Relationship To Recursive Functions
The broader class of partial recursive functions is defined by introducing an unbounded search operator. The use of this operator may result in a partial function, that is, a relation with at most one value for each argument, but does not necessarily have any value for any argument (see domain). An equivalent definition states that a partial recursive function is one that can be computed by a Turing machine. A total recursive function is a partial recursive function that is defined for every input.
Every primitive recursive function is total recursive, but not all total recursive functions are primitive recursive. The Ackermann function A(m,n) is a well-known example of a total recursive function that is not primitive recursive. There is a characterization of the primitive recursive functions as a subset of the total recursive functions using the Ackermann function. This characterization states that a function is primitive recursive if and only if there is a natural number m such that the function can be computed by a Turing machine that always halts within A(m,n) or fewer steps, where n is the sum of the arguments of the primitive recursive function.
An important property of the primitive recursive functions is that they are a recursively enumerable subset of the set of all total recursive functions (which is not itself recursively enumerable). This means that there is a single computable function f(e,n) such that:
- For every primitive recursive function g, there is an e such that g(n) = f(e,n) for all n, and
- For every e, the function h(n) = f(e,n) is primitive recursive.
However, the primitive recursive functions are not the largest recursively enumerable set of total computable functions.
Read more about this topic: Primitive Recursive Function
Famous quotes containing the words relationship to, relationship and/or functions:
“Artists have a double relationship towards nature: they are her master and her slave at the same time. They are her slave in so far as they must work with means of this world so as to be understood; her master in so far as they subject these means to their higher goals and make them subservient to them.”
—Johann Wolfgang Von Goethe (1749–1832)
“In contrast with envy, which usually occurs between two people and is focused upon another person’s qualities or possessions, jealousy occurs when a third person becomes a threat to a dyad. Jealousy involves the loss or the impending loss of a relationship that one wants to hold onto, a relationship that is vital to personal fulfillment and claimed as one’s own.”
—Carol S. Becker (b. 1942)
“Nobody is so constituted as to be able to live everywhere and anywhere; and he who has great duties to perform, which lay claim to all his strength, has, in this respect, a very limited choice. The influence of climate upon the bodily functions ... extends so far, that a blunder in the choice of locality and climate is able not only to alienate a man from his actual duty, but also to withhold it from him altogether, so that he never even comes face to face with it.”
—Friedrich Nietzsche (1844–1900)