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Linear Algebraic Characterization of Logic Programs

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Knowledge Science, Engineering and Management (KSEM 2017)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 10412))

Abstract

This paper introduces a novel approach for computing logic programming semantics based on multilinear algebra. First, a propositional Herbrand base is represented in a vector space and if-then rules in a program are encoded in a matrix. Then we provide methods of computing the least model of a Horn logic program, minimal models of a disjunctive logic program, and stable models of a normal logic program by algebraic manipulation of higher-order tensors. The result of this paper exploits a new connection between linear algebraic computation and symbolic computation, which has potential to realize logical inference in huge scale of knowledge bases.

This work is supported by NII Collaborative Research Program.

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Notes

  1. 1.

    The operator \(T_P\) of [17] is applied to Horn programs by viewing each constraint as a rule with the head \(\bot \). In this setting, every atom in \(B_P\) is derived in \(T_P(I)\) if I is inconsistent, i.e., \(\bot \in I\). Note that \(\top \in T_P(I)\) by \((\top \leftarrow \top )\in P\).

  2. 2.

    \(I=\{\top \}\) is semantically identified with \(I=\{\}\).

  3. 3.

    Here we omit the row and the column representing \(\bot \). When a program contains no constraints, the row and the column representing \(\bot \) can be removed (see Sect. 4).

  4. 4.

    Here we omit \(\top \) in each model.

References

  1. Brewka, G., Eiter, T., Truszczynski, M. (eds.): Special issue on answer set programming. AI Mag. 37(3), 5–80 (2016)

    Google Scholar 

  2. Dowling, W.F., Gallier, J.H.: Linear-time algorithms for testing the satisfiability of propositional Horn formulae. J. Logic Program. 1(3), 267–284 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  3. Fernandez, J.A., Lobo, J., Minker, J., Subrahmanian, V.S.: Disjunctive LP + integrity constraints = stable model semantic. Ann. Math. AI 8(3–4), 449–474 (1993)

    MathSciNet  MATH  Google Scholar 

  4. Gelfond, M., Lifschitz, V.: The stable model semantics for logic programming. In: Proceedings of the 5th International Conference and Symposium on Logic Programming, pp. 1070–1080. MIT Press (1988)

    Google Scholar 

  5. Grefenstette, E.: Towards a formal distributional semantics: simulating logical calculi with tensors. In: Proceedings of the 2nd Joint Conference on Lexical and Computational Semantics, pp. 1–10 (2013)

    Google Scholar 

  6. Kolda, T.G., Bader, B.W.: Tensor decompositions and applications. SIAM Rev. 51(3), 455–500 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. Lifschitz, V., Turner, H.: Splitting a logic program. In: Proceedings of the 5th International Conference on Logic Programming, pp. 23–37. MIT Press (1994)

    Google Scholar 

  8. Liu, G., Janhunen, T., Niemelä, I.: Answer set programming via mixed integer programming. In: Proceedings of the 13th International Conference on Knowledge Representation and Reasoning, pp. 32–42 (2012)

    Google Scholar 

  9. Lloyd, J.W.: Foundations of Logic Programming, 2nd edn. Springer, Heidelberg (1987)

    Book  MATH  Google Scholar 

  10. Nickel, M., Murphy, K., Tresp, V., Gabrilovich, E.: A review of relational machine learning for knowledge graphs. Proc. IEEE 104(1), 11–33 (2016)

    Article  Google Scholar 

  11. Rocktaschel, T., Bosnjak, M., Singh, S., Riedel, S.: Low-dimensional embeddings of logic. In: Proceedings of the ACL 2014 Workshop on Semantic Parsing, pp. 45–49 (2014)

    Google Scholar 

  12. Simons, P., Niemela, I., Soininen, T.: Extending and implementing the stable model semantics. Artif. Intell. 138, 181–234 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  13. Saraswat, V.: Reasoning 2.0 or machine learning and logic-the beginnings of a new computer science. Data Science Day, Kista, Sweden (2016)

    Google Scholar 

  14. Sato, T.: Embedding Tarskian semantics in vector spaces. In: Proceedings of the AAAI-2017 Workshop on Symbolic Inference and Optimization (2017)

    Google Scholar 

  15. Sato, T.: A linear algebraic approach to Datalog evaluation. TPLP 17(3), 244–265 (2017)

    MathSciNet  Google Scholar 

  16. Serafini, L., d’Avila Garcez, A.S.: Learning and reasoning with logic tensor networks. In: Adorni, G., Cagnoni, S., Gori, M., Maratea, M. (eds.) AI*IA 2016. LNCS, vol. 10037, pp. 334–348. Springer, Cham (2016). doi:10.1007/978-3-319-49130-1_25

    Chapter  Google Scholar 

  17. van Emden, M.H., Kowalski, R.A.: The semantics of predicate logic as a programming language. JACM 23(4), 733–742 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  18. Yang, B., Yih, W.-T., He, X., Gao, J., Deng, L.: Embedding entities and relations for learning and inference in knowledge bases. In: Proceedings of the International Conference on Learning Representations (2015)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Wakayama University, Wakayama, Japan

    Chiaki Sakama

  2. National Institute of Informatics, Tokyo, Japan

    Katsumi Inoue

  3. AI Research Center AIST, Tokyo, Japan

    Taisuke Sato

Authors
  1. Chiaki Sakama
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  2. Katsumi Inoue
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  3. Taisuke Sato
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Corresponding author

Correspondence to Chiaki Sakama .

Editor information

Editors and Affiliations

  1. Deakin University, Burwood, Victoria, Australia

    Gang Li

  2. University of Arizona, Tucson, Arizona, USA

    Yong Ge

  3. Southwest University, Chongqing, China

    Zili Zhang

  4. Peking University, Beijing, China

    Zhi Jin

  5. University of Technology Sydney, Sydney, New South Wales, Australia

    Michael Blumenstein

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Sakama, C., Inoue, K., Sato, T. (2017). Linear Algebraic Characterization of Logic Programs. In: Li, G., Ge, Y., Zhang, Z., Jin, Z., Blumenstein, M. (eds) Knowledge Science, Engineering and Management. KSEM 2017. Lecture Notes in Computer Science(), vol 10412. Springer, Cham. https://doi.org/10.1007/978-3-319-63558-3_44

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  • DOI: https://doi.org/10.1007/978-3-319-63558-3_44

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  • Publisher Name: Springer, Cham

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  • Online ISBN: 978-3-319-63558-3

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