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A Modal Logic for Supervised Learning

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. In this work, we develop a general framework—the supervised learning game—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 and study its properties. Broadly, this work takes a small step towards studying the interaction between graph games, logics and formal learning theory.

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Notes

  1. 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.

  2. For instance, in sabotage games, we use \(\Diamond \) to capture actions of Learner in formulas describing winning strategies (if they exist). See Gierasimczuk et al. (2009).

  3. In contrast, one extreme case of non-cooperative variants of SLG might be that Learner is allowed to stay at her current position in each round: she makes no efforts to reach the goal node.

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

  5. The four modalities used in its proof can be reduced to two by a standard argument Kracht and Wolter (1999), but we will omit the details because of the syntactic cost involved in writing the formulas.

  6. For more discussions on the applications of SG-style frameworks to paradigms of learning theory, we refer to Gierasimczuk et al. (2009), whose arguments also apply to SLG after minor modifications.

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Acknowledgements

We thank Johan van Benthem, Fenrong Liu, Fernando R. Velázquez-Quesada, Nina Gierasimczuk and Lena Kurzen for their inspiring suggestions. We also wish to thank the two anonymous reviewers for their very useful comments for improvement. Dazhu Li is supported by the Major Program of the National Social Science Foundation of China [17ZDA026].

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Baltag, A., Li, D. & Pedersen, M.Y. A Modal Logic for Supervised Learning. J of Log Lang and Inf 31, 213–234 (2022). https://doi.org/10.1007/s10849-022-09359-w

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Keywords

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