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Towards a Wide-Coverage Tableau Method for Natural Logic

Part of the Lecture Notes in Computer Science book series (LNAI,volume 9067)


The first step towards a wide-coverage tableau prover for natural logic is presented. We describe an automatized method for obtaining Lambda Logical Forms from surface forms and use this method with an implemented prover to hunt for new tableau rules in textual entailment data sets. The collected tableau rules are presented and their usage is also exemplified in several tableau proofs. The performance of the prover is evaluated against the development data sets. The evaluation results show an extremely high precision above 97 % of the prover along with a decent recall around 40 %.


  • Combinatory Categorial Grammar
  • Lambda Logical Form
  • Natural logic
  • Theorem prover
  • Tableau method
  • Textual entailment

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  1. 1.

    Hereafter we assume the following standard conventions while writing typed \(\lambda \)-terms: a type of a term is written in a subscript unless it is omitted, a term application is left-associative, and a type constructor comma is right-associative and is ignored if atomic types are single lettered.

  2. 2.

    The latter two constructions have the same semantic types in the approach of [2], who also uses the C&C parser, like us, for obtaining logical forms but of first-order logic. The reason is that both PP and N categories for prepositional phrases and nouns, respectively, are mapped to et type.

  3. 3.

    An importance of the explanations is also shown by the fact that recently SemEval-2015 introduced a pilot task interpretable STS that requires systems to explain their decisions for semantic textual similarity.

  4. 4.

    The connection between LLFs of [12] and the terms of an abstract level was already pointed out by Muskens in the project’s description “Towards logics that model natural reasoning”.

  5. 5.

    Actually the obtained CCG term is not completely a \(\lambda \)-term since it may contain type changes from lexical rules. For instance, in \((\mathbf{several}_{\mathtt{n},\mathtt{n}} \mathbf{delegate}_{\mathtt{n}})_{\mathtt{np}}\) subterm, \((.)_{\mathtt{np}}\) operator changes a type of its argument into \(\mathtt{np}\). Nevertheless, this kind of type changes are accommodated in the \(\lambda \)-term normalization calculus.

  6. 6.

    Implementation of the prover, its computational model and functionality is a separate and extensive work, and it is out of scope of the current paper.

  7. 7.

    The Fracas test suite can be found at, and the SICK trial data at

  8. 8.

    It is not true that mod_n_tr \(_1\) always gives correct conclusions for the constructions similar to (10). In case of small beer glass the rule entails small glass that is not always the case, but this can be avoided in the future by having more fine-grained analysis of phrases (that beer glass is a compound noun), richer semantic knowledge about concepts and more restricted version of the rule; currently rule mod_n_tr \(_1\) can be considered as a default rule for analyzing this kind of constructions.

  9. 9.

    Note that the phrase in (16) is wrongly analyzed by the CCG parser; the correct analysis is \(\mathbf{for}_{\mathtt{np},\mathtt{n},\mathtt{n}} C_\mathtt{np}(\mathbf{nobel}_{\mathtt{n},\mathtt{n}} \mathbf{prize}_{\mathtt{n}})\). Moreover, entailments similar to (16) are not always valid (e.g. \(\mathbf{short}_{\mathtt{n},\mathtt{n}} \big ( \mathbf{man}_{\mathtt{pp},\mathtt{n}} (\mathbf{in}_{\mathtt{np},\mathtt{pp}} \mathbf{netherlands}_\mathtt{np}) \big ) \not \Rightarrow \mathbf{short}_{\mathtt{n},\mathtt{n}} \mathbf{man}_{\mathtt{n}}\)). Since the parser and our implemented filters, at this stage, are not able to give correct analysis of noun complementation and post-nominal modification, we adopt n_pp_mod as a default rule for these constructions.

  10. 10.

    The FraCaS data contains entailment problems requiring deep semantic analysis and it is rarely used for system evaluation. We are aware of a single case of evaluating the system against this data; namely, the NatLog system [9] achieves quite high accuracy on the data but only on problems with a single premise. The comparison of our prover to it must await future research.


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I would like to thank Reinhard Muskens for his discussions and continuous feedback on this work. I also thank Matthew Honnibal, James R. Curran and Johan Bos for sharing the retrained CCG parser and anonymous reviewers of LENLS11 for their valuable comments. The research is part of the project “Towards Logics that Model Natural Reasoning” and supported by the NWO grant (project number 360-80-050).

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Correspondence to Lasha Abzianidze .

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Abzianidze, L. (2015). Towards a Wide-Coverage Tableau Method for Natural Logic. In: Murata, T., Mineshima, K., Bekki, D. (eds) New Frontiers in Artificial Intelligence. JSAI-isAI 2014. Lecture Notes in Computer Science(), vol 9067. Springer, Berlin, Heidelberg.

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