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Learning and Reasoning with Logic Tensor Networks

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Part of the book series:Lecture Notes in Computer Science ((LNAI,volume 10037))

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Abstract

The paper introduces real logic: a framework that seamlessly integrates logical deductive reasoning with efficient, data-driven relational learning. Real logic is based on full first order language. Terms are interpreted inn-dimensional feature vectors, while predicates are interpreted in fuzzy sets. In real logic it is possible to formally define the following two tasks: (i) learning from data in presence of logical constraints, and (ii) reasoning on formulas exploiting concrete data. We implement real logic in an deep learning architecture, called logic tensor networks, based on Google’s\(\textsc {TensorFlow}^{\tiny {\text {TM}}}\) primitives. The paper concludes with experiments on a simple but representative example of knowledge completion.

The first author acknowledges the Mobility Program of FBK, for supporting a long term visit at City University London. He also acknowledges NVIDIA Corporation for supporting this research with the donation of a GPU. We also thank Prof. Marco Gori and his group at the University of Siena for the generous and inspiring discussions on the topic of integrating logical reasoning and statistical machine learning.

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Notes

  1. 1.

    In logic, the term “grounding” indicates the operation of replacing the variables of a term/formula with constants, or terms that do not contains other variables. To avoid confusion, we use the synonym “instantiation” for this sense.

  2. 2.
  3. 3.

    Normally, a probabilistic approach is taken to solve this problem, and one that requires instantiating all clauses to remove variables, essentially turning the problem into a propositional one;ltn takes a different approach.

  4. 4.

    Notice how no grounding is provided about the signature of the knowledge-base.

  5. 5.

    A smoth factor\(\lambda ||\mathbf {\Omega }||^2_2\) is added to the loss to limit the size of parameters.

  6. 6.

    \(\mu (a,b) = \min (1,a+b)\).

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Author information

Authors and Affiliations

  1. Fondazione Bruno Kessler, Trento, Italy

    Luciano Serafini

  2. City University London, London, UK

    Artur S. d’Avila Garcez

Authors
  1. Luciano Serafini

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  2. Artur S. d’Avila Garcez

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Correspondence toLuciano Serafini.

Editor information

Editors and Affiliations

  1. University of Genoa , Genova, Italy

    Giovanni Adorni

  2. University of Parma , Parma, Italy

    Stefano Cagnoni

  3. University of Siena , Siena, Italy

    Marco Gori

  4. University of Genova , Genova, Italy

    Marco Maratea

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Serafini, L., d’Avila Garcez, A.S. (2016). Learning and Reasoning with Logic Tensor Networks. In: Adorni, G., Cagnoni, S., Gori, M., Maratea, M. (eds) AI*IA 2016 Advances in Artificial Intelligence. AI*IA 2016. Lecture Notes in Computer Science(), vol 10037. Springer, Cham. https://doi.org/10.1007/978-3-319-49130-1_25

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