Delaware, Benjamin × Keuchel, Steven Schrijvers, Tom Oliveira, Bruno C.d.S. #
ACM SIGPLAN Notices vol:48 issue:9 pages:319-330
This paper presents 3MT, a framework for modular
mechanized meta-theory of languages with effects. Using 3MT,
individual language features and their corresponding definitions --
semantic functions, theorem statements and proofs
-- can be built separately and then reused to create different
languages with fully mechanized meta-theory. 3MT combines
modular datatypes and monads to define
denotational semantics with effects on a per-feature basis, without fixing the
particular set of effects or language constructs.
One well-established problem with type soundness proofs for
denotational semantics is that they are notoriously brittle with
respect to the addition of new effects.
The statement of type soundness for a language depends intimately on the
effects it uses, making it particularly
challenging to achieve modularity. 3MT solves this long-standing problem by splitting these theorems into
two separate and reusable parts: a feature theorem that captures
the well-typing of denotations
produced by the semantic function of an invidual feature with respect
to only the effects used,
and an effect theorem that adapts well-typings of denotations to a
fixed superset of effects.
The proof of type soundness for a particular language simply combines these theorems for its features
and the combination of their effects.
To establish both theorems, 3MT uses two key reasoning techniques:
modular induction and algebraic laws about effects. Several
effectful language features, including references and errors, illustrate the
capabilities of 3MT. A case study reuses these features to build fully
mechanized definitions and proofs for 28 languages, including several
versions of mini-ML with effects.