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Compiler feature - useful or not?
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 Date: 2007-11-15 (13:26) From: weis@y... Subject: Re: [Caml-list] Compiler feature - useful or not?
```> >- a value of type row is in fact a concrete integer (it is not hidden in
> >any way),
>
> But you cannot apply integer operations to it, so it is not a normal
> concrete integer, right?

Right: a value of type row has type row ... not type int!

> Can you show an example where replacing all "type t = private ..."
> definitions by "type t" changes a well-typed program into an ill-typed
> one?   I understand that if private types create real subtypes (w.r.t.
> :>) then they are different than abstract types, but otherwise, I don't
> see the difference for the type-checker.

May be, I must recall what are private types in the first place: private
types has been designed to implement quotient data structure.

(***  Warning.

Wao: after re-reading this message I realize that it is really long.

You can skip it, if you already know something about mathematical quotient
structures, private types for record and variant, relational types and the
mocac compiler!

***)

What is a quotient ?
--------------------

Here quotient has to be understood in the mathematical sense of the term:
given a set S and an equivalence relation R, you consider the set S/R of the
equivalence classes modulo the relation R. S/R is named the quotient
structure of S by R. Quotient structures are fundamental in mathematics and
there properties have been largely studied, in particular to find the
relationship between operations defined on S and operations on S/R: which
operations on S can be extended to operations on S/R ? Which properties of
operations on S are still valid for there extension on S/R ? and so on.

Simple examples of quotient structures can be found everywhere in maths, for
instance consider the equivalence relation R on pairs of integer values
defined as

z_1 equiv z_2 if and only if z_1 and z_2 are both odd or are both even
(in Caml terms
z_1 equiv z_2 if and only if (z_1 mod 2) = (z_2 mod 2))

The set Z/R is named Z/2Z and it inherits properties of operations on Z: it
is an abelian group (and more).

A wonderful example of such inheritance of interesting properties by
inheritance of operations thanks to a definition by a quotient structure is
the hierarchy of sets of numbers: starting with N (the set of natural
numbers) we define Z (the set of relative integer numbers) as a quotient of
NxN), then Q (the rational numbers) as a quotient of ZxZ*, R (the set of real
numbers) as a quotient of Q^N (the sets of sequences of rational numbers), C
(the set of complex numbers) as a quotient of R[X] (the set of polynomials
with one unknown and real coefficients). Note here that at each step of the
hierarchy the quotient structure is richer (has more properties and/or more
elements) than the original structure: first we have a monoide, then a group
and a ring, then a field, then a complete field and so on.

Why quotient structures ?
-------------------------

So quotient structures are a fundamental tool to express mathematical
structures and properties. Exactly as mathematical functions, relations and
sets. As you may have noticed, programming languages are extremely related to
maths (due to their purely logical bases) even if, in some cases, the zealots
of the language at hand try to hide the mathematical fundation into a strange
anthropomorphic discurse to describe their favorite language features.

In the end, the computer programing languages try hard to incorporate
powerful features from mathematics, because these features have been polished
by mathematicians for centuries. As a consequence of this work, we know
facts, properties and theorems about the mathematical features and this is an
extremely valuable benefit.

Now, what is the next frontier ? What can we steal more to mathematics for the
benefit of our favorite programming language ?

If we review the most powerful tools of mathematicians, we found that
functions have been borrowed (hello functional programming, lambda-calculus
and friends :)), relations have been borrowed (data bases, hash tables,
association lists), sets have been more or less borrowed (hello setle,
pascal, and set definition facilities from various libraries of various
languages...). More or less, we have all those basic features.

>From the mathematical set construction tools, we have got in Caml:

- the cartesian product of sets (the * binary type constructor, the record
type definitions),
- the disjunct union of sets (the | of polymorphic variants, the sum (or
variant) type definitions).

Unfortunately, we have no powerful way to define a quotient data
structure. That what private type definitions have been designed to do.

What do we need for a quotient data structure ?
-----------------------------------------------

In the first place, we need the ``true'' thing, the real feature that is
indeed used in maths. Roughly speaking this means to assimilate the quotient
set S/R to a subset of S.

In the previous definition of quotient structures, there is a careful
distinction between the base set S and the quotient set S/R. In fact, there
always exists a canonical injection from S to S/R, and if we choose a
canonical representant in each equivalence class of S/R, we get another
canonical injection from S/R to S, so that S/R can be considered as a subset
of S (the story is a bit more complex but that's the idea). This
injection/projection trickery is intensively used in maths; for instance, in
the hierarchy of number sets, we say and consider that N is a subset of Z
that itself is a subset of Q, a subset of R, a subset of C. Rigourously, we
should say for instance, there is a subset of Z that is canonically
isomorphic to N; in fact, we abusively assimilate N to this subset of Z;
hence, we say that N is ``included'' in Z, when we should say ``the image of
N by the canonical isomorphism from N to Z'' is included in Z; then, we go
one step further in the assimilation of N to a subset of Z, by denoting the
same, the elements of N and there image in Z; for instance, ``the neutral
element of the multiplication in Z'' and the successor of 0 in N is denoted
``1''; and we now can say that 1 belongs to Z. Note here that, in the first
place, ``the neutral element of the multiplication in Z'' is an equivalence
class (as all elements in Z are). So it is a set. More exactly, the ``neutral
element of the multiplication in Z'' is the infinite set of pairs of natural
numbers (n, m) such that n - m = 1 (here ``-'' is an operation in N and ``1''
is the successor of the natural number ``0'', so that n - m = 1 is a
shorthand for n = succ m). The assimilation between N and its isomorphic
image allows to use 1 as the denotation of this infinite set of pairs of
natural numbers.

We understand why the mathematicians always write after having designed a
quotient structure: ``thanks to this isomorphism, and if no confusion may
arise, we always assimilate S to its canonical injection in S/R''.

This assimilation is what private type definitions allow.

How do we define a quotient data structure ?
--------------------------------------------

The mathematical problem:
- we have a set S and an equivalence relation R on SxS,
- we construct the quotient S/R.
- we state afterwards:
``if no confusion may arise, we always assimilate S to its canonical
injection in S/R''.

The programming problem:
- we have a data structure S (defined by a type s) and a relation R on SxS
(defined by a function r from s * s to bool).
- we construct a data structure that represents S/R.
- we have afterwards:
``No confusion may arise, we always assimilate S to its canonical injection
in S/R''.

The private data type solution:
- the data structure S is defined as any Caml data structure and the
relation R is implemented by the canonical injection(s) from S to S/R.
- the canonical projection from S/R to S is automatic.
- S (defined by s) is assimilated to S/R (which is then also s!).

We defined S/R as a subset of S, the set of canonical representants of
equivalence classes of S/R.

More exactly, the canonical injection from S to S/R maps each element of S to
its equivalent class in S/R; if we assimilate each equivalence class to its
canonical representant (an element of S), the canonical injection maps each
element of S to the canonical representant of its equivalence class. Hence
the canonical injection has type S -> S.

An example treated without private types:
-----------------------------------------

Let's take a simple example:

S is the following data structure that implements a mathematical (free) structure
generated by the constant 0, the unary symbol succ, and the binary symbol +.

type peano =
| Zero
| Succ of peano
| Plus of peano * peano

R is the (equivalence) relation that expresses that
- 0 is the neutral element for + (for all x in S, 0 + x = x and x + 0 = x),
- + is associative (for all x, y, z in S³, (x + (y + z)) = ((x + y) + z)).

The canonical injection is a function from peano -> peano that maps each value
in S (the type peano) to the canonical representant of its class in S/R (an
element of S (the type peano)):

let rec make = function
| Zero -> Zero
| Plus (Succ n, p) -> Succ (make (Plus (n, p)))
| Plus (Zero, p) -> p
| Plus (p, Zero) -> p
| Plus (Plus (x, y), z) -> make (Plus (x, make (Plus (y, z))))
| Succ p -> Succ p
;;

(This function may be wrong but you see the idea :))

So, if you want to represent S/R for peano in Caml you must:
- (1) define the type peano
- (2) always use the make function to create a value in S/R

Private data types permits (1), ensures (2), and helps for (3) concerning the
head part and ensures (3) for programs by means of compile-time type errors.

The same example with private types:
------------------------------------

To define a private data type you must define a module.
- in the implementation, you define the carrier S and its canonical injection.
- in the interface, you export the carrier and the injection.

The type checker will ensure transparent projection from the quotient to the
carrier and mandatory use of the canonical projection to build a value in
S/R.

interface peano.mli
-------------------
type peano = private
| Zero
| Succ of peano
| Plus of peano * peano
;;

val zero : peano;;
val succ : peano -> peano;;
val plus : peano * peano -> peano;;

implementation peano.ml
-----------------------
type peano =
| Zero
| Succ of peano
| Plus of peano * peano
;;

let rec make = function
...
;;

let zero = make Zero;;
let succ p = make (Succ p);;
let plus (n, m) = make (Plus (n, m));;

(See note (1) for a discussion on the design of this example.)

What is given by private types:
-------------------------------

- You cannot create a value of S/R if you do not use the canonical injection
# Zero;;
Cannot create values of the private type peano

- As a consequence, values in S (i.e. S/R) are always canonical:
# let one = succ zero;;
val one : M.peano = Succ Zero
# let three = plus (one, succ (plus (one, zero)));;
val three : M.peano = Succ (Succ (Succ Zero))

- Projection is automatic
# let rec int_of_peano = function
| Zero -> 0
| Succ n -> 1 + int_of_peano n
| Plus (n, p) -> int_of_peano n + int_of_peano p
;;
val int_of_peano : M.peano -> int = <fun>
# int_of_peano three;;
- : int = 3

----------------------------------

Private abbreviation definitions are a natural extension of private data type
definitions to abbreviation type definitions. The same notions are carried
out mutatis-mutandis:

- we have a data structure S (defined by a type s) and a relation R on SxS
(defined by a function r from s * s to bool).
- we construct a data structure that represents S/R.
- we have afterwards:
``No confusion may arise, we always assimilate S to its canonical injection
in S/R''.

In the case of abbreviations:

- the data structure S/R is defined by a type s (which is an abbreviation) and
a relation defined by a function,
- the canonical injection should be defined in the implementation file of the
private data type module,
- we always assimilate S to its canonical injection in S/R.

In pratice, as for usual private types (for which the constructive part of
the data type is not available outside the implementation), the type abbreviation
is not known outside the implementation (it is really private to its
implementation). This prevents the construction of values of S/R without
using the canonical injection.

Th noticeable difference is the projection function: it cannot be fully
implicit (otherwise, as you said Alain, the assimilation will turn to complete
confusion: we would have S identical to S/R, hence row=int and no difference
between a regular abbreviation definition and a private abbreviation
definition). If not implicit, the injection function should granted to be the
identity function (something that we would get for free, if we allow
projection via sub-typing coercion).

In short: abstract data types provide values that cannot be inspected nor
safely manipulated without using the functions provided by the module that
defines the abstract data type. In contrast, private data types are
explicitely concrete and you can freely write any algorithm you need. A good
test is printing: you can always write a function to print values of a
private type, it is up to the implementor of an abstract type to give you the
necessary primitives to access the components of the abstracted values.

Automatic generation of the canonical injection:
------------------------------------------------

You may have realized that writing the canonical injection can be a real
challenge.

The moca compiler (see http://moca.inria.fr/) helps you to write the
canonical injection by generating one for you, provided you can express the
injection at hand via a set of predefined algebraic relations (and/or rewrite
rules you specify) attached to the constructors of the private type. Private
types with constructors having algebraic relations are named relational
types. Moca generated canonical injections for relation types.

For instance, you would write the peano example as the following peano.mlm
file:

type peano = private
| Zero
| Succ of peano
| Plus of peano * peano
begin
associative
neutral (Zero)
rule Plus (Succ n, p) -> Succ (Plus (n, p))
end;;

The mocac compiler will generate the interface and implementation of the
peano module for you, including the necessary canonical injection function.

Best regards,

--
Pierre Weis

INRIA Rocquencourt, http://bat8.inria.fr/~weis/

Note (1):
- we can't just export the canonical injection since you could not build any
value of the type without previously defined values!
- we provide specialized versions of the canonical injection function
introducing the convention that the lowercase name of a value constructor is
the name of its associated canonical injection.
- we do not export the plasin true canonical injection since it would be more
confusing than useful.

```