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International Symposium on Frontiers of Combining Systems

FroCoS 2021: Frontiers of Combining Systems pp 43–62Cite as

Symbol Elimination and Applications to Parametric Entailment Problems

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Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 12941))

Abstract

We analyze possibilities of second-order quantifier elimination for formulae containing parameters – constants or functions. For this, we use a constraint resolution calculus obtained from specializing the hierarchical superposition calculus. If saturation terminates, we analyze possibilities of obtaining weakest constraints on parameters which guarantee satisfiability. If the saturation does not terminate, we identify situations in which finite representations of infinite saturated sets exist. We identify situations in which entailment between formulae expressed using second-order quantification can be effectively checked. We illustrate the ideas on a series of examples from wireless network research.

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Notes

  1. 1.

    If \({\mathcal {T}}_0\) does not allow QE but has a model completion \({\mathcal {T}}_0^*\) which does, and if we use QE in \({\mathcal {T}}_0^*\) in Alg. 1, \({\mathcal {T}}_0 \wedge \forall {\overline{x}} \Gamma _T({\overline{x}}) \cup G \models \bot \), but \(\forall {\overline{x}} \Gamma _T({\overline{x}})\) might not the weakest universal formula \(\Gamma \) with the property that \({\mathcal {T}}_0 \cup \Gamma \cup {\mathcal K}\models \bot \).

  2. 2.

    We can bring the clauses to this form using variable abstraction.

  3. 3.

    These conditions are satisfied by an LPO with an operator precedence in which the predicate symbol P (which can be regarded as function symbol with output sort \(\mathsf{bool}\)) is larger than the other operators and domain elements are minimal w.r.t. \(\succ \) which is supposed to be well-founded on the domain elements.

  4. 4.

    We can consider only models \(\mathcal {A}\) whose support of sort \(\mathsf{p}\) is infinite. The theory that formalizes this is the model completion of the theory \(\mathcal{E}\) of pure equality which allows quantifier elimination. We can then use the method for quantifier elimination in combinations of theories with QE described in [37].

  5. 5.

    If the set N of constrained P-clauses (hence the set of constrained Horn clauses \(CH_N\)) contains at least one parameter then \(\mu Z\) often returns “unknown”. In addition, if \(\mu Z\) can prove satisfiability of \(CH_N \cup \{ \lnot \mu _j({\overline{x}}) \}\) for a non-parametric problem, the model it returns is not guaranteed to be minimal in general, and cannot be used for representing the saturated set of clauses. By Theorem 7 (2), satisfiability of \(CH_N \cup \{ \lnot \mu _j({\overline{x}}) \}\) is sufficient for proving the satisfiability of N in this case.

  6. 6.

    We can iterate the application of \( HRes ^P_{\succ }\) for variables \(P^i_1, \dots , P^i_n\) (in this order). This corresponds to a variant of ordered resolution which we denote by \( HRes ^{P^i_1,\dots ,P^i_n}_{\succ }\); if saturation terminates the conjunction of clauses not containing \(P^i_1,\dots , P^i_n\) is equivalent to \(\exists P^i_1,\dots ,P^i_n~ N_{F_i}\), where \(N_{F_i}\) is the clause form of \(F_i\).

  7. 7.

    To check that the inclusion holds in one given model \(\mathcal {A}\) we can choose \({\mathcal {T}}= \mathsf{Th}(\mathcal {A})\).

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Acknowledgments

We thank Hannes Frey and Lucas Böltz for the numerous discussions we had on the problems in wireless networks discussed in Sect. 1.1, Renate Schmidt for maintaining a website where one can run SCAN online and for sending us the executables and instructions for running them, and to the anonymous reviewers for their helpful comments.

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  1. University Koblenz-Landau, Koblenz, Germany

    Dennis Peuter & Viorica Sofronie-Stokkermans

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  1. Dennis Peuter
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Correspondence to Viorica Sofronie-Stokkermans .

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  1. University of Liverpool, Liverpool, UK

    Boris Konev

  2. The University of Manchester, Manchester, UK

    Giles Reger

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Peuter, D., Sofronie-Stokkermans, V. (2021). Symbol Elimination and Applications to Parametric Entailment Problems. In: Konev, B., Reger, G. (eds) Frontiers of Combining Systems. FroCoS 2021. Lecture Notes in Computer Science(), vol 12941. Springer, Cham. https://doi.org/10.1007/978-3-030-86205-3_3

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