Partial function
A partial function of the set by the set is a right unambiguous relation , that is, a relation in which each element of the set is assigned at most one element of the set . The concept of partial function is widespread in theoretical computer science , especially in computability theory .
The concept of partial function is a generalization of the concept of function . A function from to is understood to be a left-total , right-unambiguous relation, i.e. a relation in which each element of exactly one element of is assigned. Every function from to is in particular a partial function from to , namely a (left-) total partial function, but not vice versa. In this respect, the term partial function can be misleading. To express that a partial function is actually a function in the strict sense, one sometimes says that it is a total function . The difference between partial functions and (total) functions is: For partial functions holds true , for (total) functions holds true .
The domain of definition of the partial function is the set of all those elements from which an element from is assigned. A partial function is a function if and only if holds.
A partial function from to can be modeled as a function in two ways:
- as a function or
- as a function
- The value ("undefined") must not be in .
Spellings
For “ is a partial function from to ” one writes: or , alternatively also , or . Not recommended are a. the notations as well as , because the former is defined as a (total) function and the latter is easy to be confused with, which means, however, that no (total) function is from to . However, like the former, this is generally not the case.
The spelling “ is undefined” or even “ ” is problematic because the expression is then not allowed. It is clearer to say " is undefined at the point " or as a formula " ".
Examples
- The partial function is undefined at this point because division by is not permitted in the real numbers. One can educate
- or
- partially recursive functions
- an unbounded linear operator
Applications
If an algorithm accepts inputs from the set and delivers outputs from the set , then it computes a partial function of after . The domain of this function is the set of all elements for which the algorithm returns a value. In order to deliver a value, it has to come to an end ( terminate ) with its calculation .
Individual evidence
- ↑ Technical University of Braunschweig Partial and total functions (PDF; 112 kB).
- ↑ ^{a } ^{b} Thomas Holder: partial map classifier , on: nLab, July 3, 2015