Μ-recursive function

The strong equality operator can be used to compare partial μ-recursive functions.

Equivalent definitions can be given using μ-recursive functions, Turing machines or λ-calculus as the formal representation of algorithms.

However, not every μ-recursive function is a primitive recursive function-the most famous example is the Ackermann function.

In the remainder of this article, functions from natural numbers to natural numbers are used (as is the case for, e.g., the μ-recursive functions).

Particular models of computability that give rise to the set of computable functions are the Turing-computable functions and the μ-recursive functions.

In fact, in computability theory it is shown that the μ-recursive functions are precisely the functions that can be computed by Turing machines.

The most widely-studied models of computability are the Turing-computable and μ-recursive functions, and the lambda calculus, all of which have computationally equivalent power.

The μ-recursive functions are closely related to primitive recursive functions, and their inductive definition (below) builds upon that of the primitive recursive functions.

It should be no surprise that we can achieve this encoding given the existence of a Gödel number and computational equivalence between Turing machines and μ-recursive functions.

In mathematical logic and computer science, the μ-recursive functions are a class of partial functions from natural numbers to natural numbers which are "computable" in an intuitive sense.

The 'μ-recursive functions' (or 'partial μ-recursive functions') are partial functions that take finite tuples of natural numbers and return a single natural number.

In the first half of the 20th century, various formalisms were proposed to capture the informal concept of a computable function, with μ-recursive functions, Turing Machines and the lambda calculus possibly being the best-known examples today.

The definition in terms of μ-recursive functions as well as a different definition of rekursiv functions by Gödel led to the traditional name recursive for sets and functions computable by a Turing machine.

The use of Turing machines here is not necessary; there are many other models of computation that have the same computing power as Turing machines; for example the μ-recursive functions obtained from primitive recursion and the μ operator.

The number e is called an index or Gödel number for the function f. A consequence of this result is that any μ-recursive function can be defined using a single instance of the μ operator applied to a (total) primitive recursive function.

If we eschew the Minsky approach of one monster number in one register, and specify that our machine model will be "like a computer" we have to confront this problem of indirection if we are to compute the recursive functions (also called the μ-recursive functions )-both total and partial varieties.