Bartosz Witkowski - Blog.

Mercury basics

Execution model

To have an understanding what mercury and logic programming is about I’ll try to introduce a mental model of execution of mercury programs. If you’re familiar with how logic programming works (e.g. by knowing prolog) you can safely skim this section.

Because functional programming and logic programming are so closely related I’ll try to explain logic programming using functional programming. If you don’t know functional programming this may be little off-putting but I hope that the descriptions are good enough that you will be able to get a feeling for logic programming.

So, if you are familiar with functional programming you probably know that the main execution model is lambda calculus. I won’t try to introduce such a formal model for logic programming but one does exists - instead I’ll talk about something called equational reasoning.

In a functional programming language we can always think of the program as a step of substitutions. Using Lisp as an example the output of the following program:

(append (list 1 2 3) (append (list 4 5) (list 6 7)))

is equivalent to this program:

(append (list 1 2 3) (list 4 5 6 7))

and to this program:

(list 1 2 3 4 5 6 7)

I will talk more about Lisp in the series but for now let’s stop here.

Hopefully you see how each of the steps relate to each other. This process is not necessarily how the computer will interpret this program but the results will always be equivalent.

In logic programming we also use equational reasoning to think about the program execution. But logic programming adds another dimension to this process.

Logic programs are based on the concept of unification (trying to make “two sides equal”), a logic program consists of two things:

  1. Facts
  2. Query

One can view execution of a functional program as a list of expressions undergoing simplification. A logic program is a list of facts and a query.

The query and facts can be thought as the “opposite sides of an equation”. The act of trying to check if both sides are equal is the program.

This seems to imply two things:

  1. By only changing the query (and not the facts) you can get a completely different program.
  2. Word like “trying” and “querying” imply that a program can fail.

Indeed both above statements are true.

The first point is seen as a strength of logical programming where programs can be used in multiple ways - we will see what this means shortly.

The second point is also important - some completely normal logic programs will fail to unify. For a brain dead example:

X = 1, X = 2, X.

Will fail to unify. ‘,’ in both mercury and prolog is just ∧ (conjunction) from mathematics. Logical variables always start with an Upper case letter or an underscore _.

The facts for the above program are:

  1. X = 1
  2. X = 2

At a glance we see that the facts given are nonsensical. Querying for X must fail to unify.

Here we see the additional dimension I mentioned earlier - apart from thinking about simplifying terms we also must think about if they fail or not.

Variables

For simplification I’ll say that a logical variable can be in two states

  1. ground - having a set value and immutable
  2. free - yet unset and “waiting” for a value

More about unification

Unification works by trying to make “two sides” equal. For now I’ll list a couple of rules of unification (there are more but they will be introduced as we go):

  1. two ints unify only if they are equal
  2. lists will unify with each other if all of the elements of the list unify (position-wise)
  3. two ground variables will unify only if their values will unify
  4. a free variable will always unify with a ground variable, grounding it with that value.
  5. two free variables technically cannot unify - this differs from prolog where two free variables will become aliased. Aliasing “entwines” two variables so that unifying one unifies with the other aliased one - i.e X = Y, Y = 2 will unify both X and Y to 2. As long as all the variables are ground in the scope of one predicate (see below) we can think of them as aliased. What will happen behind the scenes is that mercury will reorder the facts so that the “groundings” happen first.

Quiz

Which of the examples will unify:

  1. 1 = 1.
  2. A = B, A = 1.
  3. A = B, A = 1, B = 2.
  4. A = B, A = 1, B = C, C = 2.
  5. [1, 2] = [1, 2].
  6. [2, 1] = [1, 2].
  7. X = 1, X = 2.
  8. X = 1, 2 = Y.
  9. X = Y, Y = X, X = [1, 2].

Answers:

  1. Y
  2. Y
  3. N
  4. N
  5. Y
  6. N
  7. N
  8. Y
  9. Y

Predicates

We can decompose functional programs into smaller units called… functions.

A function of arity is a relation between input arguments and the output. The output should be the same for the same inputs. Or more formally:

A predicate is the smallest unit of decomposition available to the logical programmer.

A predicate of arity - is a relationship between between variables and true, false - the result of unification.

Predicates can have input and output variables - this differs from functions that only have input variables and one (implicit) output.

An output variable will be free before unifying with the predicate and ground after.

An input variable will be ground (because we must “know” it at unification time) and ground after (in other word it won’t change it’s ground state).

While mercury can sometimes infer if the variable is an input variable or an output one we will be explicit about it during this tutorial. To mark some variables as input we use a mode declaration in the form of:

:- mode predicate_name(in, out). 

Where :- is a token that starts the mode declaration, mode is the keyword, predicate_name is the programmer-readable name of the predicate followed by a comma separated list of modes. There are more modes then just in and out and they will be introduced further along in the series.

Knowing all this we can rewrite the above scheme program to mercury:

% mode declaration
:- mode not_hello_world(out).
% facts
not_hello_world(X) :- 
  list.append([4, 5], [6, 7], A),
  list.append([1, 2, 3], A, X).
% query
not_hello_world(Z), Z.

First of all a % starts a comment in mercury - everything until the end of the line will be ignored.

Predicates consist of two parts - the head:

not_hello_world(X)

which consists of the predicate name (the programmer readable name of the predicate) and a comma separated list of variables - in this case we only have one variable called X. The names of the variables don’t matter outside of the scope of the predicate.

The second part of the predicate is the predicate body, in this case:

  list.append([4, 5], [6, 7], A),
  list.append([1, 2, 3], A, X)

Predicates are separated from the body by the :- token (sometimes called the neck) and terminated by the . token.

The predicate body is is composed (by conjunction or disjunction) of smaller facts or other predicates.

To interpret the above program we need to know that:

1. list.append(A, B, AB) - unifies AB to with concatenation of lists A and B - when used in the (in, in, out) mode. 1. [1, 2, 3] is a list comprehension

We can model the computation in our head by a series of steps

not_hello_world(Z), Z. 
% let's expand what not_hello_world stands for
not_hello_world(Z) ---> 
list.append([4, 5], [6, 7], A), list.append([1, 2, 3], A, X) -->
% We can potentially simplify either
%  (1) list.append([4, 5], [6, 7], A) or 
%  (2) list.append([1, 2, 3], A, X) 
% for (2) we don't know how list.append(in, out, out) works - that means we
% can only simplify it after A is ground we can however simplify (1)
A = [4, 5, 6, 7], list.append([1, 2, 3], A, X) --->
list.append([1, 2, 3], [4, 5, 6, 7], X),
X = [1, 2, 3, 4, 5, 6, 7]
% comming back to our query we have:
X = [1, 2, 3, 4, 5, 6, 7], X.
% which is true

The predicate above will unify Z with a list - again while mercury can infer the type of the variable it is good form to write it out explicitly:

:- pred not_hello_world(list(int)).

This declares the not_hello_world predicate to unify with a list of ints. We will learn more about types further along in the series.

Now for an example of a query which will fail:

not_hello_world(Z), Z = [1, 2, 3].

We would expand not_hello_world(Z) as before and “ground” it to the value [1, 2, 3, 4, 5, 6, 7] - our query would fail however because we cannot unify ` [1, 2, 3, 4, 5, 6, 7] with [1, 2, 3]`.

This query on the other hand is not possible at all:

Z = [1, 2, 3], not_hello_world(Z).

Why? Terms in mercury are by default unified top to bottom (from main), and from left to right ( see not_hello_world has only the not_hello_world(out) mode declared so we cannot use a ground variable. We need to declare another mode for not_hello_world(Z):

:- mode not_hello_world(in).

That will actually fail to compile - because not_hello_world(in) can fail the possibility of failing must be explicitly marked. The correct mode declarations are actually:

:- mode not_hello_world(out) is det.
:- mode not_hello_world(in) is semidet.

det and semited are two determinism modes that we will use for the time being. If a predicate is declared as det it means that it cannot fail. If there’s a possibility that the predicate could fail with this determinism declaration the compiler will signal an error (at compile time).

semidet predicates can either fail (to unify) or will succeed (unify)

Quizz

Try to infer all posible mode and type declarations for the following predicates:

1.

q1(A, B, C) :-
  A = 5,
  B = A,
  C = [B].

2.

q2(A, B, C) =
  A = [B, C],
  B = 1,
  C = 1.

3.

q3(A, B) =
  A = 5,
  B = list(list(A)).

Answers:

1.

:- pred q1(int, int, list(int)).
:- mode q1(out, out, out) is det.
:- mode q1(in, out, out) is semidet.
:- mode q1(out, in, out) is semidet.
:- mode q1(out, out, in) is semidet.
:- mode q1(in, in, out) is semidet.
:- mode q1(in, out, in) is semidet.
:- mode q1(out, in, in) is semidet.
:- mode q1(in, in, in) is semidet.

2.

:- pred q2(list(int), int, int).
:- mode q2(out, out, out) is det.
% all modes with at least one **in** argument will be semidet; same as in q1

3.

:- pred q3(int, list(list(int))).
:- mode q3(out, out) is det.
% all modes with at least one **in** argument will be semidet; same as in q1 and
% q2 but only for two variables

Closing words

In practice you cannot write programs like X = 1, X = 2, X. - while this is style of querying is possible in the prolog console mercury lacks that mechanism.

Nevertheless this method of analyzing mercury programs is applicable everywhere. Check out the next part for our very first mercury program.

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