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Combining Induction and Saturation-Based Theorem Proving

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Abstract

A method is devised to integrate reasoning by mathematical induction into saturation-based proof procedures based on resolution or superposition. The obtained calculi are capable of handling formulas in which some of the quantified variables range over inductively defined domains (which, as is well-known, cannot be expressed in first-order logic). The procedure is defined as a set of inference rules that generate inductive invariants incrementally and prove their validity. Although the considered logic itself is incomplete, it is shown that the invariant generation rules are complete, in the sense that if an invariant (of some specific form) is deducible from the considered clauses, then it is eventually generated.

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Notes

  1. Provided usual ordering restrictions are used, blocking any inference involving the first literal of the first clause since it is not maximal.

  2. Usual structural transformations (see, e.g., [28, 29]) can be used to avoid an exponential blow-up – the symbols \(\mathbf{T}_{\alpha }\) can be used for this purpose, together with the axioms in Sect. 3.2, which ensures that equivalence is preserved in \(\mathtt{I}\)-normal interpretations.

  3. Because the inference trees are usually finite, it is sufficient to check that no clause is one of its own ancestors.

  4. The order of the premises is arbitrary.

  5. In practice, the inference tree does not have to be computed explicitly: we only have to be able to compute the candidate invariant \(\mathfrak {I}^{\delta }({\mathcal {C}},{\mathcal {X}},p)\) defined in Sect. 3.5. This can be done by generating renamings on demand.

  6. The reader can check that the implication does not hold if \({\mathtt {s}}\) is not an inductive sort.

  7. Formula \(\exists x\, (x \prec t \wedge \lnot \alpha (x))\) may be encoded as a finite disjunction of cnf formulas \(\lnot \alpha (t_1) \vee \cdots \vee \lnot \alpha (t_n)\), with \(t_1,\dots ,t_n \triangleleft t\).

  8. As we shall see, this assertion is derived by a form of case splitting, performed by the Domain Decomposition rule defined in Sect. 3.5, Page 24.

  9. This corresponds to an application of Tseitin’s extension rule with the formula \(\alpha \). Note that in the definition of \({\varGamma }_{\mathbf{T}_{}}\), \(\llbracket \alpha (\mathbf {x}) \vee \lnot \mathbf{T}_{\alpha }(\mathbf {x})\;\big |\; \top \rrbracket \) denotes a set of c-clauses, as explained in Sect. 2.3.1.

  10. Note that we do not have \(H_1,\dots ,H_n \models _{fol} C\sigma \) in general, because our definition of logical entailment does not assume that free variables are implicitly universally quantified. Instead we have \(\forall ^* H_1,\dots ,\forall ^* H_n \models _{fol} \forall ^* C\sigma \).

  11. We recall that \({\mathcal R}\) denotes the set of rules associated with the selectors, see Page 7 for details.

  12. The two parts are not necessarily disjoint.

  13. Formally, \(A^{\delta }({\mathcal {C}},{\mathcal {X}},p)\), \(\mathfrak {I}^{\delta }({\mathcal {C}},{\mathcal {X}},p)\) and \(\mathfrak {X}^{\delta }({\mathcal {C}},{\mathcal {X}},p)\) are thus also parameterized by the variable \(z\), however this dependency is left implicit for the sake of readability.

  14. In practice, the condition will be checked simply by testing the rule applied to derive \({\mathcal {C}}\).

  15. For the sake of readability the predicate N describing natural numbers in the original formulation is omitted.

  16. Note that we also generate the clause \(\lnot p(x) \vee p(x)\), which is a tautology. Intuitively, the invariant \(p(x) \wedge q(x,y)\), is proven by induction on y, under the condition p(x).

  17. See Sect. 3.2 for the definition of \({\varGamma }\) and Definition 10 on Page 10 for the definition of \(S_{\top }\).

  18. Note that u is not necessarily a constant (\(\triangleleft \)-minimal constants of the same sort as \(t|_p\) do not always exist). We may, however, assume w.l.o.g. that the signature contains a constant of every sort, but this constant is not necessarily a constructor.

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    M. Echenim & N. Peltier

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Echenim, M., Peltier, N. Combining Induction and Saturation-Based Theorem Proving. J Autom Reasoning 64, 253–294 (2020). https://doi.org/10.1007/s10817-019-09519-x

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