Examples of using Computable function in English and their translations into Serbian
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That reduction function must be a computable function.
This argument provides a computable function which is not primitive recursive.
Enderton[1977] gives the following characteristics of a procedure for computing a computable function;
This means that there is a single computable function f(e, n) such that.
Enderton goes on to list several clarifications of these 3 requirements of the procedure for a computable function.
This argument provides a total computable function that is not primitive recursive.
Every computable function has a finite procedure giving explicit, unambiguous instructions on how to compute it.
Some coding system must be developed to allow a computable function to take an arbitrary word in the language as input;
There are Turing computable partial functions that have no extension to a total Turing computable function.
Not every total computable function is provably total in Peano arithmetic, however;
Primitive recursive functions tend to correspond very closely with our intuition of what a computable function must be.
If g were a total computable function extending f then g would be computable by some Turing machine;
Many equivalent models of computation are known, andthey all give the same definition of computable function(or a weaker version, in some instances).
Thus every computable function must have a finite program that completely describes how the function is to be computed.
Every Turing machine computes a certain fixed partial computable function from the input strings over its alphabet.
If g were a total computable function extending f then g would be computable by some Turing machine; fix e as the index of such a machine.
This set is recursively enumerable,which means there is a computable function that lists all of the pairs(i, x) it contains.
As with the concept of a computable function relative computability can be given equivalent definitions in many different models of computation.
Enderton goes on to list several clarifications of these 3 requirements of the procedure for a computable function: The procedure must theoretically work for arbitrarily large arguments.
Not every total computable function is provably total in Peano arithmetic, however; an example of such a function is provided by Goodstein's theorem.
However the set of primitive recursive functions does not include every possible computable function--- this can be seen with a variant of Cantor's diagonalization argument.
The partial function f defined so that f(n)= m if and only if the Turing machine with index n halts on input 0 with output m has no extension to a total computable function.
Some coding system must be developed to allow a computable function to take an arbitrary word in the language as input; this is usually considered routine.
Displaystyle T_{1},\ldots T_{2},\ldots}of Turing machines that compute total functions and so that every total computable function is computable by one of the machines Ti.
Before the precise definition of computable function mathematicians often used the informal term effectively calculable to describe functions that are computable by paper-- pencil methods.
A recursively enumerable language is a formal language for which there exists a Turing machine(or other computable function) which will enumerate all valid strings of the language.
Nowadays these are often considered as a single hypothesis, the Church- Turing thesis,which states that any function that is computable by an algorithm is a computable function.
A language is called computable(synonyms: recursive, decidable)if there is a computable function f such that for each word w over the alphabet, f(w)= 1 if the word is in the language and f(w)= 0 if the word is not in the language.
Can every partial function computable by a partial Turing machine be extended(that is, have its domain enlarged)to become a total computable function?
We shall use the expression"computable function" to mean a function calculable by a machine, and we let"effectively calculable" refer to the intuitive idea without particular identification with any one of these definitions".