Examples of using Recursive function in English and their translations into Serbian
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Or(with a return type declared for a recursive function).
A recursive function is a function which uses itself.
Let fn denote the unary primitive recursive function given by this definition.
The recursive function is a function which calls by itself.
Tetration is neither an elementary function nor an elementary recursive function.[1].
A recursive function is a function that makes a call to itself.
The Paris- Harrington theorem involves a total recursive function that is not primitive recursive. .
Recursive function is a function which contains a call to itself.
This means that the n-th definition of a primitive recursive function in this enumeration can be effectively determined from n.
A total recursive function is a partial recursive function that is defined for every input.
An equivalent definition states that a partial recursive function is one that can be computed by a Turing machine.
It is possible to define the arithmetical hierarchy of formulas using a language extended with a function symbol for each primitive recursive function.
The function that takes m to Ackermann(m, m) is a unary total recursive function that is not primitive recursive. .
However, not every μ-recursive function is a primitive recursive function- the most famous example is the Ackermann function. .
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.
Every primitive recursive function is total recursive, but not all total recursive functions are primitive recursive. .
In the equivalence of models of computability, a parallel is drawn between Turing machines that do not terminate for certain inputs andan undefined result for that input in the corresponding partial recursive function.
Thus, the defining characteristic of a structurally recursive function is that the argument to each recursive call is the content of a field of the original input.
This extreme growth can be exploited to show that f, which is obviously computable on a machine with infinite memory such as a Turing machine and so is a computable function, grows faster than any primitive recursive function and is therefore not primitive recursive. .
A universal Turing machine can calculate any recursive function, decide any recursive language, and accept any recursively enumerable language.
A structurally recursive function uses the same idea to define a recursive function:"base cases" handle each minimal structure and a rule for recursion.
Clearly ev is total and computable, since one can effectively determine the definition of fi, andbeing a primitive recursive function fi is itself total and computable, so fi(j) is always defined and effectively computable.
In a properly designed recursive function, with each recursive call, the input problem must be simplified in such a way that eventually the base case must be reached.
Such a proof establishes that the consistency of a theory T implies the consistency of a theory S by producing a primitive recursive function that can transform any proof of an inconsistency from S into a proof of an inconsistency from T.
Every recursive function can be transformed into an iterative function by replacing recursive calls with iterative control constructs and simulating the call stack with a stack explicitly managed by the program.[11][12].
Such a proof establishes that the consistency of a theory T implies the consistency of a theory S by producing a primitive recursive function that can transform any proof of an inconsistency from S into a proof of an inconsistency from T. One sufficient condition for a consistency proof to be finitistic is the ability to formalize it in PRA.
That is, a lower elementary recursive function must be a zero, successor, or projection function, a composition of other lower elementary recursive functions, or the bounded sum of another lower elementary recursive function. .
In contemporary use, the term"computable function" has various definitions: according to Cutland(1980),it is a partial recursive function(which can be undefined for some inputs), while according to Soare(1987) it is a total recursive(equivalently, general recursive) function.
Wrapper functions can be used to validate parameters(so the recursive function can skip these), perform initialization(allocate memory, initialize variables), particularly for auxiliary variables such as"level of recursion" or partial computations for memoization, and handle exceptions and errors.
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).