Refinement Types in ML: Lattice-Based Type Inference and Applications

1. Freeman-Pfenning: Refinement Types Robert Harper Fall Semester, 2003

2. Refinement Types • First system of type refinements for ML. • Inspired by Yardeni & Shapiro type system for Prolog. • Main idea: use regular trees to isolate subsets of a datatype. • Use lattice-based abstract interpretation methods for type inference. 15-814 Type Refinements

3. Lists Example • Start with an ML datatype:datatype  list = nil j cons of  *  list • Introduce rectype’s that define refinements (subsets / properties) of interest.rectype  singleton = cons of  * nil • Infer refined types for functions based on these.cons 2 *  nil ! singleton Æ *  list ! list 15-814 Type Refinements

4.  list singÇnil sing nil ? Refinement Types • The declared refinements of a datatype determine a finite lattice of subsets. 15-814 Type Refinements

5. Refinement Types • Peculiarity: the lattice is supposed to consist of declared types only (rather than, say, all possible subsets). • But some points are declared ( sing) and some not ( nil). • I’m not sure I understand how this is supposed to work. 15-814 Type Refinements

6. Declared Refinements • If a constructor contains a function, the domain cannot be refined! • Example (I think):datatype D = I of int | F of D! Drectype R = I of Int | F of D ! R • This seems essential to preserve regularity of the subset. 15-814 Type Refinements

7. Refinement Types in General • Empty (?) and total (>). • Intersection (1Æ2) and union (1Ç2). • Function space (1!2). • Declared refinements ( r). • Bounded refinement variables (::). • Pre-order induced by lattice structure. 15-814 Type Refinements

8. Bit Strings • Type of bit strings (lsb outermost).datatype bs = e | z of bs | o of bs • No leading zero bits:rectype std = e | stdposand stdpos = o(e) | z(stdpos) | o(stdpos) • For example, o(e) is not std, but e is. 15-814 Type Refinements

9. Bit Strings • Bit-wise addition function:fun add e m = m | add n e = n | add (z m) (z n) = z (add m n) | add (o m) (z n) = o (add m n) | add (z m) (o n) = o (add m n) | add (o m) (o n) = z (add (add (o e) m) n) 15-814 Type Refinements

10. Bit Strings • Refinement checker infers many types for add. • Propagate consequences of all refinements for arguments. • Three declared (?) refinements: std, stdpos, e. • Nine total clauses, includingstd ! std ! std. 15-814 Type Refinements

11. Refinement Propagation • Each constructor has a conjunct for each recursive type in which it appears.o 2 e! stdpos Æ stdpos! stdpos • Case analysis checks each arm against all possible matches. • eg, for z(x) ) p, x could be e or stdpos. • multiple matches lead to disjunctive result • eg, if arg is stdpos, result could be result of either z or o case. 15-814 Type Refinements

12. Extending the Lattice • The lattice structure on refinements of a datatype extends to function types:1!1·2!2 if 2·2 and 1·2 . • Requires variance annotations on type constructors. • eg, + list states that list preserves order • How to decide subtyping? 15-814 Type Refinements

13. Deciding Subtyping • For two refinements of a datatype, determined by the finite lattice. • For refinements of a function type, seems to employ an ad hoc rule. • Put types into DNF using 1Æ(2Ç3) ´ (1Æ2)Ç(2Æ3)(1Ç2)!´ (1!)Æ(2!) • Conjuncts are all CNF! DNF, or all named refinements of a datatype, or all variables with a common bound. 15-814 Type Refinements

14. Deciding Subtyping • DNF subtyping:Çii·Çjj iff 8i9j st i·j. • CNF subtyping: • For named refinements, consult lattice ordering. • For functions, see following. • For variables??? 15-814 Type Refinements

15. Deciding Subtyping • To decide =Æi (i!i) ·Æj (’j!’j)=’, decide whether app(,) · app(’,) for every  = i or ’j. • Define app(,) = Æ·ii, the most precise answer for argument of type . • I’m not clear on the motivation for this definition. 15-814 Type Refinements

16. Refinement Inference • Invariant: assumed and derived refinements are in DNF. • Ignore polymorphism here, but see paper. • Applications:` e1(e2) 2Çi,j app(i,j), where ` e12Çii, and  ` e22Çjj . 15-814 Type Refinements

17. Refinement Inference • Abstractions: ` x.e 2Æi (i!i), where, x2i` e 2i for each refinement of the domain type. • Question: how do we determine all such i’s in general? 15-814 Type Refinements

18. Refinement Inference • Fixed points by recursion: • Start with ? for recursive variable. • Infer refinement of argument. • Refine recursive variable to inferred type. • Iterate until you reach a fixed point (must exist, by finiteness). 15-814 Type Refinements

19. Refinement Inference • Considerfun inc e = o(e) | inc (z m) = (o m) | inc (o m) = z (inc m) • Use inference rules to infer thatinc 2 e! std Æ std! stdpos Æ stdpos! stdpos. 15-814 Type Refinements