Intersection Types and Their Effects in Refinement Type Systems

Davies-Pfenning: Intersection Types and Effects Robert Harper Fall Semester, 2003

Intersection Types • Intersections are fundamental to refinement types:nat! pos Æ pos! ps • Conventional intersection types are incompatible with languages with effects. • Related to unsoundness in the original ML type system. 15-814 Type Refinements

Unsoundness • The classic ML mistake:let x : ‘a list ref = ref nil inlet y = (x := [true]) inlet z = hd(!x)+1 in z • Clearly unsound! But why? • nil has type ‘a list • so (?) ref nil has type ‘a list ref. • Boom. 15-814 Type Refinements

Substitution Principle • Let polymorphism in ML is justified by substitution: • But replication of ref nil allocates two cells, not one! 15-814 Type Refinements

Substitution Principle • Solution: ensure that CBV = CBN. • Replication or removal doesn’t change meaning. • Conservative approximation: the value restriction. • Require bound expression to be a syntactic value (constant, lambda, pair of values, etc). 15-814 Type Refinements

Substitution Principle • But, unfortunately, nil @ nil is not a value! • So it’s not polymorphic. • Annoying, especially interactively. • Valuability: • Every value is valuable. • Application of a total function to a valuable argument is valuable. • Requires a notion of totality. 15-814 Type Refinements

Unsoundness • A related problem:let x = ref (1) : nat ref Æ pos ref inlet y = (x := ) inlet z = !x in(z:pos) • The bit string  1 has type nat Æ pos. • So (?) ref(1) should be nat ref Æ pos ref. • But the value is , which is not pos! 15-814 Type Refinements

Substitution Principle • Typing rule for intersections replicates. • Must restrict subject to be a value (or valuable). • Elim “chooses” which version to use. 15-814 Type Refinements

Subtyping • Subtyping is least pre-order containing: • pos· nat, nat· bits • Co- and contra-variant function subtyping. • Intersections are meets (glb’s). • Ref’s are invariant: 15-814 Type Refinements

Non-distributivity • Do not have the principle • Consider  x.ref(). • OK: unit! nat ref Æ unit! pos ref • Not OK: unit ! (nat ref Æ pos ref) 15-814 Type Refinements

Algorithmic Subtyping • Make transitivity and reflexivity admissible. • Related to confluence in term rewriting. • Precludes searching for arbitrary mediating type: A· B if 9 C A· C· B • Main changes: • No transitivity. • Axioms such as pos· bits. • Special treatment of meets. 15-814 Type Refinements

Algorithmic Subtyping • Key algorithmic rules for meets: • Upper bound is not an intersection. • Without loss of generality. • Forces search order to handle these first. 15-814 Type Refinements

Typing Rules • Judgement form: • Context  governs locations. • Context  governs variables. • Most rules are standard. 15-814 Type Refinements

Constructor Rules 15-814 Type Refinements

Case Rules 15-814 Type Refinements

Intersection, Subsumption 15-814 Type Refinements

Safety • Without going into details, this version of the language is safe. • Define a state machine with a store. • Prove the usual progress and preservation theorems. • See the paper for details; nothing especially surprising. 15-814 Type Refinements

Bidirectional Type Checking • Non-goal oriented type synthesis is infeasible. • Generates more than you’d want to know. • e.g., nine clauses for the addition function. • No principal types, so no “best” answer. • Want some form of type inference. • Just impractical to specify everything. • No good way to specify intersection types. 15-814 Type Refinements

Bidirectional Type Checking • Distinguish synthesizable from checkable terms. • Synthesizable: can infer a type. • Checkable: can be tested against a type. • Add explicit “cast” (ascription) operation. • Specify type of an arbitrary expression. • No run-time overhead! 15-814 Type Refinements

Bidirectional Type Checking • Inferable terms: • Checkable terms: 15-814 Type Refinements

Bidirectional Typing • Synthesis: • Analysis: 15-814 Type Refinements

Bidirectional Typing • Analyzing ’s: • Synthesizing ascriptions: 15-814 Type Refinements

Bidirectional Typing • The inclusion of inferable into checkable is tricky. • Can only check a value against an intersection (for soundness). • Checking inferables requires distinguishing values from non-values. 15-814 Type Refinements

Bidirectional Typing 15-814 Type Refinements

Polymorphism • Essentially, treat as an infinite intersection. • 8 . A · {B/}A • A· B ¾ A·8 B, provided  not free in A • No right-distributivity of 8 over !. • Rest goes through analogously. • Value-restricted polymorphism as in SML. 15-814 Type Refinements

Increment: val inc : (bits! bits)Æ (nat! pos) = fix inc.  n. case n of) 1 j x0 ) x1 j x1) (inc x) 0 • Checker pushes declared type onto function to verify typing. 15-814 Type Refinements

Examples • Addition:val add : (nat ! nat ! nat) Æ (pos! nat! pos) Æ(nat! pos! pos) =fix plus  m  n . case m of) n | x 0 ) …plus… | x 1 ) …inc…plus… 15-814 Type Refinements

Intersection Types and Their Effects in Refinement Type Systems