Defining the material conditional in Prolog

Question

I have been trying to acclimate to Prolog and Horn clauses, but the transition from formal logic still feels awkward and forced. I understand there are advantages to having everything in a standard form, but:

What is the best way to define the material conditional operator --> in Prolog, where A --> B succeeds when either A = true and B = true OR B = false ? That is, an if->then statement that doesn't fail when if is false without an else .

Also, what exactly are the non-obvious advantages of Horn clauses?

Answer 1

What is the best way to define the material conditional operator --> in Prolog

When A and B are just variables to be bound to the atoms true and false , this is easy:

cond(false, _).
cond(_, true).

But in general, there is no best way because Prolog doesn't offer proper negation, only negation as failure , which is non-monotonic. The closest you can come with actual propositions A and B is often

(\+ A ; B)

which tries to prove A , then goes on to B if A cannot be proven (which does not mean that it is false due to the closed-world assumption ).

Negation, however, should be used with care in Prolog.

Also, what exactly are the non-obvious advantages of Horn clauses?

That they have a straightforward procedural reading. Prolog is a programming language , not a theorem prover. It's possible to write programs that have a clear logical meaning, but they're still programs.

To see the difference, consider the classical problem of sorting. If L is a list of numbers without duplicates, then

sort(L, S) :-
    permutation(L, S),
    sorted(S).
sorted([]).
sorted([_]).
sorted([X,Y|L]) :-
    X < Y,
    sorted([Y|L]).

is a logical specification of what it means for S to contain the elements of L in sorted order. However, it also has a procedural meaning, which is: try all the permutations of L until you have one that it sorted. This procedure, in the worst case, runs through all n ! permutations, even though sorting can be done in O( n lg n ) time, making it a very poor sorting program.

See also this question .