Wikipedia / Elementary recursive function
Wikipedia / Elementary recursive function
The term elementary was originally introduced by László Kalmár in the context of computability theory. He defined the class of elementary recursive functions ("Kalmár elementary functions") as a subset of the primitive recursive functions — specifically, those that can be computed using a limited set of operations such as composition, bounded sums, and bounded products. These functions grow no faster than a fixed-height tower of exponentiation). Not all primitive recursive functions are elementary; for example, tetration of the Grzegorczyk hierarchy.
In computational complexity theory, the term ELEMENTARY refers to a class of decision problems solvable in elementary time — that is, within time bounded by some fixed number of exponentials. Formally:
denotes a k).
Although the name comes from the same historical origin, the ELEMENTARY complexity class deals with decision problems and Turing machine runtime, rather than total functions.
Definition
The definitions of elementary recursive functions are the same as for primitive recursive functions, except that primitive recursion is replaced by bounded summation and bounded product. All functions work over the natural numbers. The basic functions, all of them elementary recursive, are:
Zero function.
Successor function. Via repeated application of a successor function, one can achieve addition.
Projection functions is a projection function.
Subtraction function. This function is used to define conditionals and iteration.
From these basic functions, we can build other elementary recursive functions.
Composition is elementary recursive.
Bounded summation is elementary recursive.
Bounded product is elementary recursive.
Superposition bases for elementary functions
In the context of computability theory, superposition is a method of constructing new functions from existing ones by functional composition. It allows the outputs of one or more functions to serve as the inputs to another function.
More formally, suppose:
-ary function, and
-ary functions.
-ary function:
.
The class of elementary recursive functions coincides with the closure under superposition of the projection functions and one of the following sets of initial functions:
denotes truncated subtraction (monus).
Mihai Prunescu, Lorenzo Sauras-Altuzarra and Joseph M. Shunia proved that the class of Kalmár elementary functions can be inductively generated from addition), integer remainder) and base-two exponentiation), improving previous results by Mazzanti and Marchenkov. They further proved that the substitution basis defined by these three operations is minimal. is an alternative basis.
Example 1:
by superposition alone. This shows how functions like squaring can be expressed using only addition, integer remainder, and base-two exponentiation through superposition, without requiring explicit recursion.
Example 2:
Another example of an elementary recursive function is the Kronecker delta otherwise.
Further examples
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Lower elementary recursive functions
Lower elementary recursive functions follow the definitions as above, except that bounded product is disallowed. 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.
Lower elementary recursive functions are also known as Skolem elementary functions.
Whereas elementary recursive functions have potentially more than exponential growth, the lower elementary recursive functions have polynomial growth.
The class of lower elementary functions has a description in terms of composition of simple functions analogous to that we have for elementary functions. is a bitwise AND of and .
See also
Notes
References
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Further reading
External links