In mathematics, a partial function from X to Y is a function ƒ: X' → Y, where X' is a subset of X. It generalizes the concept of a function by not forcing f to map every element of X to an element of Y (only some subset X' of X). If X' = X, then ƒ is called a total function and is equivalent to a function. Partial functions are often used when the exact domain, X', is not known (e.g. many functions in computability theory).
Specifically, we will say that for any x ∈ X, either:
- ƒ(x) = y ∈ Y (it is defined as a single element in Y) or
- ƒ(x) is undefined.
For example we can consider the square root function restricted to the integers
Thus g(n) is only defined for n that are perfect squares (i.e. 0, 1, 4, 9, 16, ...). So, g(25) = 5, but g(26) is undefined.
Other articles related to "functions, partial functions, function, partial function, partial":
... the computability of "all computable functions" by the Turing machine model and its equivalents ... "computable" by casting the net wider—by allowing into the notion of "functions" both "total functions" and "partial functions" ... A total function is one that is defined for all natural numbers (positive integers including 0) ...
... In computability theory, a semicomputable function is a partial function that can be approximated either from above or from below by a computable function ... More precisely a partial function is upper semicomputable, meaning it can be approximated from above, if there exists a computable function, where is the desired parameter ...
... In other words, it is an abuse of language to refer to partial functions from E × E to E as "functions from E × E to E that are not everywhere defined." To clarify this, it ... A partial function from A to B is a function f A' → B, where A' is a subset of A ... A function not everywhere defined from A to B is a function f A' → B, where A' is a subset of A ...
... In some automated theorem proving systems a partial function is considered as returning the bottom type when it is undefined ... In computer science a partial function corresponds to a subroutine that raises an exception or loops forever ... In a programming language where function parameters are statically typed, a function may be defined as a partial function because the language's type system cannot express the exact domain of ...
... A general Turing machine will compute a partial function ... Two questions can be asked about the relationship between partial Turing machines and total Turing machines Can every partial function computable by a partial ... The following theorem shows that the functions computable by machines that always halt do not include extensions of all partial computable functions, which implies the first ...
Famous quotes containing the words function and/or partial:
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