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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

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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?

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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)

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

Double exponential exponential

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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

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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 intint to true – a run-time error

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One possible solution: lazy evaluation of let :

let x = e // create binding xe

….

… 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

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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

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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.

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Object-oriented languages – some concepts

A well known feature of OO pl’s is

sub-type polymorphism

We concentrate on this subject

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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)

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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

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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

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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

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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)

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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

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Using these two rules, we can prove:

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

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

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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

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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

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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

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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

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