Primitive Recursive Functions

1
Primitive Recursive Functions

• Russell Centanni

2
Overview

• Computability Review
• Primitive Recursive Functions
• Addition Example
• Conclusion

3
Computability

• Modern Computers can perform Turing machine
  computable operations.
• Turing machines present computability in a
  mechanical perspective.
• Elementary operations of a machine.
• Primitive Recursive Functions present
  computability in a functional perspective
• Functions are the elementary operations.

4
Primitive Recursive

• Turing Computability and Primitive Recursivity
  are equivalent.
• Turing machine can be simulated with arithmetic
  operations (arithmetization)

5
Primitive Recursive Chomsky Hierarchy
6
Primitive Recursive

• Primitive Recursivity is composed of three
  functions.
• Zero function
• Successor function
• Projection function

7
Zero Function

• z z(x) 0
• Resolves to zero for all inputs, x.
• z(0) 0
• z(1) 0
• z(2) 0
• z(3) 0

8
Successor Function

• s s(x) x 1
• Requires the ability to add 1 to a natural
  number.
• s(1) 1 1 2
• s(2) 2 1 3
• s(3) 3 1 4

9
Projection Function

• Function selects the ith argument out of n
  arguments.

10
Primitive Functions

• Primitive recursive functions are constructed
  from the zero, successor, and projection
  functions by two operations.
• functional composition.
• primitive recursion.
• Both operations preserve computability.
• If a function is composed entirely of primitive
  recursive functions, then it is primitive
  recursive.

11
Functional Composition
12
Primitive Recursion

• Primitive Recursion can be defined by the
  following
• xis are the parameters.
• y is the recursive variable.

13
Primitive Recursion Example
14
Addition

• Base case x 0 x
• Recursive Step
• Next, we will re-write this in terms of the
  Primitive Recursive Functions.

15
Primitive Recursive Addition

• I(x) is an identity function.
• functions s,p are the primitive recursive
  functions as defined before.

16
Addition Example
17
Addition Example
18
Conclusion

• A Turing machine can be simulated by arithmetic
  operations.
• Primitive Recursive functions can be used to
  perform arithmetic operations.
• Primitive Recursive functions can perform the
  operations of a Turing machine.
• Turing Computability and Primitive Recursivity
  are equivalent.

19
References

• Chomsky Hierarchy April 10, 2005.
  lthttp//www.aistudy.co.kr/linguistics/chomsky_hier
  archy.htmgt
• Dewdney, A. K. The New Turing Omnibus. Computer
  Science Press/Freeman, New York, 1993.
• Sudkamp, Thomas A. Languages and Machines An
  Introduction to the Theory of Computer Science,
  Second Edition. Addision Wesley Longman, Inc.
  1997.