A Formally Specified Type System and Operational Semantics for Higher-Order Procedural Variables

We formally specified the type system and operational semantics of Loop with Ott and Isabelle/HOL proof assistant. Moreover, both the type system and the semantics of Loop have been tested using Isabelle/HOL program extraction facility for inductively defined relations. In particular, the program that computes the Ackermann function type checks and behaves as expected. The main difference (apart from the choice of an Ada-like concrete syntax) with Loop comes from the treatment of parameter passing. Indeed, since Ott does not currently fully support α-conversion, we rephrased the operational semantics with explicit aliasing in order to implement the out parameter passing mode. Introduction We formally specified the type system and operational semantics of Loopω as described in [CPV09] with Ott [SNO+07] and Isabelle/HOL proof assistant [NPW02]. Moreover, both the type system and the semantics of Loopω have been tested using Isabelle/HOL program extraction facility for inductively defined relations [BN02]. In particular, the program that computes the Ackermann function (reproduced below) type checks and behaves as expected. The main difference (apart from the choice of an Ada-like concrete syntax) with the description given in [CPV09] comes from the treatment of parameter passing. Indeed, since Ott does not currently fully support α-conversion, we rephrased the operational semantics with explicit aliasing in order to implement the out parameter passing mode (instead of a simpler substitution-based semantics as in [CPV09]). On the other hand, the in parameter passing mode is implemented exactly as in [CPV09] and relies on Ott generated substitution (see the Isabelle/HOL code given in appendix). Section 1 contains the description of an Ada-like grammar for Loopω. We then present the type system in Section 2 and the structural operational semantic in section 3. Finally, in the appendix we include the Isabelle/HOL theory generated by Ott (all source files are available on request). Example: the Ackermann function procedure Ack(M : in int; N : in int; R : out int) is P : proc(in int, out int) := Incr; begin for I in 1 . . M loop declare Q : constant proc(in int, out int) := P; procedure Aux(S : in int; R : out int) is X : int := 0; begin Q(1, X); for J in 1 . . S loop Q(X, X); end loop; R := X; end; begin P := Aux; end; end loop; P(N, R); end; 1 ha l-0 03 85 41 6, v er si on 1 19 M ay 2 00 9 1 Syntax index , i , j , l , n indices ident , x , y , z , p, f idents number , q terminals ::= | −→ | → | ⇒ | ← | %→ | ! | & | ∅ | × | (= | := | 〈 | 〉 | ∼ | (∈ | ! mode, m ::= modes: | S | in | out | in out integer , k ::= | q | { k1 + k2 } | { k1 − k2 } | { k1 × k2 } boolean, b ::= | true | false | { b1 and b2 } | { b1 or b2 } | { not b } | { k1 = k2 } | k1 > k2 | k1 < k2 exp, e ::= terms: | x var | v value | e1 + e2 addition | e1 − e2 subtraction | e1 × e2 multiplication | e1 = e2 equality | e1 > e2 greater | e1 < e2 less | e1 and e2 conjunction | e1 or e2 disjunction | not e negation | ( e ) S parentheses