A correspondence between type checking via reduction and type checking via evaluation

Abstract

We describe a derivational approach to proving the equivalence of different representations of a type system. Different ways of representing type assignments are convenient for particular applications such as reasoning or implementation, but some kind of correspondence between them should be proven. In this paper we address two such semantics for type checking: one, due to Kuan et al., in the form of a term rewriting system and the other in the form of a traditional set of derivation rules. By employing a set of techniques investigated by Danvy et al., we mechanically derive the correspondence between a reductionbased semantics for type-checking and a traditional one in the form of derivation rules, implemented as a recursive descent. The correspondence is established through a series of semantics-preserving functional program transformations.

From type checking by recursive descent to type checking with an abstract machine

“…For this purpose, appropriate completeness and soundness theorems need to be proven. As it was shown recently by Sergey and Clarke [25], a correspondence between a traditional type system and a corresponding reduction-based semantics for type inference can be provided by the construction and inter-derivation of their computational counterparts in the spirit of the current work and recently described systematic functional transformations [6].…”

“…extend (x , arg_type , gamma ))) in T_ARR ( arg_type , body_type ) end | check0 ( APP (e1 , e2 ), gamma ) = let val T_ARR (t1 , t2 ) = check0 (e1 , gamma ) val arg_type = check0 (e2 , gamma ) in if arg_type = t1 then t2 else raise ( TYPING_ERROR " type mismatch ") end ( * type_check : term -> typ * ) fun type_check t = check0 (t , TEnv . empty ) 2 In order to keep to the uniform approach for different semantics for type inference [17,25], we leave environments parametrized by the type parameter 'a, which is instantiated with typ in this case.…”

From Outermost Reduction Semantics to Abstract Machine

“…In particular, it always works for innermost reduction, and has given rise to a 'syntactic correspondence' between reduction semantics and abstract machines [2,3]. This syntactic correspondence has proved successful to reconstruct many preexisting abstract machines as well as to construct new ones [1,7,17,30], even in the presence of control operators [3,8]. For a class of examples, it applies to all the reduction semantics of Felleisen et al's latest textbook [15].…”

A logical correspondence between natural semantics and abstract machines

“…These relations are stated using a diversity of methods and methodologies. To the best of the authors' knowledge, only one, the Reynolds functional correspondence [1], [2] has seen repeated use outside the work of its inventors [3], [4]. Our goal [5], [6] is to develop a logical counterpart.…”