Praline a programming language for primitive recursive arithmetic
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PRALine A programming language for primitive recursive arithmetic. Ben Braun, Joe Rogers The University of Texas at Austin November 28, 2012. Why primitive recursive arithmetic?. Primitive recursive arithmetic is consistent. Many functions over natural numbers are primitive recursive:.

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PRALine A programming language for primitive recursive arithmetic

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Praline a programming language for primitive recursive arithmetic

PRALineA programming language for primitive recursive arithmetic

Ben Braun, Joe Rogers

The University of Texas at Austin

November 28, 2012


Why primitive recursive arithmetic

Why primitive recursive arithmetic?

  • 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).


Objective

Objective

  • Develop a language, PRALine, for expressing primitive recursive arithmetic

  • Develop tools which output proofs for primitive recursive arithmetic queries


Example

Example

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


Praline a programming language for primitive recursive arithmetic

Example

Defadd as

(x 0 x)

(x y' (x y add)')

end

eval (2 3 add)


Abstract syntax tree

Abstract Syntax Tree


Implementation

Implementation


Live demo

Live demo

DefLT as

(x 0 0)

(0 x 1)

(x' y' (x y LT))

end


Relational notation pr

Relational notation ⊇ PR

  • 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.


Conclusions

Conclusions

  • 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


Future work

Future Work

  • 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


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