Towards a semantic model for Java wildcards

Abstract

Wildcard types enrich the types expressible in Java, and extend the set of typeable Java programs. Syntactic models and proofs of soundness for type systems related to Java wildcards have been suggested in the past, however, the semantics of wildcards has not yet been studied.In this paper we propose a semantic model for Java wildcards, inspired by work on semantic subtyping, which traditionally interprets types as sets of possible values. To easily reflect the nominal type system of Java, our model is defined in terms of runtime types (closed class types) rather than the structure of the runtime values themselves. To reflect the variance introduced by wildcards, our model interprets types as sets of such closed class types.Having defined our model, we employ a standard semantic notion of subtyping. We show soundness of syntactic subtyping with respect to the semantic version, and demonstrate that completeness fails in the general case. We identify a restricted (but nonetheless rich) type language for which the syntactic notion of subtyping is both sound and complete.

Taming wildcards in Java's type system

“…We also did a manual investigation for implicitly infinite proofs and found none. These findings are significant because Cameron et al proved soundness of wildcards assuming all wildcards and subtyping proofs translate to finite traditional existential types and subtyping proofs [3], and Summers et al gave semantics to wildcards under the same assumptions [14], so although we have determined these assumptions do not hold theoretically our survey demonstrates that they do hold in practice. Nonetheless, we expect that Cameron et al and Summers et al would be able to adapt their work to implicitly infinite types and proofs now that the problem has been identified.…”

“…super Number> is not a subtype of Numbers<Number>, which would be subtypes should one use a complete equivalence algorithm. While from a theoretical perspective this seems to be a weakness, as Summers et al have argued [14], from a practical perspective it is a strength since it forces programmers to use the more precise type whenever they actually rely on that extra precision rather than obscure it through implicit equivalences. Plus, our weaker notion of equivalence is still strong enough to achieve our goal of allowing equivalent types to be used interchangeably (provided they satisfy all applicable restrictions).…”

Towards a Java Subtyping Operad

“…Separately, probably as the most complex feature of Java generics, the addition of "wildcards" (i.e., wildcard type parameters) to Java (in [49], which is based on the earlier research in [29]) also generated some research that is particularly focused on modeling wildcards and variance annotations [48,12,31,11,46]. This significant and substantial work points yet to the need for more research on wildcards and generics.…”

Constrained kinds

“…Summers and Cameron et al [6,43] characterized wildcards in terms of existential types. Our encoding of wildcards in FXG similarly uses existentials, over constraint terms rather than types, however.…”

“…Our encoding of wildcards in FXG similarly uses existentials, over constraint terms rather than types, however. Summers et al [42,43] observe that care must be taken to model assignment to avoid an unsoundness. We leave this extension for future work.…”