Editing Chaitin's constant (section)

== Background ==
The definition of a halting probability relies on the existence of a prefix-free universal computable function. Such a function, intuitively, represents a program in a programming language with the property that no valid program can be obtained as a proper extension of another valid program.

Suppose that {{mvar|F}} is a [[partial function]] that takes one argument, a finite binary string, and possibly returns a single binary string as output. The function {{mvar|F}} is called [[Computable function|computable]] if there is a [[Turing machine]] that computes it, in the sense that for any finite binary strings {{mvar|x}} and {{mvar|y}},  {{math|1=''F''(''x'') = ''y''}} if and only if the Turing machine halts with {{mvar|y}} on its tape when given the input {{mvar|x}}.

The function {{mvar|F}} is called [[Computational universality|universal]] if for every computable function {{mvar|f}} of a single variable there is a string {{mvar|w}} such that for all {{mvar|x}}, {{math|1=''F''(''w''&nbsp;''x'') = ''f''(''x'')}}; here {{math|''w''&nbsp;''x''}} represents the [[concatenation]] of the two strings {{mvar|w}} and {{mvar|x}}.  This means that {{mvar|F}} can be used to simulate any computable function of one variable.  Informally, {{mvar|w}} represents a "script" for the computable function {{mvar|f}}, and {{mvar|F}} represents an "interpreter" that parses the script as a prefix of its input and then executes it on the remainder of input.

The [[Domain of a function|domain]] of {{mvar|F}} is the set of all inputs {{mvar|p}} on which it is defined. For {{mvar|F}} that are universal, such a {{mvar|p}} can generally be seen both as the concatenation of a program part and a data part, and as a single program for the function {{mvar|F}}.

The function {{mvar|F}} is called prefix-free if there are no two elements {{mvar|p}}, {{mvar|p&prime;}} in its domain such that {{mvar|p&prime;}} is a proper extension of {{mvar|p}}. This can be rephrased as: the domain of {{mvar|F}} is a [[prefix-free code]] (instantaneous code) on the set of finite binary strings. A simple way to enforce prefix-free-ness is to use machines whose means of input is a binary stream from which bits can be read one at a time. There is no end-of-stream marker; the end of input is determined by when the universal machine decides to stop reading more bits, and the remaining bits are not considered part of the accepted string. Here, the difference between the two notions of program mentioned in the last paragraph becomes clear: one is easily recognized by some grammar, while the other requires arbitrary computation to recognize.

The domain of any universal computable function is a [[computably enumerable set]] but never a [[computable set]].  The domain is always [[Turing degree|Turing equivalent]] to the [[halting problem]].