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Logic Tensor Networks

Paper · 2020 · Samy Badreddine, Artur d'Avila Garcez, Luciano Serafini et al.

Summary

Deep networks learn well from raw data but need a lot of it, generalize poorly outside their training distribution and are hard to interpret. Symbolic AI reasons over readable knowledge but handles noise and incomplete information badly. This paper gives the full formal account of Logic Tensor Networks (LTN), which unites the two approaches in a single neurosymbolic framework. Its core is Real Logic, a first-order language whose truth values lie anywhere between 0 and 1. In Real Logic, constants, functions and predicates are grounded as tensors and neural computational graphs, so every formula is differentiable. Learning means maximizing how well a knowledge base is satisfied, which in practice turns logical constraints into a loss function. Reasoning works semantically: gradient descent searches for counterexamples to a candidate consequence. A set of small illustrative examples, rather than large benchmarks, shows that one language covers classification, clustering, regression, relational and semi-supervised learning, embedding learning and query answering. The result is a single, efficient TensorFlow 2 tool for injecting structured knowledge into deep models.

Key points

  • Real Logic uses fuzzy first-order semantics with typed domains. Variables are fixed-length sequences of tensors, and relations are interpreted as real-valued functions rather than fuzzy set memberships.
  • The paper adds three features to earlier LTN versions:
    • explicit domain typing, including Cartesian-product domains;
    • guarded quantifiers, which restrict quantification by a Boolean condition;
    • diagonal quantification over aligned tuples.
  • The product t-norm setup gains a generalized mean aggregator and fixes for vanishing and exploding gradients, inspired by Differentiable Fuzzy Logic (DFL).
  • A refutation-based reasoning procedure is formally defined. Example 4.8 shows that it captures logical consequences better than querying formulas after training.
  • The library and all example code are implemented in TensorFlow 2 and released at github.com/logictensornetworks/logictensornetworks.

Our summary of the paper, in our own words — not its text. The work belongs to its authors; read it at the links above.

 Institutions

🔬Fondazione Bruno KesslerTrento · Italy35 works · 43,945 citations

Also on the byline: City, University of London · Sony Computer Science Laboratories.

 Fields, topics and concepts

📚 Shelves 2The field of science this work belongs to (OpenAlex fields).Computer Science · 16,667 worksArtificial Intelligence · 16,194 works

💡 Concepts 1Key ideas and methods explained on our concept pages.Artificial intelligence · 49 works

Cite

Cite as Samy Badreddine, Artur d'Avila Garcez, Luciano Serafini and Michael Spranger. 2020. Logic Tensor Networks. arXiv:2012.13635 · Artificial Intelligence, Volume 303, February 2022, 103,649.

The paper itself, not this page. BibTeX from the library's record of the work. DOI 10.1016/j.artint.2021.103649 · arXiv 2012.13635

@misc{badreddine2020logic,
  title = {{Logic Tensor Networks}},
  author = {Samy Badreddine, Artur d'Avila Garcez, Luciano Serafini and Michael Spranger},
  year = {2020},
  doi = {10.1016/j.artint.2021.103649},
  eprint = {2012.13635},
  archivePrefix = {arXiv},
  url = {https://doi.org/10.1016/j.artint.2021.103649}
}

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