Subtyping can have a simple semantics

Abstract

Consider a rst order typed language, with semantics ] ] for expressions and types. Adding subtyping means that a partial order on types is de ned and that the typing rules are extended to the e ect that expression e has type whenever e has type and . We s h o w h o w to adapt the semantics ] ] i n a simple set-theoretic way, obtaining a semantics f g ] that satis es, in addition to some obvious requirements, also the property: f g] f g], whenever .whether there exists a (mathematical) semantics for types (and subtyping). Let us denote the semantics of closed expressions e and types by e] ] and ] ], respectively. It would be nice if the semantics ] ] o f t ype is merely a set such t h a t e] ] 2 ] ] w h e n e v er e has type . However, only for simple (so-called rst order, non-recursive) types such a simple set-theoretic semantics seems possible. Most often one nds types interpreted as \domains" (continuous lattices or the like) and sometimes a set-theoretic interpretation is proved to beimpossible Reynolds 1984]. The semantics of subtyping is our prime concern in this paper.We set out to construct, by simple set-theoretic means, a semantics for types {in the presence of subtyping{ such t h a t ] ]] ] whenever This poses serious semantical problems. Consider for example the following situation: Assume that ( ! )] ] = some non-empty set of functions that have domain ] ] and co-domain ] ]Assume that int] ] real] ] Assume that int real, so that, as motivated in Section 3, (real! ) (int ! ) We then nd that the desire (real! )] ] (int ! )] ] c o n tradicts the following two observations:Functions f 2 (real! )] ] cannot belong to (int ! )] ] because the domain of f di ers from int] ] Our solution, on the contrary, is as simple as e ective, and can bestated in a single line. Given a semantics ] ] for the language without subtyping, we form a new semantics f g] when subtyping is added, by de ning f g] = ] ]For now w e h a ve, when , that f g] = S ] ] S ] ] (transitivity of , )Note that we have used only elementary, primary school set-theoretic constructions in the de nition of f g] for types. However, this still leaves us with the problem of de ning f g] for expressions in such a w ay that

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