Two comments on let polymorphism

1. Two comments on let polymorphism I. What is the (time, space) complexity of type reconstruction? In practice – executes “fast” (seems linear time) But, some bad cases exist last

2. consider: let f1 = fun x  (x,x);; let f2 = fun y  f1(f1 y);; let f3 = fun y  f2(f2 y);; ….. …. fn …. fn(some expression that uses fn) How do the types of these functions look like? last

3. let f1 = fun x  (x,x);; ‘a  ‘a * ‘a (1  2) let f2 = fun y  f1(f1 y);; ‘a  (‘a * ‘a) * (‘a * ‘a) (1  4) let f3 = fun y  f2(f2 y);; ‘a ( [(‘a * ‘a)*(‘a * ‘a)]*[(‘a * ‘a)*(‘a * ‘a)] )* ( [(‘a * ‘a)*(‘a * ‘a)]*[(‘a * ‘a)*(‘a * ‘a)] ) (1  16) let fn = … (double exponential) last

4. One can save a lot of space by representing types as graphs, instead of trees (common sub expression elimination) Double exponential  exponential last

5. Theorem: Type reconstruction for core ML is exptime-complete This means that worst-case complexity is bad, but in practice it is sufficiently efficient Note: extending type reconstruction to the full calculus with universal types is impossible --- type reconstruction for this calculus is undecidable last

6. II. polymorphic references are problematic: let c = ref (lambda x.x);; here, the type for c is c:= lambda x. x+5;; type-checker allows the assignment, type is unit (!c) true;; type checker accepts but, at run-time we apply a function of type intint to true – a run-time error last

7. One possible solution: lazy evaluation of let : let x = e // create binding xe …. … x // substitute e for x, and continue evaluation In the example: let c = ref (lambda x. x) // bind c to the expression c:= lambda x. x+5 // substitute binding for x  (ref lambda x.x) := lambda x.x+5 // one cell created) (!c) true // substitute binding for x  (!(ref lambda x. x)) true // another cell created last

8. But • Nobody really knows how to specify or implement lazy evaluation for languages with imperative features (side-effects) – how to order the side-effects? • The examples shows this leads to a semantics that is not very useful last

9. The ML solution: In let x = e in … Allow to generalize the type for x only if e is a syntactic value is is not is not Statistics collected on systems w/o this restriction (a more liberal but complex solution)  there are almost no programs where this restriction hurts. last

10. Object-oriented languages – some concepts A well known feature of OO pl’s is sub-type polymorphism We concentrate on this subject last

11. What is sub-type ? Two possible answers: A type t is (denotes) a set of values With the second, int<: float holds; Compiler inserts the coercion during type-checking (so a bit more complexity of type-checking is expected) We use the first (simpler intuition) last

12. The basic intuition of sub-typing: List to the type-checker level : an expression of a sub-type can be safely used in a context where an expression of a type is expected use is defined by the operations available on the two types  sub-typing is not a new independent feature, it interacts with the other components of a type system last

13. Sub-typing with records and cells Objects are similar to records  convenient to introduce sub-typing in the context of a language with : base types, records, functions, ref cells We assume some (possibly none) sub-type axioms are given for the base types last

14. From the basic intuition : The interaction of sub-typing with the type-checker : the subsumption rule: Wherever the type-checker expects a type, it allows a sub-type Note: algorithmically, this rule is problematic last

15. Rules independent of the given type system: From the basic intuition, sub-typing is reflexive and transitive Second rule also looks a bit problematic (algorithmically) last

16. Rules for records : Example : let f = lambda x : {a:int} . x.a;; seems reasonable to apply f also to {a=4, b=“john”}, since f uses only x.a But, also make sense to allow a sub-type in a field last

17. Using these two rules, we can prove: Using reflexivity, we can refine a single field, rather than all last

18. By combining the two record rules with transitivity, we can change both the number of fields and their types last

19. Can combine to one comprehensive record rule: Note: we assume that in a record, order of fields is irrelevant, so in all the rules, the label-type pairs are assumed to be a set. This can be emphasized by a rule that allows to change position of fields  Can “add” fields anywhere in a record last

20. Rules for functions : These can be applied, be passed as arguments/return values If a context requires a function that for an argument of type return a value of type , then a function that returns a value of a sub-type is ok last

21. But, for the input type: In a context that expects a function of this type, you also accept a function that has this guarantee for a larger set last

22. The two rules are typically combined : Contra-variance for the input type is difficult to swallow; convince yourself that There are also applications where a restriction of the in-type in a co-variant fashion seems desirable Some languages (e.g., Eiffel) also co-variant change on in-type, and leave a hole in the type system last