Bartosz Witkowski - Blog.

Algebraic Data Types and the Base of Our Interpreter

In this part we will define the core data types used in the interpreter such as the strings, numbers, atoms etc.

Algebraic Data Type

Sum types, disjunct types, tagged unions or variant types… are actually a pretty simple concept - a value that is a sum type may be “matched” (deconstructed) against it’s possible constructors.

Consider a simple boolean sum type:

:- type boolean ---> yes ; no.

The syntax is pretty easy boolean is the name of the sum type and on the right from the arrow (note the number of - in it) there’s a ;-delimited list of constructors.

Now a predicate that operates on booleans may use this the following way:

:- pred say(boolean::in, string::out) is det.
say(Boolean, String) :-
  ( Boolean = yes,
    String = "YES :)!"
  ; Boolean = no,
    String = "NO! ;("

The ( X = constructor1 ; X = constructor2 ; ... ; X = constructorN ) construct is how mercury matches on values. Matches are guaranteed to be exhaustive (check every possible constructor) - not matching on some value is a compile time error.

As a sidenote - ; in mercury or prolog is the disjunction - you’ve seen it used in if-then-else and now in sum types and matches. It may be easier to understand if you know to read the types as “yes OR no”, “Boolean unifies with yes, String unifies with “YES :)!”, OR …” or in if-then-else “Some condition THEN something OR something else”.

Constructors in sum types may be parametrized by values e.g:

:- type shape ---> circle(float) 
                 ; square(float) 
                 ; triangle(float, float, float)

Now consider this area predicate, for example on how to match constructors with parameters:

:- pred area(shape::in, int::out) is det.
area(Shape, Area) :-
  ( Shape = circle(Radius)
    Area = math.pi * Radius * Radius
  ; Shape = square(Side)
    Area = Side * Side
  ; Shape = triangle(A, B, C)
    P = (A + B + C) / 2
    Area = math.sqrt(P * (P - A) * (P - B) * (P - C))

Like everywhere if we don’t care about a variable we can name it either _ or give it a _ prefix.

Now algebraic data type are sum types that can also be recursive - the standard example is the list:

:- type list(T) ---> nil ; cons(T, list(T)).

Note that this list definition is generic - types, like predicates and functions, can also have generic parameters. So the above definition states “A list T is either nil OR is cons containing a value and another list T

Multi module compilation

Before we only wrote in one (main) module from now on we’ll divide the project into multiple modules - for now it may not be so useful - but organizing it this way will be less of a hassle in the long run.

Mercury comes out of the box with a nice build system in the form of mmc. To compile a library use:

$ mmc --make lib[LIBRARY_MODULE]

and the main module is just:

$ mmc --make [MAIN_MODULE]

If they’re in one directory mercury compiler will work out the dependencies by itself.

integers and bools

integers in mercury are like ints but with arbitrary precision - we’ll use them to represent lisp numbers.

bool is a type that can store boolean values - its constructors are exactly like the ones we’ve seen in the adt examples:yes and no.


Create an adts module with the following declarations:

:- type lisp_val ---> lisp_atom(string)
                    ; lisp_list(list(lisp_val))
                    ; lisp_dotted_list(list(lisp_val), lisp_val)
                    ; lisp_number(integer)
                    ; lisp_string(string)
                    ; lisp_bool(bool.bool)

For some motivation:

  1. In scheme atoms are one of the two fundamental data type (the other being the list) - you can think of them as a string with an unique identity (two atoms with the same name are always equal).
  2. Lists are THE data structure that comes to my mind when I think about Lisp (it’s LISt Processing for a reason) - we’ll back one up will a mercury list.
  3. A dotted list represents the scheme form (a b . c) - we store the all elements “untill the dot” in a list and after the dot in another field.
  4. A number backed by the integer type.
  5. A string backed by the string type.
  6. A boolean backed by the bool type.


The implementation for this part is available here:

It’s not that exciting honestly - the adt is spelled out, and everything was explained above. So let’s see the two changed files.


:- module adts.

:- interface.

:- import_module string, list, integer, bool.

:- type lisp_val ---> lisp_atom(string)
                    ; lisp_list(list(lisp_val))
                    ; lisp_dotted_list(list(lisp_val), lisp_val)
                    ; lisp_number(integer)
                    ; lisp_string(string)
                    ; lisp_bool(bool.bool)

And the Makefile:

	mmc --make libadts
	mmc --make main


In the next part we’ll add parsers returning the values above and introduce a way to make writing/reading parsers a little easier.