Termination



In Prolog, there are two kinds of termination: Although both properties are undecidable in general, we can often derive interesting insights already by observation alone.

In the following, we assume that no exceptions arise during the computation.

Existential termination

A Prolog query Q terminates existentially iff we receive an answer when posting Q.

For example, the following query terminates existentially, because an answer is reported:
?- length(Ls, L).
Ls = [],
L = 0 .
    
If Prolog were a functional programming language, this would be the end of the story, because a (partial) function returns at most one result.

However, Prolog is a relational programming language, and relations can hold between multiple entities that are reported on backtracking. Therefore, existential termination does not fully characterize the procedural behaviour of a Prolog query.

Universal termination

A Prolog query Q terminates universally iff the query
?- Q, false.
    
terminates existentially.

Declaratively, the query ?- Q, false. is always false. For this reason, if an answer is eventually reported, then the answer is necessarily false.

More about termination

Consider again the definition of list_list_together/3, which is true if the third argument is the concatenation of the lists in the first and second argument:
list_list_together([], Bs, Bs).
list_list_together([A|As], Bs, [A|Cs]) :-
        list_list_together(As, Bs, Cs).
    
Notice that the simpler of the two cases, i.e., the fact that occurs in this definition, has no influence on the universal termination properties of this predicate. It is therefore wrong to call this fact a "termination condition", "stopping criterion" or anything related.

The following fragment alone already makes the most general query nonterminating in this case:
list_list_together([], Bs, Bs) :- false.
list_list_together([A|As], Bs, [A|Cs]) :-
        list_list_together(As, Bs, Cs).
    
Adding further pure clauses can never prevent nontermination.

See Reading Prolog Programs and Nontermination for more information about this technique, which is called program slicing.

Importantly, program slices can be generated automatically to help you find the reason for nontermination. See cTI and especially GUPU.


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