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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 . consider: let f1 = fun x  (x,x);; let f2 = fun y  f1(f1 y);; let f3 = fun y  f2(f2 y);; …..

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two comments on let polymorphism
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



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?


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)


One can save a lot of space by representing types as graphs, instead of trees (common sub expression elimination)

Double exponential  exponential



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


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


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


  • 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


The ML solution:

In let x = e in …

Allow to generalize the type for x only if e is a syntactic value


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.


object oriented languages some concepts
Object-oriented languages – some concepts

A well known feature of OO pl’s is

sub-type polymorphism

We concentrate on this subject


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)


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


sub typing with records and cells
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


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


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)


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


Using these two rules, we can prove:

Using reflexivity, we can refine a single field, rather than all


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


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


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


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


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