Notes on Abstract Syntax 15 - 312 : Foundations of Programming Languages

Syntax L3.3 k nat num(k) expr t1 expr t2 expr plus(t1, t2) expr t1 expr t2 expr times(t1, t2) expr Now we specify the proper relation between concrete and abstract syntax through several simultaneously inductive judgments. Perhaps the easiest way to generate these judgments is to add the corresponding abstract syntax terms to each of the inference rules defining the concrete syntax. 0 D←→ 0 nat · · · 9 D←→ 9 nat s D←→ k nat s N←→ k nat s1 N←→ k1 nat s2 D←→ k2 nat s1 s2 N←→ 10k1 + k2 nat s T←→ t expr s E←→ t expr s1 E←→ t1 expr s2 T←→ t2 expr s1+s2 E←→ plus(t1, t2) expr s F←→ t expr s T←→ t expr s1 T←→ t1 expr s2 F←→ t2 expr s1*s2 T←→ times(t1, t2) expr s N←→ k nat s F←→ num(k) expr s E←→ t expr (s) F←→ t expr When giving a specification of the form above, we should verify that the basic properties we expect, actually hold. In this case we would like to check that related strings and terms belong to the correct (concrete or abstract, respectively) syntactic classes. Theorem 1 (i) If s E←→ t expr then s E and t expr. (ii) If s E then there exists a t such that s E←→ t expr. Proof: For each part, by rule induction on the given derivation. In each case we can immediately appeal to the induction hypothesis on all subderivations and construct a derivation of the desired judgment from the SUPPLEMENTARY NOTES SEPTEMBER 3, 2002 L3.4 Abstract Syntax