Unions and free data types

1.Introductory example

Mathemagix classes provide a simple way to introduce new data types with specified data fields, methods, constructors and an optional destructor. Mathemagix structures provide a convenient tool for the definition of unions and more general free data types.

One typical example of a structure is the type of finite integer sequences, defined as follows:

structure Sequence == {
  null ();
  cons (head: Integer, tail: Sequence);
}

Any such sequence is a formal expression of the form

where are integers. The symbols null and cons are called the constructors for the structure and any structure of the type Sequence is always either of the form null () or cons (h, t), with h: Integer and t: Sequence. The declaration of the structure Sequence automatically gives rise to predicates

null?: Sequence -> Boolean;
cons?: Sequence -> Boolean;

which allow the user to determine the kind of structure. The symbols head and tail induce partially defined accessors

postfix .head: Sequence -> Integer;
postfix .tail: Sequence -> Sequence;

If a structure s: Sequence is of the form cons (h, t), then cons? s holds and we may obtain h and t via the expressions s.head and s.tail respectively.

For instance, the following function can be used in order to compute the length of a sequence:

prefix # (s: Sequence): Int ==
  if null? s then 0 else #s.tail + 1;

Pattern matching provides us with an alternative way to do this:

prefix # (s: Sequence): Int ==
  match s with {
    case null () do return 0;
    case cons (_, t: Sequence) do return #t + 1;
  }

Mathemagix also provides the following syntax for doing the same thing:

prefix # (s: Sequence): Int == 0;
prefix # (cons (_, t: Sequence)): Int == #t + 1;

Notice that this syntax relies on the mechanism of conditional overloading (see the section on conditional overloading).

Likewise classes, structures can be parameterized. The List T container is a typical example of a parameterized structure which generalizes the structure Sequence:

structure List (T: Type) == {
  null ();
  cons (head: T, tail: List T);
}

2.Structures

The general syntax for the declaration of a structure is the following:

structure S == {
  cons_1 (field_11: T_11, ..., field_1n1: T_1n1);
  ...
  cons_k (field_k1: T_k1, ..., field_knk: T_knk);
}

This declaration in particular induces the declaration of functions

cons_1: (T_11, ..., T_1n1) -> S;
...
cons_k: (T_k1, ..., T_knk) -> S;

These functions cons_1, , cons_k are called the constructors for S, and all instances of S are expressions which are built up freely using these constructors. Hence, any instance x has the form cons_i (y_1, ..., y_ni) for exactly one index i and y_1: T_i1, , y_ni: T_ini. The declaration of S also declares inspection predicates

cons_1?: S -> Boolean;
...
cons_k?: S -> Boolean;

such that cons_i? x holds if and only if x is of the form cons_i (y_1, ..., y_ni). The declaration of S finally induces the declaration of partially defined accessors

postfix .field_11 : S -> T_11;
... ...
postfix .field_knk: S -> T_knk;

Whenever x: S is of the form cons_i (y_1, ..., y_ni) then y_j can be retrieved via the expression x.field_ij. Whenever x is not of this form, the expression x.field_ij is undefined and may provoke an error or worse.

Example 1. A structure of the above kind is called a union precisely then when each constructor takes exactly one argument; in that case, S corresponds to the union of the types T_11, , T_k1.

Parameterized structures are defined in a similar way as ordinary structures using the syntax

structure S (Param_1: C_1, ..., Param_p: C_p) == {
  cons_1 (field_11: T_11, ..., field_1n1: T_1n1);
  ...
  cons_k (field_k1: T_k1, ..., field_knk: T_knk);
}

Such declarations induce declarations of constructors, inspection predicates and accessors in the same way as their unparameterized homologues.

In additional to the constructors, inspection predicates and accessors, declarations of (parameterized) structures also give rise to the following additional functions and constants for more low level introspection:

postfix .kind: S -> Int;
cons_1_kind?: Int -> Boolean;
...
cons_k_kind?: Int -> Boolean;
cons_1_kind: Int;
...
cons_k_kind: Int;

If x is of the form cons_i (y_1, ..., y_iki), then x.kind is just the integer i-1. The predicate cons_i_kind? just checks whether the input integer is equal to i-1, so that cons_i? x holds if and only if cons_i_kind? x.kind holds. Finally, cons_i_kind is equal to the integer i-1.

3.Extensible structures

Under some circumstances, it is useful to add additional constructors to structures a posteriori.

For instance, assume that we are writing a compiler for some language and that the expressions of the language are represented by a structure. It might be that someone else would like to define a language extension but still use as much as possible the existing functions in the compiler for all kinds of manipulations of expressions. The simplest solution would then be to extend the expression type a posteriori with some new constructors provided by the extended language, modulo customization of existing functions on expressions whenever necessary.

Another important example is the design of a flexible type for symbolic expressions. Symbolic expression types are usually unions of various basic expression types, such as literals, integers, function applications, sums, products, etc. Whenever someone develops a library for a new kind of mathematical objects, say differential operators, then it is useful if these new objects can be seen as a new kind of symbolic expressions.

In Mathemagix, extensible structures can be declared using the syntax

structure S := {
  cons_1 (field_11: T_11, ..., field_1n1: T_1n1);
  ...
  cons_k (field_k1: T_k1, ..., field_knk: T_knk);
}

The only distinction with respect to the mechanism from the previous section is that we are now allowed to further extend the structure using the following syntax:

structure S += {
  extra_1 (added_11: X_11, ..., added_1n1: X_1n1);
  ...
  extra_l (added_l1: X_l1, ..., added_lnl: X_lnl);
}

An arbitrary number of such extensions can be made after the initial declaration of S and these extensions may occur in different files (provided that the file with the initial declaration is (directly or indirectly) included in each of these files).

Whenever we extend a structure in the above way the corresponding new constructors, inspection predicates and accessors are automatically defined. The lower level inspection routines are also automatically extended, although the actual values of extra_1_kind, , extra_l_kind now depend on the initialization order, and are therefore harder to predict. Nevertheless, the set of all these kinds is always of the form , where is the total number of constructors.

4.Pattern matching

The constructors of a structure can also be used as building bricks for so called patterns. For instance, given the structure

structure Sequence == {
  null ();
  cons (head: Integer, tail: Sequence);
}

from the beginning of this section, the expression

cons (_, cons (_, tail_tail: Sequence))

is a pattern which corresponds to all sequences of length at least two, and with an explicit name tail_tail for the typed name of the tail of the tail of the sequence.

More generally, there are six kinds of patterns:

  1. Structural patterns, of the form cons (p_1, ..., p_n) where cons is a constructor with arity n of some structure, and p_1, , p_n are other patterns.

  2. User defined patterns, to be described in the next section.

  3. Wildcards of the form var: T, where var is the name of a variable and T a type.

  4. Unnamed wildcards of the form _: T, where T is a type.

  5. The unnamed and untyped wildcard _.

  6. Ordinary expressions.

One may define a natural relation “matches” on expressions and patterns, and if an expression matches a pattern, then there exists a binding for the wildcards which realizes the match. For instance, the expression cons (1, cons (2, cons (3, null ())) is said to match the pattern cons (_, cons (_, tail_tail: Sequence)) for the binding tail_tail cons (3, null ()).

Patterns can be used as arguments of the case keyword inside bodies of the match keyword. Whenever the pattern after case matches the expression after match, the wildcards are bound using the bindings of the match, and can be used inside the body of the case statement. For instance, we may use the following function in order to increase all terms of a sequence with an even index:

even_increase (s: Sequence): Sequence ==
  match s with {
    case cons (x: Integer, cons (y: Integer, t: Sequence)) do
      return cons (x, cons (y+1, even_increase t));
    case _ do
      return s;
  }

Patterns can also be used as generalized arguments inside function declarations. In that case, the declaration is considered as a special kind of conditionally overloaded function declaration. For instance, the declaration

prefix # (cons (_, t: Sequence)): Int == #t - 1;

is equivalent to

prefix # (s: Sequence | cons? s): Int == {
  t: Sequence == s.tail;
  #t - 1;
}

Again, the bindings of potential matches are used as values for the wildcards, which become local variables inside the function body.

5.User defined patterns

Besides the patterns which are induced by constructors of structures, new patterns may be defined explicitly by the user. The syntax for pattern declaration is as follows:

pattern pat_name (sub_1: T_1, ..., sub_n: T_n): PT == pat_body

where the body pat_body consist of a finite number of cases of the form

case pat_case do {
  sub_1 == expr_1;
  ...
  sub_n == expr_n;
}

where pat_case is a pattern. Assuming this declaration of pat_name, any patterns p_1, , p_n of types T_1, , T_n give rise to a pattern pat_name (p_1, ..., p_n) of type PT. This pattern is matched if and only if one of the patterns pat_case in the various cases for pat_name is matched (with all occurrences of sub_1, , sub_n in pat_case replaced by p_1, , p_n). In that case, we privilege the first case which matches, and bind var_i to expr_i whenever sub_i is of the form var_i: T_i.

In a similar way as for structures, replacing == by != or += in the declaration of the above pattern allows for the declaration of an extensible pattern resp. an actual extension of it. The above declaration of the pattern pat_name also gives rise to a generalized inspection predicate

pat_name?: PT -> Boolean;

and functions

postfix .sub_1: PT -> T_1;
...
postfix .sub_n: PT -> T_n;

which behave in a similar way as accessors of structures.

For instance, for sequences which really represent association lists , we may use the following pattern for retrieving the first association :

pattern assoc (key: Integer, val: Integer, next: Sequence): Sequence ==
  case cons (k: Integer, cons (v: Integer, n: Sequence)) do {
    key  == k;
    val  == v;
    next == n;
  }

The implementation of the predicate assoc? is equivalent to

assoc? (s: Sequence): Boolean == cons? s and cons? s.next;

and the implementations of the accessors postfix .key, postfix .val and postfix .next are equivalent to

postfix .key (s: Sequence): Integer == s.head;
postfix .val (s: Sequence): Integer == s.tail.head;
postfix .next (s: Sequence): Sequence == s.tail.tail;

6.Syntactic sugar for symbolic expressions

As mentioned in the introduction of section ?, one important special case where extensible structures are useful is for the definition of a flexible type Symbolic for symbolic expressions. This type is essentially a union type. For instance, we might start with

structure Symbolic := {
  sym_undefined ();
  sym_boolean (boolean: Boolean);
  sym_literal (literal "literal": Literal);
  sym_compound (compound: Compound);
}

and further extend Symbolic whenever necessary:

structure Symbolic += {
  sym_integer (integer: Integer);
}

structure Symbolic += {
  sym_rational (rational: Rational);
}

The prefix sym_ provides us with a clean syntatic distinction between a “symbolic expression which contains an object of type T” and a mere object of type T. However, it would be simpler to write integer? instead of sym_integer? as an inspection predicate, and it is somewhat cumbersome to implement addition of symbolic integers using

infix + (sym_integer (i: Integer), sym_integer (j: Integer)): Symbolic ==
  sym_integer (i + j);

For this reason, Mathemagix provides us with a special operator :: to be used for structure constructors with one argument. For instance, the constructor sym_integer would rather be declared using

structure Symbolic += {
  sym_integer (integer :: Integer);
}

This declaration automatically declares the synonym integer? for syn_integer? and also introduces the simplified notation var :: Integer for the pattern sym_integer (var: Integer). Assuming the further implementation of a constructor

convert (i: Integer): Symbolic == sym_integer i;

we may then simplify the declaration of our addition on symbolic integers into

infix + (i :: Integer, j :: Integer): Symbolic == i + j;

In a similar way, Mathemagix provides support for an operator ::> which allows us to mimick the notation :> used for a priori conversions for their symbolic wrappers. This notation is best illustrated with the example of an “symbolic converter from symbolic integers to symbolic rational numbers”:

pattern sym_as_rational (as_rational ::> Rational): Symbolic := {
  case sym_rational (x: Rational) do as_rational == x;
  case sym_integer (x: Integer) do as_rational == x :> Rational;
}

This allows us for instance to write

infix + (i :: Integer, x ::> Rational): Symbolic == i + x;

7.Fast dispatching

We already noticed that the mechanism of conditional overloading induces a performance overhead (see the section on performance issues concerning conditional overloading). In the case of basic operations on symbolic expressions, such as additions, which are overloaded dozens if not hundreds of times, this performance penalty is too significant.

Fortunately, the structure Symbolic as described in the previous section is essentially an extensible union type. Given a unary conditionally overloaded operation foo on symbolic expressions, the appropriate routine to be called for a given input x can often be determined as a function of the integer x.kind only. This makes it possible to use a fast lookup table instead of the usual conditional overloading mechanism in this particular case. In order to do so, we have to declare the operation foo using

foo (x: dispatch Symbolic): Symbolic := default_implementation

Operations with higher arities and operations which involve other types are treated in a similar way: the type of each argument involved in the fast table lookup should be preceded by the keyword dispatch. For instance,

infix + (x: dispatch Symbolic, y: dispatch Symbolic): Symbolic := ...;
postfix [] (x: dispatch Symbolic, i: Int): Symbolic := ...;

Notice that the size of the dispatch table is equal to the product of the number of structure constructors of all arguments on which we perform our dispatching. For this reason, we do not allow dispatching on more than two arguments.

Mathemagix also provides the keyword disjunction as a special case of user defined patterns which is compatible with the above dispatching mechanism. A typical example of a disjunction is

disjunction sym_scalar (scalar :: Scalar): Symbolic := {
  sym_boolean _;
  sym_integer _;
  sym_rational _;
}

Permission is granted to copy, distribute and/or modify this document under the terms of the GNU General Public License. If you don't have this file, write to the Free Software Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA 02111-1307, USA.