Examples of using Partial computable in English and their translations into Portuguese
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These researchers also use terminology such as"partial computable function" and"computably enumerable""c.e.
Note however that the"partial" computable functions(those that need not be defined for all arguments) can be explicitly enumerated, for instance by enumerating Turing machine encodings.
By the church thesis any theoretic approach to the class of partial computable functions comes to exactly the same class of functions.
If F is a subset of the set of all partial computable function from N{\displaystyle\mathbb{N}} to itself such that F and its complement are both non-empty, then the problem of deciding whether or not a given Turing machine computes a function from F is not decidable on any exponentially generic subset of I. The following theorems are from: Theorem 2 The set of formal languages which are generically computable has measure zero.
Every Turing machine computes a certain fixed partial computable function from the input strings over its alphabet.
Some of these are weak 1-randomness, Schnorr randomness,computable randomness, partial computable randomness.
For any recursive operator Ψ there is a partial computable function φ such that Ψ(φ) φ and φ is the smallest partial computable function with this property.
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 question above has a negative answer.
We write formula_5 for the"i"-th partial computable function under the Gödel numbering formula_2, and formula_7 for the partial computable function formula_8.
The following pseudocode illustrates a straightforward way to compute g: procedure compute g i if f(i,i)== 0 then return 0 else loop forever Because g is partial computable, there must be a program e that computes g, by the assumption that the model of computation is Turing-complete.
The function h{\displaystyle h} can be constructed from the partial computable function g( x, y){\displaystyle g(x, y)} described above and the s-m-n theorem: for each x{\displaystyle x}, h( x){\displaystyle h(x)} is the index of a program which computes the function y↦ g( x, y){\displaystyle y\mapsto gx, y.
Definitions==A Blum complexity measure is a tuple formula_1 with formula_2 a Gödel numbering of the partial computable functions formula_3 and a computable function: formula_4which satisfies the following Blum axioms.
These researchers also use terminology such as partial computable function and computably enumerable(c.e.) set instead of partial recursive function and recursively enumerable(r.e.) set.
Given a natural number formula_14,the function formula_13 outputs the index of the partial computable function that performs the following computation:: Given an input formula_16, first attempt to compute formula_17.
A famous Rice's theorem states that if F is a subset of the set of partial computable functions from N{\displaystyle\mathbb{N}} to{ 0, 1}{\displaystyle\{0,1\}}, then unless F or its complement is empty, the problem of deciding whether or not a particular Turing machine computes a function in F is undecidable.
Given a Gödel numbering φ{\displaystyle\varphi} of recursive functions, there is a primitive recursive function s of two arguments with the following property:for every Gödel number p of a partial computable function f with two arguments, the expressions φ s( p, x)( y){\displaystyle\varphi_{s(p, x)}(y)} and f( x, y){\displaystyle f(x, y)} are defined for the same combinations of natural numbers x and y, and their values are equal for any such combination.
The following pseudocode illustrates a straightforward way to compute"g":Because"g" is partial computable, there must be a program"e" that computes"g", by the assumption that the model of computation is Turing-complete.
Here, a property of partial functions is called"trivial" if it holds for all partial computable functions or for none, and an effective decision method is called"general" if it decides correctly for every algorithm.
Given a natural number x{\displaystyle x},the function h{\displaystyle h} outputs the index of the partial computable function that performs the following computation: Given an input y{\displaystyle y}, first attempt to compute φ x( x){\displaystyle\varphi_{x}x.
Reiterating, the set K{\displaystyle K} referenced above and defined as the domain of the function d( x)( x)+ 1{\displaystyle d(x)=(x)+1}that takes the diagonal of all enumerated 1-place computable partial functions and adds 1 to them is an example of a creative set.
There are Turing computable partial functions that have no extension to a total Turing computable function.
Ultimately, these ideas are connected to Church's thesis that says the mathematical notion of computable partial functions is the correct formalization of an effectively calculable partial function, which can neither be proved or disproved.
Two questions can be asked about the relationship between partial Turing machines and total Turing machines:Can every partial function computable by a partial Turing machine be extended(that is, have its domain enlarged) to become a total computable function?