zlacker

>>matt_d+(OP)
Abstract. Logic programming has a long history. The representative of logic programming in practice, the language Prolog, has been introduced more than 50 years ago. The main features of Prolog are still present today: a Prolog program is a set of predicate definitions executed by resolution steps with a backtracking search strategy. The use of back- tracking was justified by efficiency reasons when Prolog was invented. However, its incompleteness destroys the elegant connection of logic pro- gramming and the underlying Horn clause logic and causes difficulties to teach logic programming. Moreover, the restriction to predicates hinders an adequate modeling of real world problems, which are often functions from input to output data, and leads to unnecessarily inefficient exe- cutions. In this paper we show a way to overcome these problems. By transforming predicates and goals into functions and nested expressions, one can evaluate them with a demand-driven strategy which might re- duce the number of computation steps and avoid infinite search spaces. Replacing backtracking by complete search strategies with new imple- mentation techniques closes the gap between the theory and practice of logic programming. In this way, we can keep the ideas of logic program- ming in future programming systems.

>>xelxeb+th2
isn't backtracking a complete search strategy?

>>cerved+Mt2
That phrase was badly written.

Backtracking is a complete search of the problem-space.

What is incomplete is the Horn-SAT problem space, which is a subset of SAT, that can be solved in polynomial time, and is what Prolog is based on.

A complete logic system would have to solve SAT, which is NP-complete.

At least that's what I understood they meant by that.

>>desden+YH2
Yeah, it's confusing. The article is referring to the incompleteness of Prolog implemented using Depth First Search. That's what the author means by "backtracking". I know this because I know "backtracking" is used in the logic programming community to stand for DFS, but if you don't know the jargon you'd be right to be confused. You can kind of, er, glean, that meaning in the article if you see how they refer to a "fixed" search strategy, and also notice that "backtracking" is not normally a search strategy since it can't search on its own. "Backtracking" is really "DFS with backtracking".

The article is pointing out that Prolog with backtracking DFS is incomplete with respect to the completeness of SLD-Resolution. To clarify, SLD-Resolution is complete for refutation, or with subsumption. Prolog is an implementation of SLD-Resolution using DFS with backtracking. DFS is incomplete in the sense that it gets stuck in infinite loops when an SLD tree (the structure searched by DFS in Prolog) has cycles, especially left-recursive cycles. The article gives an example of a program that loops forever when executed with DFS with backtracking, in ordinary Prolog.

SLD-Resolution's completeness does not violate the Church-Turing thesis, so it's semi-decidable: SLD-trees may have infinite branches. To be honest I don't know about the equivalence with Horn-SAT, but Resolution, restricted to definite clauses, i.e. SLD-Resolution, is complete (by refutation and subsumption, as I say above, and respecting some structural constraints to do with the sharing of variables in heads and bodies of clauses). We got several different proofs of its completeness so I think we can trust it's true.

Edit: where does this knowledge about Horn-Sat come from? Do you have references? Gimme gimme gimme.