alternative characterizations of recursive functions
The class of recursive functions may be characterized by considerably weaker conditions than those given in the entry “recursive function (http://planetmath.org/RecursiveFunction)” of this encyclopaedia. This entry will discuss several such characterizations.
Criteria 2 and 3 in the list may be replaced by the considerably weaker criterion:
2’) The successor function defined as is a recursive function.
By means of a pairing function, the definition may be simplified considerably. Using such a function and its inverses, the set of recursive functions of variables may be put in one-to-one correspondence with recursive functions of variables for any pair of non-zero positive integers and . Hence one can focus attention on recursive functions of a small fixed number of variables. One characterization of recursive functions of not more than two variables is the following:
The class of recursive functions is the smallest class of positive integer valued functions of not more than two positive integers which satisfies the following criteria:
-
1’
The constant function defined by for all is a recursive function.
-
2’
The successor function defined as is a recursive function.
-
3’
The projection functions , , and defined as
are recursive functions.
-
4’
If , , ,, and are recursive functions, then , , and , defined by
are recursive functions.
-
5’
If , are recursive functions, then the function defined by the recursion
with the initial condition
is a recursive function.
-
6’
If is a recursive function then is a recursive function, where is defined to equal if there exists a such that
-
(a)
are all defined,
-
(b)
when , and
-
(c)
.
Otherwise, is undefined.
-
(a)
The criterion 5’ may be shown to follow from the remaining criteria, and hence it may be dropped.
By further exploiting the marvelous properties of the pairing function, criterion 6’ may be replaced by the following:
6”) If is a recursive function then is a recursive function, where is defined to equal if there exists a such that
-
a
are all defined,
-
b
when , and
-
c
.
Otherwise, is undefined.
The operation introduced in this new criterion is called minimized inversion and will be denoted as . Note that there is no conflict with the usual notion of inverse of a function because, if is invertible, the minimized inverse of is the same as the inverse of in the usual sense; otherwise the notion of minimized inverse extends the definition of inverse to a larger class of functions.
Finally, by taking these ideas even further, Czirmaz showed that recursive functions of a single variable had the following simple characterization:
The class of recursive functions is the smallest class of positive integer valued functions of a positive integer which satisfies the following criteria:
-
1”
The constant function defined by for all is a recursive function.
-
2”
The successor function defined as is a recursive function.
-
3”
The function defined as is a recursive function. In words, is the difference between and the largest square number smaller than .
-
4”
If and are recursive functions, then is a recursive function, where
-
6”
If is a recursive function then is a recursive function, where is defined to equal if there exists a such that
-
a
are all defined,
-
b
when , and
-
c
.
Otherwise, is undefined.
-
a
Title | alternative characterizations of recursive functions |
---|---|
Canonical name | AlternativeCharacterizationsOfRecursiveFunctions |
Date of creation | 2013-03-22 14:34:42 |
Last modified on | 2013-03-22 14:34:42 |
Owner | rspuzio (6075) |
Last modified by | rspuzio (6075) |
Numerical id | 12 |
Author | rspuzio (6075) |
Entry type | Topic |
Classification | msc 03D20 |
Defines | minimized inversion |