[brian_cloutier]

Transitive closure is the first thing nearly every introduction to datalog (or Prolog, for that matter) will show you. All you had to do was click the link and scroll down:

    Edge("a", "b").
    Edge("b", "c").

    Path(x, y) :-
        Edge(x, y).

    Path(x, z) :-
        Path(x, y),
        Edge(y, z).

Symmetric closure, assuming I'm understanding correctly, is also trivial:

    SymmetricEdge(Left, Right) :-
        Edge(Left, Right).
    SymmetricEdge(Left, Right) :-
        Edge(Right, Left).

[fspeech]

I was wondering about termination. With finite ground facts that appears not to be an issue. Complexity is another matter.

[cfallin]

Re: complexity, read up on "semi-naïve evaluation" for the usual execution strategy, which basically involves keeping lists of newly-added tuples in each iteration of the fixpoint loop and only processing deltas related to those new additions. This, combined with good index inference for each relation (predicate), so that e.g. the node-neighbor lookup is O(log |V|), should result in efficient execution...

[JadeNB]

Termination is achieved in brian_cloutier's example (https://news.ycombinator.com/item?id=19423320) by the use of different clauses (e.g., `Path` vs. `Edge`), so that all recursion is productive.

[maweki]

Datalog always terminates because there is no way to derive new atoms.

[brian_cloutier]

Yes, but there's more to it than just that. It's possible to write Prolog programs which never derive new atoms yet still fail to terminate.

It might be more accurate to say that Datalog programs cannot derive new atoms and this allows Datalog interpreters to use a search strategy which is guaranteed to terminate.

EDIT: Thinking about this more, I'm not sure why Prolog couldn't also use breadth-first search. So maybe both are necessary: Datalog not only disallows creating new atoms, but it also has a better search strategy; the combination of the two results in guaranteed termination.

[maweki]

Datalog does not have a search strategy per se. The general "strategy" (or definition) is the fixpoint of the T_p operator that derives all one-step-derivable facts from the facts and the rules.

The minimal model (result of derivation) is defined as the fixpoint of this operation. And the minimal model is finite because of the fixed number of atoms.

Every other evaluation strategy must be equivalent to this. Both in terms of termination and minimal model.

Prolog could use breadth first instead of SLD (and there are other Prolog evaluation strategies that, for example, use tabling for derived facts) but then you might get an infinite number of backtracking points instead of terms of infinite depth. You haven't really won anything by doing that.

[YeGoblynQueenne]

My understanding is that Datalog terminates because it does not allow functions as arguments to predicates. No functions means it's not possible to create infinite terms:

  P(f(x)).
  p(f(f(x)).
  p(f(f(f(x)))).
  ... etc

In fact I understand that termination is guaranteed even if a datalog program is executed by a Prolog interpreter. Or in other words, it's a result of the language semantics, not its implementation.

(but I might be wrong about this- corrections welcome).

[maweki]

You are wrong.

p(X) :- p(X). does usually not terminate in a top-down (Prolog) system but does in a bottom-up (Datalog) system. Datalog doesn't even allow terms as predicate arguments so of course you can't construct them of infinite size. But also this isn't allowed (where it is in Prolog)

p(X) :- p(Y), X is Y + 1.

Given a fact p(0) you get p(X) true for any integer X or generate all positive integers. This doesn't terminate for some binding patterns.

[YeGoblynQueenne]

>> p(X) :- p(Y), X is Y + 1.

is/2 is not Datalog though- it takes an arithmetic function as an argument; +/2 in your example. So the above would not terminate because it's Prolog, not because it's a Datalog program evaluated by Prolog.

>> Datalog doesn't even allow terms as predicate arguments so of course you can't construct them of infinite size.

Usual confusion: I think you mean "terms" as in Prolog terms, i.e. atoms of predicates (as opposed to terms in logic programming, i.e. variables, constants and functions). Datalog accepts constants and variables so with a finite vocabulary you can't create infinite atoms. I don't think that has to do with bottom-up or top-down evaluation.

To be honest, I haven't read any Datalog papers, but I'd be surprised if the termination guaranteed rested upon a specific implementation rather than the language semantics. Maybe not surprised- but it would be less interersting if you can only guarantee termination if you implement the language just so, vs. if it's a property of the language.