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On the Right Path: A Modal Logic for Supervised Learning

Part of the Lecture Notes in Computer Science book series (LNTCS,volume 11813)

Abstract

Formal learning theory formalizes the process of inferring a general result from examples, as in the case of inferring grammars from sentences when learning a language. Although empirical evidence suggests that children can learn a language without responding to the correction of linguistic mistakes, the importance of Teacher in many other paradigms is significant. Instead of focusing only on learner(s), this work develops a general framework—the supervised learning game (SLG)—to investigate the interaction between Teacher and Learner. In particular, our proposal highlights several interesting features of the agents: on the one hand, Learner may make mistakes in the learning process, and she may also ignore the potential relation between different hypotheses; on the other hand, Teacher is able to correct Learner’s mistakes, eliminate potential mistakes and point out the facts ignored by Learner. To reason about strategies in this game, we develop a modal logic of supervised learning (SLL). Broadly, this work takes a small step towards studying the interaction between graph games, logics and formal learning theory.

Keywords

  • Formal learning theory
  • Modal logic
  • Dynamic logic
  • Undecidability
  • Graph games

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Fig. 1.

Notes

  1. 1.

    Due to page-limit constraints, in this Proceedings version all the proofs are omitted, as well as some auxiliary definitions. The reader interested in these details is encouraged to communicate with the authors.

  2. 2.

    Generally speaking, to define the existence of winning strategies for players, we need to extend SLG with some fixpoint operators. We leave this for future inquiry.

  3. 3.

    By abuse of notation, for any \(\varphi \in \mathcal {L}_{\blacklozenge \langle + \rangle }\), \(\varphi ^\star \) is a formula of the bridge modal logic.

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Acknowledgments

We thank Johan van Benthem and Fenrong Liu for their inspiring suggestions. We also wish to thank three anonymous LORI-VII referees for improvement comments. Dazhu Li is supported by China Scholarship Council and the Major Program of the National Social Science Foundations of China [17ZDA026].

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Baltag, A., Li, D., Pedersen, M.Y. (2019). On the Right Path: A Modal Logic for Supervised Learning. In: Blackburn, P., Lorini, E., Guo, M. (eds) Logic, Rationality, and Interaction. LORI 2019. Lecture Notes in Computer Science(), vol 11813. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-60292-8_1

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  • DOI: https://doi.org/10.1007/978-3-662-60292-8_1

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