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Craig Interpolation with Clausal First-Order Tableaux

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Abstract

We develop foundations for computing Craig-Lyndon interpolants of two given formulas with first-order theorem provers that construct clausal tableaux. Provers that can be understood in this way include efficient machine-oriented systems based on calculi of two families: goal-oriented such as model elimination and the connection method, and bottom-up such as the hypertableau calculus. We present the first interpolation method for first-order proofs represented by closed tableaux that proceeds in two stages, similar to known interpolation methods for resolution proofs. The first stage is an induction on the tableau structure, which is sufficient to compute propositional interpolants. We show that this can linearly simulate different prominent propositional interpolation methods that operate by an induction on a resolution deduction tree. In the second stage, interpolant lifting, quantified variables that replace certain terms (constants and compound terms) by variables are introduced. We justify the correctness of interpolant lifting (for the case without built-in equality) abstractly on the basis of Herbrand’s theorem and for a different characterization of the formulas to be lifted than in the literature. In addition, we discuss various subtle aspects that are relevant for the investigation and practical realization of first-order interpolation based on clausal tableaux.

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Notes

  1. The development of Craig’s interpolation theorem is described in [25].

  2. This does not preclude to represent equality as a predicate with axioms that express reflexivity, symmetry, transitivity and substitutivity.

  3. Symbols that appear only in F or \(G\) are called F -colored and \(G\) -colored, respectively, or colored in general (also local has been used synonymously to colored), while the other symbols are called transparent (or grey). It appears that the association with occurrences is in particular necessary to take account of predicate polarity required by Craig-Lyndon (in contrast to just Craig) interpolants. Another reason for the occurrence-based labeling is that the possibility that some clause C is an instance of a clause in F as well as an instance of a clause in \(G\) should be retained. An occurrence of C then can be associated with either side label. A perhaps confusing aspect of the color-based terminology is that the term colored on the one hand indicates for a symbol that it appears only in one of F or \(G\), a property called isolated in [24], while, on the other hand, the colored property for compound structures, that is, terms and formulas, permits occurrences of transparent symbols. We apply the color-based terminology in Sect. 5 and discuss it further in the context of first-order interpolation at the end of Sect. 7. Limitations of the color-based approaches are also discussed in [14, Sect. 3].

  4. The \({\mathsf {ipol}}\) function is also defined on non-ground tableaux, as it is independent of occurrences of variables. However, the association with properties relevant for interpolation is simplified if we assume a ground tableau, which is without loss of generality: A calculus may construct as proof a closed free-variable tableau with occurrences of free rigid variables. Any ground instantiation of these variables yields a ground tableau that provides a proof of the same formula. In particular, an instantiation of each variable by a dedicated constant. For the considered properties, tableaux with variables can be represented by ground tableaux with such constants. The restriction that the tableau is for ground formulas provides another simplification that, however, does not restrict the applicability of the lemma in a first-order context: A ground tableau that provides a proof of a clausal first-order formula also provides a proof of a clausal ground formula, the conjunction of the tableau clauses. In the setting of two-sided tableaux this can be stated more precisely as: A leaf-closed two-sided ground tableau for two clausal first-order formulas is also a leaf-closed two-sided ground tableau for two clausal ground formulas, the conjunction of the tableau clauses with side \({\mathsf {F}}\) and the conjunction of the tableau clauses with side \({\mathsf {G}}\).

  5. Exhaustively rewriting with \(F \wedge \bot \equiv \bot \), \(F \vee \top \equiv \top \), \(F \wedge \top \equiv F\), \(F \vee \bot \equiv F\), modulo commutativity.

  6. An argument for this is that even clausal tableaux with atomic cut, which can polynomially simulate tree resolution, cannot polynomially simulate unrestricted resolution where proofs may have the form of dags [49].

  7. That semantic trees and tree resolution can simulate each other polynomially has been shown already in [69], where also relationships to many other propositional systems are investigated.

  8. To take account of merging duplicate literals, the method as presented in [8] has to be supplemented by the explicit consideration of factoring steps with the dedicated assignment of partial interpolants from [14, Def. 15], or, alternatively, it must be possible to label an occurrence with both provenance values at the same time, as noted in [36], and retaining Huang’s original definition for case (iii).(c) for the subcase where the resolution step is upon two literals which each have both provenances. As an example where merging is necessary consider \(F = (p \vee q) \wedge (\lnot p \vee r)\) and \(G= (p \vee \lnot q) \wedge (\lnot p \vee \lnot r)\).

  9. The argument in the proof of Theorem 12 to show that in case (i) for any leaf with label \(L_i\) there exists a target with the same side \({\mathsf {F}}\) can also be applied here, but the analogy for case (ii) has to be shown differently. It follows since in case (ii) a leaf with label \(L_i\) must have an ancestor with complementary literal that was introduced by the encoding of a resolution step upon \(L_i\) and its complement. If the label of the involved occurrence of the complement is \({\mathsf {F}}\), then the encoding of the resolution step would result in two successive nodes with the complement of \(L_i\), one with side \({\mathsf {F}}\) and one with side \({\mathsf {G}}\).

  10. The symbol \(\lhd \) is an adaptation of \(t \unrhd s\) for s is a subterm of t, common in the literature on term rewriting [27]. We use it in the strict version and in flipped direction because it is then in direct correspondence with ordering constraints on terms that determine the order of quantifications in our Theorem 21.

  11. \(C_1, \ldots , C_n\) can be arbitrary quantifier-free formulas, with clauses as a special case.

  12. In the case where \(F'\) and \(G'\) are clausal formulas, there also may exist smaller formulas \(F_{\textsc {exp}}\) and \(G_{\textsc {exp}}\) obtained as conjunction of variants of clauses of \(F'\) and by negating a conjunction of variants of clauses of \(G'\), respectively.

  13. This contrast to approaches like [21], where instantiation is performed specifically for interpolation.

  14. Inspired by the Prolog Technology Theorem Prover [74] and SETHEO [51], CMProver was originally written in 1993 but had been revived in 1996 [26] and in 2016. It was evaluated in 2018 on all suitable TPTP problems, that is, problems that have a distinguished theorem, are not classified as satisfiable and are in clausal or quantified first-order form: Of these, it can solve about 76% of the 2143 problems without equality (in 9 configurations) and about 26% of the 11321 problems with equality (in 4 configurations). The timeout was 600s, the TPTP version was 7.1.0. See http://cs.christophwernhard.com/pie/cmprover/ for details.

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Acknowledgements

This research was in part supported by Deutsche Forschungsgemeinschaft (DFG) with grant WE 5641/1-1. The author thanks anonymous reviewers of previous versions for numerous very helpful suggestions and remarks.

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Wernhard, C. Craig Interpolation with Clausal First-Order Tableaux. J Autom Reasoning 65, 647–690 (2021). https://doi.org/10.1007/s10817-021-09590-3

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