PRALine A programming language for primitive recursive arithmetic

1. PRALineA programming language for primitive recursive arithmetic Ben Braun, Joe Rogers The University of Texas at Austin November 28, 2012

2. 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).

3. Objective • Develop a language, PRALine, for expressing primitive recursive arithmetic • Develop tools which output proofs for primitive recursive arithmetic queries

4. 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

5. Example Defadd as (x 0 x) (x y' (x y add)') end eval (2 3 add)

6. Abstract Syntax Tree

7. Implementation

8. Live demo DefLT as (x 0 0) (0 x 1) (x' y' (x y LT)) end

9. 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.

10. 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

11. 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