Examples of using Primitive recursive in English and their translations into Serbian
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Cyrillic
It follows that it is difficult to devise a computable function that is not primitive recursive, although some are known(see the section on Limitations below).
Primitive recursive functions tend to correspond very closely with our intuition of what a computable function must be.
This means that the n-th definition of a primitive recursive function in this enumeration can be effectively determined from n.
A sketch of the proof is as follows: The primitive recursive functions of one argument(i.e., unary functions)
There is a characterization of the primitive recursive functions as a subset of the total recursive functions using the Ackermann function.
However, the set of primitive recursive functions is not the largest recursively enumerable subset of the set of all total recursive functions.
In fact, it is difficult to devise a total recursive function that is not primitive recursive, although some are known(see the section on Limitations below).
The primitive recursive functions are among the number-theoretic functions,
PR is the complexity class of all primitive recursive functions- or, equivalently, the set of all formal languages that can be decided by such a function.
However, not every μ-recursive function is a primitive recursive function-the most famous example is the Ackermann function.
Most number-theoretic functions which can be defined using recursion on a single variable are primitive recursive.
The primitive recursive functions are the basic functions
is a unary total recursive function that is not primitive recursive.
However, the set of primitive recursive functions does not include every possible total computable function-this can be seen with a variant of Cantor's diagonal argument.
Every primitive recursive function is total recursive,
Skolem(1923) sets out his primitive recursive arithmetic, a very early contribution to the theory of computable functions,
The functions that can be programmed in the LOOP programming language are exactly the primitive recursive functions.
A consequence of this result is that any μ-recursive function can be defined using a single instance of the μ operator applied to a(total) primitive recursive function.
However a diagonal argument will show that the function ev of two arguments is not primitive recursive.
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.