Vladimir Aleksić
Department of Computer Science, King’s College, Strand, London WC2R 2LS, U.K.
vladimir@dcs.kcl.ac.uk
Abstract. For first-order Horn clauses without equality, resolution is complete with an arbitrary selection of a single literal in each clause [dN 96]. Here we extend this result to the case of clauses with
equality for superposition-based inference systems. Our result is a generalization of the result given in
[BG 01]. We answer their question about the completeness of a superposition-based system for general
clauses with an arbitrary selection strategy, provided there exists a refutation without applications of
the factoring inference rule.
1
Introduction
Since the appearance of paramodulation as a development of resolution for first-order logic with equality,
there has been a lot of research in the direction of improving the efficiency of paramodulation-based inference systems. It resulted in numerous refinements of paramodulation, which all aimed at restricting the
applicability of the paramodulation inference rule. In this paper, we deal with one such refinement, namely
superposition on constrained clauses with constraint inheritance [NR 95], or basic superposition.
It is possible to further reduce the search space by applying selection strategies. The key idea is to restrict
the application of inference rules by allowing inference only on selected literals. Some of the known complete
selection strategies for basic superposition are the maximal strategy (where only maximal literals are selected
in each clause) and the positive strategy (where a single negative literal is selected, whenever there is one in
a clause).
There has been a few attempts to generalize the completeness results for different selection strategies (for
example, see [DV 95]). The latest result is the one of Bofill and Godoy [BG 01], where they prove that
arbitrary selection strategies are complete for a basic superposition calculus on Horn clauses, if it is compatible
with the positive strategy. Here we strengthen up their result (and answer a question they poised) by
proving that a basic superposition calculus for general first-order clauses is complete with arbitrary selection
strategies, provided that there exists a refutation without factoring inferences. A similar result, under the
same restriction for factoring inferences, was proved in [dN 96] (Theorem 6.7.4) for resolution calculi, and
our result means its generalization to basic superposition calculi.
2
Completeness for refutations without factoring
Our result is given for the following system BS for constrained clauses, which is motivated by strict superposition given in [NR 95]. We drop their factoring inference rule and, for left and right superposition inferences,
the literal ordering requirements.
Left superposition
Γ1 → l ≈ r, ∆1 | T1
Γ2 , s[l ] ≈ t → ∆2 | T2
Γ1 , Γ2 , s[r] ≈ t → ∆1 , ∆2 | T1 ∧ T2 ∧ l = l ∧ l r ∧ s t
where l is not a variable.
Right superposition
Γ1 → l ≈ r, ∆1 | T1
Γ2 → s[l ] ≈ t, ∆2 | T2
Γ1 , Γ2 → s[r] ≈ t, ∆1 , ∆2 | T1 ∧ T2 ∧ l = l ∧ l r ∧ s t
where l is not a variable.
Equality solution
Γ, s ≈ t → ∆
Γ →∆ | T ∧s=t
Theorem 1. Let C be a set of constrained first-order clauses that has a refutation by BS. Then there exists
a refutation compatible with any selection strategy, where exactly one literal is selected in each clause.
transformation is essentially different from the one used in [BG 01], for two reasons. First, we address derivations from general clauses, whereas they restrict themselves to the Horn case. Secondly, their transformation
method is constrained by the condition that a superposition-based calculus is complete with the positive
selection strategy, while we do not assume any such requirement.
Since in the case of derivations with Horn clauses the factoring inference never appears, the following statement easily follows from the previous theorem.
Corollary 1. Basic superposition with equality and ordering constraints for Horn clauses is complete with
arbitrary selection.
Our result cannot be further generalized. In the case of general clauses where all refutations involve factoring,
incompleteness for arbitrary selection strategies already appears in the propositional case (see [Ly 97]).
3
Future work
In case we apply a selection strategy by which only equality literals are selected some tautologies have to be
kept, as it has been shown by the following example from [Ly 97].
Example 1.
→ P (c, b, b)
P (c, c, b), P (c, b, c) → b ≈ c
P (x, y, y) → P (x, y, x)
P (x, y, y) → P (x, x, y)
P (c, c, c) →
Say that b c. It is possible to make only two superposition inferences, and in either case a tautology is
derived.
However, this set of clauses can be transformed to a logically equivalent set of clauses, where all arguments
of predicate clauses are variables (flat clauses).
x ≈ c, y ≈ b → P (x, y, y)
x ≈ c, y ≈ b, P (x, x, y), P (y, x, y) → b ≈ c
P (x, y, y) → P (x, y, x)
P (x, y, y) → P (x, x, y)
x ≈ c, P (x, x, x) →
In case we apply the same selection strategy to the modified set of clauses, the empty clause can be derived
without tautologies. Here the question arises – is it possible to eliminate all tautologies from derivations from
flat clauses if the selection strategy picks only equality literals from each clause?
References
[Al 05] V. Aleksić On arbitrary selection strategies for superposition. Technical report TR–05–03, King’s College
London, June 2005.
[BG 01] L. Bofill, G. Godoy. On the completeness of arbitrary selection strategies for paramoduletion. In Proceedings
ICALP 2001, pages 951–962, 2001.
[dN 96] H. de Nivelle. Ordering refinements of resolution. Dissertation, Technische Universiteit Delft, Delft, 1996.
[NR 95] R. Nieuwenhuis and A. Rubio. Theorem proving with ordering and equality constrained clauses. Journal of
Symbolic Computations, 19:321–351, 1995.
[Ly 97] C. Lynch. Oriented Equational Logic is Complete. Journal of Symbolic Computations, 23(1):23–45, 1997.
[DV 95] Anatoli Degtyarev, Yuri Koval and Andrei Voronkov. Handling Equality in Logic Programming via Basic
Folding. Technical report 101, Uppsala University, Computing Science Department, 1995.