valid programs. Their translations to C++ are as follows:
In this case, 3 programs halt and 2 don't, therefore the Chaitin constant for this programming language is .
The definition of a halting probability relies on the existence of a prefix-free universal computable function. Such a function, intuitively, represents a programming language with the property that no valid program can be obtained as a proper extension of another valid program.
Suppose that F is a partial function that takes one argument, a finite binary string, and possibly returns a single binary string as output. The function F is called computable if there is a Turing machine that computes it (in the sense that for any finite binary strings x and y, F(x) = y if and only if the Turing machine halts with y on its tape when given the input x).
The function F is called universal if the following property holds: for every computable function f of a single variable there is a string w such that for all x, F(w x) = f(x); here w x represents the concatenation of the two strings w and x.
Official source
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