PR (complexity)

PR (complexity), in the context of computational complexity theory, stands for Primitive Recursive. It refers to the class of functions that can be defined using primitive recursion. Primitive recursive functions are a subset of total computable functions, meaning that they always halt and return a value for any input.

Definition:

A function is considered primitive recursive if it can be obtained from a basic set of functions through a finite number of applications of the following operations:

• Basic Functions:

  • Zero Function: A constant function that always returns 0. Often denoted as zero(x) = 0.
  • Successor Function: A function that increments its input by 1. Often denoted as succ(x) = x + 1.
  • Projection Function: A function that returns one of its arguments. Often denoted as proj(x1, x2, ..., xn) = xi, where 1 <= i <= n.

• Composition: If f is a primitive recursive function of m variables, and g1, g2, ..., gm are primitive recursive functions of n variables, then the function h defined as h(x1, x2, ..., xn) = f(g1(x1, x2, ..., xn), g2(x1, x2, ..., xn), ..., gm(x1, x2, ..., xn)) is also primitive recursive.

• Primitive Recursion: If f is a primitive recursive function of n variables, and g is a primitive recursive function of n+2 variables, then the function h of n+1 variables defined by:

  • h(0, x1, x2, ..., xn) = f(x1, x2, ..., xn)
  • h(y+1, x1, x2, ..., xn) = g(y, h(y, x1, x2, ..., xn), x1, x2, ..., xn) is also primitive recursive. This provides a means to define a function by specifying its value at 0 and then recursively defining its value at y+1 in terms of its value at y.

Significance:

The class of primitive recursive functions is a fundamental concept in computability theory. While relatively simple to define, it encompasses a broad range of commonly used functions in mathematics and computer science, including addition, multiplication, exponentiation, and many logical operations. However, it is important to note that not all computable functions are primitive recursive. The existence of functions that are computable but not primitive recursive demonstrates the limitations of primitive recursion as a sole means of defining all computable processes. The Ackermann function is a classic example of a computable function that is not primitive recursive. This function grows faster than any primitive recursive function.

Limitations:

The key limitation of primitive recursive functions is that the depth of recursion is fixed in advance. The value used in the recursion step is always the immediately previous value. This prevents the expression of certain functions that require more complex forms of recursion. The Ackermann function escapes this limitation by using its own function result as an argument in recursive calls, which cannot be directly achieved via the primitive recursion scheme.

Relationship to Other Complexity Classes:

Primitive recursive functions are a subset of computable functions (also known as recursive functions). All primitive recursive functions are total computable functions. Computable functions include functions definable by Turing machines. The class of primitive recursive functions is strictly smaller than the class of computable functions; there exist computable functions that are not primitive recursive.