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Certifying Standard and Stratified Datalog Inference Engines in SSReflect

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Interactive Theorem Proving (ITP 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10499))

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Abstract

We propose a SSReflect library for logic programming in the Datalog setting. As part of this work, we give a first mechanization of standard Datalog and of its extension with stratified negation. The library contains a formalization of the model theoretical and fixpoint semantics of the languages, implemented through bottom-up and, respectively, through stratified evaluation procedures. We provide corresponding soundness, termination, completeness and model minimality proofs. To this end, we rely on the Coq proof assistant and SSReflect. In this context, we also construct a preliminary framework for dealing with stratified programs. We consider this to be a necessary first step towards the certification of security-aware data-centric applications.

This work was supported by the Datacert project (ANR-15-CE39-0009) of the French ANR.

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Notes

  1. 1.

    Arithmetic predicates and skolem functions destroy this guarantee.

  2. 2.

    http://math-comp.github.io/math-comp/.

  3. 3.

    Term sequences \(t_{1},\ldots ,t_{n}\) are abbreviated as \(\vec {t}\) and \(|\vec {t} |= n\) denotes their length.

  4. 4.

    We call language constructs that are variable-free, ground and, otherwise, open.

  5. 5.

    The set of program constants is also called its active domain, denoted adom(P).

  6. 6.

    Also called clause instantiation.

  7. 7.

    \(\sqcup \) denotes the pairwise disjoint set union.

  8. 8.

    A program can have multiple stratifications.

  9. 9.

    As proven by Apt et al. [3], \(M_n\) is independent from the choice of stratification.

  10. 10.

    By abuse of notation, we use the same \(\omega \) for the different numbers of \(T_P\) iterations needed to reach a fixpoint, when evaluating each program slice.

  11. 11.

    Since Datalog does not have function symbols and interpretations are ground, we can restrict substitution codomains to the set of program constants, w.l.o.g.

  12. 12.

    Groundings can be coerced to substitutions and substitutions can be lifted to groundings, by padding with a default element def.

  13. 13.

    We establish corresponding reflexivity, antisymmetry and transitivity properties.

  14. 14.

    We use the boolean quantifier, as the ordinal type of variables is finite.

  15. 15.

    gr_atom_def lifts substitutions to groundings, by padding with the def constant.

  16. 16.

    Thanks to using the bigcup operator from the SSReflect bigop library.

  17. 17.

    We state this as the fwd_chainP reflection lemma.

  18. 18.

    “Positive” interpretations are sets of ground atoms with a true flag.

  19. 19.

    This corresponds to the set of all “positive” ground atoms.

  20. 20.

    This is the top element of interp cf. Sect. 3.1.

  21. 21.

    The dashed encodep arrow marks the partiality of the cancellation lemma.

  22. 22.

    i.e, \(\mathrm {str}_\le \) stratifies \(\mathrm {p}_{\mathrm {str}_\le }\) and \(\mathrm {str}_>\) stratifies \(\mathrm {p}_{\mathrm {str}_>}\).

  23. 23.

    http://www.cs.nott.ac.uk/types06/slides/NathanWhitehead.pdf.

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Authors and Affiliations

  1. Université Paris Sud, LRI, 91 405, Orsay, France

    Véronique Benzaken

  2. CNRS, LRI, Université Paris Sud, 91 405, Orsay, France

    Évelyne Contejean

  3. LIRIS, Université Lyon 1, Lyon, France

    Stefania Dumbrava

Authors
  1. Véronique Benzaken
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  2. Évelyne Contejean
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  3. Stefania Dumbrava
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Correspondence to Stefania Dumbrava .

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Editors and Affiliations

  1. University of Brasília, Brasília D.F., Brazil

    Mauricio Ayala-Rincón

  2. NASA, Hampton, VA, USA

    César A. Muñoz

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Benzaken, V., Contejean, É., Dumbrava, S. (2017). Certifying Standard and Stratified Datalog Inference Engines in SSReflect. In: Ayala-Rincón, M., Muñoz, C.A. (eds) Interactive Theorem Proving. ITP 2017. Lecture Notes in Computer Science(), vol 10499. Springer, Cham. https://doi.org/10.1007/978-3-319-66107-0_12

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  • DOI: https://doi.org/10.1007/978-3-319-66107-0_12

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