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PRALine A programming language for primitive recursive arithmeticPowerPoint Presentation

PRALine A programming language for primitive recursive arithmetic

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PRALine A programming language for primitive recursive arithmetic

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PRALine A programming language for primitive recursive arithmetic

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PRALineA programming language for primitive recursive arithmetic

Ben Braun, Joe Rogers

The University of Texas at Austin

November 28, 2012

- Primitive recursive arithmetic is consistent.
- Many functions over natural numbers are primitive recursive:

ADD, MULTIPLY, EXPONENT, LESS-THAN, GREATER-THAN, FACTORIAL, PREDECESSOR, MINIMUM, MAXIMUM, ABS, SIGN, EQUALS, MOD,

DIVISIBLE, IS-PRIME, …

Proof in G. Gentzen, 1936. 'Die Widerspruchfreiheit der reinenZahlentheorie'. MathematischeAnnalen, 112:493–565. Translated as 'The consistency of arithmetic', in (M. E. Szabo 1969).

- Develop a language, PRALine, for expressing primitive recursive arithmetic
- Develop tools which output proofs for primitive recursive arithmetic queries

Output

PRALine code

Defadd as

(x 0 x)

(x y' (x y add)')

end

eval (2 3 add)

(2 0 2) in add

(2 1 3) in add

(2 2 4) in add

(2 3 5) in add

Successor

Example

Defadd as

(x 0 x)

(x y' (x y add)')

end

eval (2 3 add)

DefLT as

(x 0 0)

(0 x 1)

(x' y' (x y LT))

end

- Relational notation can describe functions which are not strictly primitive recursive, such as the Ackermann function:

DefAckermann as

(0 n n’)

(m’ 0 (m 1 Ackermann))

(m’ n’ (m (m’ n Ackermann) Ackermann))

end

Proof that the Ackermann function is not PR: Doetzel, G. “A function to End All Functions.” Algorithm: Recreational Programming 2.4, 16-17, 1991.

- The PRALine language expresses functions in Relational Notation
- We present tools which can verifiably compute a query expressed in PRALine
- The code is designed to be extensible

- Check that all functions are strictly primitive recursive
- Add lists to the language (where list lengths known at compile time)
- Emitting to different languages, especially those that support tail recursion, to allow larger computations