 μrecursive function

In mathematical logic and computer science, the μrecursive functions are a class of partial functions from natural numbers to natural numbers which are "computable" in an intuitive sense. In fact, in computability theory it is shown that the μrecursive functions are precisely the functions that can be computed by Turing machines. The μrecursive functions are closely related to primitive recursive functions, and their inductive definition (below) builds upon that of the primitive recursive functions. However, not every μrecursive function is a primitive recursive function — the most famous example is the Ackermann function.
Other equivalent classes of functions are the λrecursive functions and the functions that can be computed by Markov algorithms.
The set of all recursive functions is known as R in computational complexity theory.
Contents
Definition
The μrecursive functions (or partial μrecursive functions) are partial functions that take finite tuples of natural numbers and return a single natural number. They are the smallest class of partial functions that includes the initial functions and is closed under composition, primitive recursion, and the μ operator.
The smallest class of functions including the initial functions and closed under composition and primitive recursion (i.e. without minimisation) is the class of primitive recursive functions. While all primitive recursive functions are total, this is not true of partial recursive functions; for example, the minimisation of the successor function is undefined. The set of total recursive functions is a subset of the partial recursive functions and is a superset of the primitive recursive functions; functions like the Ackermann function can be proven to be total recursive, and not primitive.
The first three functions are called the "initial" or "basic" functions: (In the following the subscripting is per Kleene (1952) p. 219. For more about some of the various symbolisms found in the literature see Symbolism below.)
 Constant function: For each natural number and every :
 .
 Alternative definitions use compositions of the successor function and use a zero function, that always returns zero, in place of the constant function.
 Successor function S:
 Projection function (also called the Identity function ): For all natural numbers such that :
 .
 Composition operator (also called the substitution operator): Given an mary function and m kary functions :
 .
 Primitive recursion operator : Given the kary function and k+2 ary function :
 .
 Minimisation operator : Given a (k+1)ary total function :
 Intuitively, minimisation seeksbeginning the search from 0 and proceeding upwardsthe smallest argument that causes the function to return zero; if there is no such argument, the search never terminates.
The strong equality operator can be used to compare partial μrecursive functions. This is defined for all partial functions f and g so that
holds if and only if for any choice of arguments either both functions are defined and their values are equal or both functions are undefined.
Equivalence with other models of computability
In the equivalence of models of computability, a parallel is drawn between Turing machines which do not terminate for certain inputs and an undefined result for that input in the corresponding partial recursive function. The unbounded search operator is not definable by the rules of primitive recursion as those do not provide a mechanism for "infinite loops" (undefined values).
Normal form theorem
A normal form theorem due to Kleene says that for each k there are primitive recursive functions and such that for any μrecursive function with k free variables there is an e such that
 .
The number e is called an index or Gödel number for the function f. A consequence of this result is that any μrecursive function can be defined using a single instance of the μ operator applied to a (total) primitive recursive function.
Minsky (1967) observes (as does BoolosBurgessJeffrey (2002) pp. 94–95) that the U defined above is in essence the μrecursive equivalent of the universal Turing machine:
 To construct U is to write down the definition of a generalrecursive function U(n, x) that correctly interprets the number n and computes the appropriate function of x. to construct U directly would involve essentially the same amount of effort, and essentially the same ideas, as we have invested in constructing the universal Turing machine. (italics in original, Minsky (1967) p. 189)
Symbolism
A number of different symbolisms are used in the literature. An advantage to using the symbolism is a derivation of a function by "nesting" of the operators one inside the other is easier to write in a compact form. In the following we will abbreviate the string of parameters x_{1}, ..., x_{n} as x:
 Constant function: Kleene uses " C_{q}^{n}(x) = q " and BoolosBurgessJeffry (2002) (BBJ) use the abbreviation " const_{n}( x) = n ":

 e.g. C_{13}^{7} ( r, s, t, u, v, w, x ) = 13
 e.g. const_{13} ( r, s, t, u, v, w, x ) = 13
 Successor function: Kleene uses x' and S for "Successor". As "successor" is considered to be primitive, most texts use the apostrophe as follows:

 S(a) = a +1 =_{def} a', where 1 =_{def} 0', 2 =_{def} 0 ' ', etc.
 Identity function: Kleene (1952) uses " U_{i}^{n} " to indicate the identity function over the variables x_{i}; BBJ use the identity function id_{i}^{n} over the variables x_{1} to x_{n}:
 U_{i}^{n}( x ) = id_{i}^{n}( x ) = x_{i}
 e.g. U_{3}^{7} = id_{3}^{7} ( r, s, t, u, v, w, x ) = t
 Composition (Substitution) operator: Kleene uses a boldface S_{n}^{m} (not to be confused with his S for "successor" ! ). The superscript "m" refers to the m^{th} of function "f_{m}", whereas the subscript "n" refers to the n^{th} variable "x_{n}":
 If we are given h( x )= g( f_{1}(x), ... , f_{m}(x) )
 h(x) = S_{m}^{n}(g, f_{1}, ... , f_{m} )
 In a similar manner, but without the sub and superscripts, BBJ write:
 h(x')= Cn[g, f_{1} ,..., f_{m}](x)
 Primitive Recursion: Kleene uses the symbol " R^{n}(base step, induction step) " where n indicates the number of variables, BBJ use " Pr(base step, induction step)(x)". Given:

 base step: h( 0, x )= f( x ), and
 induction step: h( y+1, x ) = g( y, h(x,y),x )
 Example: primitive recursion definition of a + b:

 base step: f( 0, a ) = a = U_{1}^{1}(a)
 induction step: f( b' , a ) = ( f ( b, a ) )' = g( b, f( b, a), a ) = g( b, c, a ) = c' = S(U_{2}^{3}( b, c, a )
 R^{2} { U_{1}^{1}(a), S [ (U_{2}^{3}( b, c, a ) ] }
 Pr{ U_{1}^{1}(a), S[ (U_{2}^{3}( b, c, a ) ] }

Example: Kleene gives an example of how to perform the recursive derivation of f(b, a) = b + a (notice reversal of variables a and b). He starting with 3 initial functions

 S(a) = a'
 U_{1}^{1}(a) = a
 U_{2}^{3}( b, c, a ) = c
 g(b, c, a) = S(U_{2}^{3}( b, c, a )) = c'
 base step: h( 0, a ) = U_{1}^{1}(a)
 induction step: h( b', a ) = g( b, h( b, a ), a )
He arrives at:

 a+b = R^{2}[ U_{1}^{1}, S_{1}^{3}(S, U_{2}^{3}) ]
Examples
See also
External links
References
 Stephen Kleene (1952) Introduction to Metamathematics. WaltersNoordhoff & NorthHolland, with corrections (6th imprint 1971); Tenth impression 1991, ISBN 0720421039.
 Soare, R. Recursively enumerable sets and degrees. SpringerVerlag 1987.
 Marvin L. Minsky (1967), Computation: Finite and Infinite Machines, PrenticeHall, Inc. Englewood Cliffs, N.J.
 On pages 210215 Minsky shows how to create the μoperator using the register machine model, thus demonstrating its equivalence to the general recursive functions.
 George Boolos, John Burgess, Richard Jeffrey (2002), Computability and Logic: Fourth Edition, Cambridge University Press, Cambridge, UK. Cf pp. 70–71.
Categories:  Constant function: For each natural number and every :
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