Type-Outference with Label-Listeners: Foundations for Decidable Type-Consistency for Nominal Object-Oriented Generics

We present the first sound and complete decision procedure capable of validating nominal object-oriented programs without type annotations in expressions even in the presence of generic interfaces with irregularly-recursive signatures. Such interface signatures are exemplary of the challenges an algorithm must overcome in order to scale to the needs of modern major typed object-oriented languages. We do so by incorporating event-driven label-listeners into our constraint system, enabling more complex aspects of program validation to be generated on demand while still ensuring termination. Furthermore, we define type-consistency as a novel declarative notion of program validity that ensures safety without requiring a complex grammar of types to perform inference within. While type-inferability ensures type-consistency, the converse does not necessarily hold. Thus our algorithm decides program validity with out inferring the types missing from the program, and so we instead call it a type-outference algorithm. By bypassing type-inference, the proofs involving type-consistency more directly connect the design of and reasoning about constraints to the operational semantics of the language, simplifying much of the design and verification process. We mechanically formalize and verify these concepts and techniques—type-consistency, type-outference, and label-listeners—in Rocq to provide a solid foundation for scaling decidability beyond the limitations of structural types.