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Title
Abstract
Key Takeaways
Introduction
Different Nature of Data and Conclusions
Class of Hypotheses
Infinite Procedures
Positive, True and Readable Data
Summary
Hypotheses as Protocols. A Note on a Temporal View
Conclusions and Further Work
References
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Philosophy
Logic

Bridging learning theory and dynamic epistemic logic

Profile image of Nina GierasimczukNina Gierasimczuk

2009, Synthese

https://doi.org/10.1007/S11229-009-9549-1
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14 pages

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Abstract

This paper discusses the possibility of modelling inductive inference in dynamic epistemic logic (see e.g. ). The general purpose is to propose a semantic basis for designing a modal logic for learning in the limit. First, we analyze a variety of epistemological notions involved in identification in the limit and match it with traditional epistemic and doxastic logic approaches. Then, we provide a comparison of learning by erasing and iterated epistemic update (Baltag and Moss 2004) as analyzed in dynamic epistemic logic. We show that finite identification can be modelled in dynamic epistemic logic, and that the elimination process of learning by erasing can be seen as iterated belief-revision modelled in dynamic doxastic logic. Finally, we propose viewing hypothesis spaces as temporal frames and discuss possible advantages of that perspective.

Key takeaways
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AI

  1. Dynamic epistemic logic (DEL) serves as a framework for modeling inductive inference and finite identification.
  2. Learning by erasing can be modeled in dynamic doxastic logic (DDL), emphasizing hypothesis preference.
  3. This paper proposes a semantic basis for designing modal logic for learning in the limit.
  4. Learning processes involve interactions between hypotheses, environments, and data streams.
  5. Hypothesis spaces can be viewed as temporal frames, enriching the model with historical context.

Related papers

Connecting dynamic epistemic and temporal epistemic logics
Hans van Ditmarsch

Logic Journal of IGPL, 2013

We give a relation between a logic of knowledge and change, with a semantics on Kripke models, and a logic of knowledge and time, with a semantics on interpreted systems. In particular, given an epistemic state (pointed Kripke model with equivalence relations) and a formula in a dynamic epistemic logic (a logic describing the consequences of epistemic actions), we construct an interpreted system relative to that epistemic state and that formula that satisfies the translation of the formula into a temporal epistemic logic. The construction involves that the protocol that is implicit in the dynamic epistemic formula, i.e., the set of sequences of actions being executed to evaluate the formula, is made explicit. We first focus on the logic of knowledge and change that is known as public announcement logic, then generalize our results to a dynamic epistemic logic. When compared to Kripke (possible worlds) models, interpreted systems have at least two appealing features: a natural accessibility relation between domain objects, and an equally natural notion of dynamics, modelled by runs. The accessibility relation as we know it from the possible worlds model is in this case grounded; it has a direct and natural interpretation, as follows. In an interpreted system, the role of possible worlds is performed by global states, which are constituted by the agents' local states and the state of the environment. Each agent knows exactly its own local state and the possible local states of other agents: two global states are indistinguishable for an agent if his local compartment is the same. Secondly, an interpreted system defines a number of runs through such global states (i.e., a sequence of global states). Each run corresponds to a possible computation allowed by a protocol. In an object language with temporal and epistemic operators one can then express temporal properties such as liveness and temporal epistemic properties such as perfect recall.

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A Proof-Theoretic Semantic Analysis of Dynamic Epistemic Logic
Giuseppe Greco

The present paper provides an analysis of the existing proof systems for dynamic epistemic logic from the viewpoint of proof-theoretic semantics. Dynamic epistemic logic is one of the best known members of a family of logical systems which have been successfully applied to diverse scientific disciplines, but the proof theoretic treatment of which presents many difficulties. After an illustration of the proof-theoretic semantic principles most relevant to the treatment of logical connectives, we turn to illustrating the main features of display calculi, a proof-theoretic paradigm which has been successfully employed to give a proof-theoretic semantic account of modal and substructural logics. Then, we review some of the most significant proposals of proof systems for dynamic epistemic logics, and we critically reflect on them in the light of the previously introduced proof-theoretic semantic principles. The contributions of the present paper include a generalisation of Belnap's cut elimination metatheorem for display calculi, and a revised version of the display-style calculus D.EAK [30]. We verify that the revised version satisfies the previously mentioned proof-theoretic semantic principles, and show that it enjoys cut elimination as a consequence of the generalised metatheorem.

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DDL as an " Internalization " of Dynamic Belief Revision
Virginie Fiutek, Sonja Smets

2014

In this paper we re-evaluate Segerberg's " full DDL " (Dynamic Doxas-tic Logic) from the perspective of Dynamic Epistemic Logic (DEL), in its belief-revision-friendly incarnation. We argue that a correct version of full DDL must give up the Success Postulate for dynamic revision. Next, we present (an appropriately generalized and simplified version of) full DDL, showing that it is a generalization of the so-called Topo-logic of Moss and Parikh. We construct AGM-friendly versions of full DDL, corresponding to various revising/contracting operations considered in the Belief Revision literature. We show that DDL can internalize inside one model the " external " doxastic dynamics of DEL, as well as the evidential dynamics investigated by van Benthem and Pacuit. In our Conclusions section, we compare three styles of modeling doxastic dynamics: DDL, DEL and PDL/ETL (the Propo-sitional Dynamic Logic approach, with its Epistemic Temporal Logic variant).

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Dynamic Epistemic Logic
Jelle Gerbrandy

1998

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The Dynamic Turn in Epistemic / Doxastic Logic
Jan van Eijck

2013

Dynamic epistemic logic, broadly conceived, is the study of rational social interaction in context, the study, that is, of how agents update their knowledge and change their beliefs on the basis of pieces of information they exchange in various ways. The information that gets exchanged can be about what is the case in the world, about what changes in the world, and about what agents know or believe about the world and about what others know or believe. This chapter gives an overview of dynamic epistemic logics, and traces some connections with propositional dynamic logic, with planning and with probabilistic updating.

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Dynamic Epistemic Logic and Knowledge Puzzles
Wiebe Van Der Hoek

Lecture Notes in Computer Science, 2007

We briefly give an overview of Dynamic Epistemic Logic (DEL), mainly in semantic terms. We focus on the simplest of epistemic actions in DEL, called public announcements. We also sketch the effect of more complex epistemic actions, and briefly show how als factual change can be modelled in the same framework. We then apply the logic of public announcements in DEL to the analysis of a knowledge puzzle, called 'What Sum'.

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Modeling truly dynamic epistemic scenarios in a partial version of DEL
Jens Ulrik Hansen

The Logica Yearbook 2013, pages 63-76, College Publications, 2014. ISBN 978-1848901445., 2014

Dynamic Epistemic Logic is claimed to be a dynamic version of epistemic logic. While this being true, there are several dynamical aspects that cannot be reasoned about in Dynamic Epistemic Logic. When a scenario is fixed and a possible world model representing the scenario is constructed, the possible future ways the system can evolve are in some sense already determined. For instance no new agents can enter the scenario and no new propositional facts can become relevant. This modeling perspective is the main motivation for the partial version of Dynamic Epistemic Logic introduced in this paper, which in particular, allows for the set of agents and the set of propositional variables to change.

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Dynamic Logics of Belief Change
Sonja Smets, Johan van Benthem

2015

This chapter gives an overview of current dynamic logics that describe belief update and revision, both for single agents and in multi-agent settings. We employ a mixture of ideas from AGM belief revision theory and dynamic-epistemic logics of information-driven agency. After describing the basic background, we review logics of various kinds of beliefs based on plausibility models, and then go on to various sorts of belief change engendered by changes in current models through hard and soft information. We present matching complete logics with dynamic-epistemic recursion axioms, and develop a very general perspective on belief change by the use of event models and priority update. The chapter continues with three topics that naturally complement the setting of single steps of belief change: connections with probabilistic approaches to belief change, long-term temporal process structure including links with formal learning theory, and multi-agent scenarios of information flow and belief revision in games and social networks. We end with a discussion of alternative approaches, further directions, and windows to the broader literature, while links with relevant philosophical traditions are discussed throughout.

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A Dynamic Informational-Epistemic Logic
Yuri David Santos

Dynamic Logic. New Trends and Applications - Lecture Notes in Computer Science, 2017

Epistemic logic is usually employed to model two aspects of a situation: the ontic and the epistemic aspects. Truth, however, is not always attainable, and in many cases we are forced to reason only with whatever information is available to us. In this paper, we will explore a four-valued epistemic logic designed to deal with situations of this sort. The technical results include a set of reduction axioms for public announcements, correspondence proofs, and a complete tableau system.

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Dynamic logic for belief revision
Johan van Benthem

Journal of Applied Non-Classical Logics, 2007

We show how belief revision can be treated systematically in the format of dynamicepistemic logic, when operators of conditional belief are added. The core engine consists of definable update rules for changing plausibility relations between worlds, which have been proposed independently in the dynamic-epistemic literature on preference change. Our analysis yields two new types of modal result. First, we obtain complete logics for concrete mechanisms of belief revision, based on compositional reduction axioms. Next, we show how various abstract postulates for belief revision can be analyzed by standard modal frame correspondences for model-changing operations.

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References (21)

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  2. Angluin, D. (1980). Inductive inference of formal languages from positive data. Information and Control, 45(2), 117-135.
  3. Baltag, A., & Moss, L. (2004). Logics for epistemic programs. Synthese, 139(2), 165-224.
  4. Batlag, A., Moss, L. S., & Solecki, S. (1998). The logic of public announcements and common knowledge and private suspicions. In Proceedings of the 7th TARK (pp. 43-56).
  5. Costa Florêntio, C. (2002). Learning generalized quantifiers. In Proceedings of the 7th ESSLLI Student Session.
  6. Dégremont, C., & Gierasimczuk, N. (2009). Can doxastic agents learn? On the temporal structure of learning. ILLC Prepublication (PP) Series PP-2009-14, Amsterdam.
  7. Freivalds, R., & Zeugmann, T. (1995). Co-learning of recursive languages from positive data, RIFIS. Tech- nical report, RIFIS-TR-CS-110, RIFIS, Kyushu University 33.
  8. Gierasimczuk, N. (2007). The problem of learning the semantics of quantifiers. In Proceedings of the 6th TbiLLC, Vol. 4363 of LNAI (pp. 117-126). Springer.
  9. Gierasimczuk, N. (2009). Identification through inductive verification. In Proceedings of the 7th TbiLLC, Vol. 5422 of LNAI (pp. 193-205). Springer.
  10. Gierasimczuk, N., Kurzen, L., & Velázquez-Quesada, F. (2009). Learning as interaction (manuscript).
  11. Gold, E. (1967). Language identification in the limit. Information and Control, 10, 447-474.
  12. Hintikka, J. (1962). Knowledge and belief. An introduction to the logic of the two notions. Ithaca: Cornell University Press.
  13. Jain, S., Osherson, D., Royer, J. S., & Sharma, A. (1999). Systems that learn. Chicago: MIT Press.
  14. Kelly, K. (1996). The logic of reliable inquiry. Oxford: Oxford University Press.
  15. Lange, S., Wiehagen, R., & Zeugmann, T. (1996). Learning by erasing. In Proceedings of the 7th interna- tional workshop on algorithmic learning theory (pp. 228-241). Springer-Verlag.
  16. Martin, E., & Osherson, D. (1998). Elements of scientific inquiry. Cambridge: MIT Press.
  17. Osherson, D., de Jongh, D., Martin, E., & Weinstein, S. (1997). Formal learning theory. In J. van Benthem & A. Ter Meulen (Eds.), Handbook of logic and language. Amsterdam: North Holland.
  18. Tiede, H.-J. (1999). Identifiability in the limit of context-free generalized quantifiers. Journal of Language and Computation, 1, 93-102.
  19. van Benthem, J. (1986). Essays in logical semantics. Dordrecht: D. Reidel.
  20. van Benthem, J., Gerbrandy, J., & Pacuit, E. (2007). Merging frameworks for interaction: DEL and ETL. In Proceedings of the 11th TARK (pp. 72-81).
  21. van Ditmarsch, H., van der Hoek, W., & Kooi, B. (2007). Dynamic epistemic logic. Springer Netherlands.

Related papers

Learning by Erasing in Dynamic Epistemic Logic
Nina Gierasimczuk

Lecture Notes in Computer Science, 2009

This work provides a comparison of learning by erasing and iterated epistemic update [2] as analyzed in dynamic epistemic logic (see e.g.

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Finite identification from the viewpoint of epistemic update
Nina Gierasimczuk

Information and Computation, 2011

Finite identifiability of sets from positive data Dynamic epistemic logic Epistemic temporal logic Epistemic update Formal learning theory constitutes an attempt to describe and explain the phenomenon of learning, in particular of language acquisition. The considerations in this domain are also applicable in philosophy of science, where it can be interpreted as a description of the process of scientific inquiry. The theory focuses on various properties of the process of hypothesis change over time. Treating conjectures as informational states, we link the process of conjecture-change to epistemic update. We reconstruct and analyze the temporal aspect of learning in the context of dynamic and temporal logics of epistemic change. We first introduce the basic formal notions of learning theory and basic epistemic logic. We provide a translation of the components of learning scenarios into the domain of epistemic logic. Then, we propose a characterization of finite identifiability in an epistemic temporal language. In the end we discuss consequences and possible extensions of our work.

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A Dynamic Logic for Learning Theory
Nina Gierasimczuk

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Building on previous work [4, 5] that bridged Formal Learning Theory and Dynamic Epistemic Logic in a topological setting, we introduce a Dynamic Logic for Learning Theory (DLLT), extending Subset Space Logics [17, 9] with dynamic observation modalities [o]ϕ, as well as with a learning operator L(#» o), which encodes the learner's conjecture after observing a finite sequence of data #» o. We completely axiomatise DLLT, study its expressivity and use it to characterise various notions of knowledge, belief, and learning.

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Dynamic Epistemic Logics
Jan van Eijck

Outstanding Contributions to Logic, 2014

Dynamic epistemic logic, broadly conceived, is the study of rational social interaction in context, the study, that is, of how agents update their knowledge and change their beliefs on the basis of pieces of information they exchange in various ways. The information that gets exchanged can be about what is the case in the world, about what changes in the world, and about what agents know or believe about the world and about what others know or believe. This chapter gives an overview of dynamic epistemic logics, and traces some connections with propositional dynamic logic, with planning and with probabilistic updating.

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Indicative Conditionals and Dynamic Epistemic Logic
Wesley Holliday

Proceedings of the Sixteenth Conference on Theoretical Aspects of Rationality and Knowledge (TARK XVI), 2017

Recent ideas about epistemic modals and indicative conditionals in formal semantics have significant overlap with ideas in modal logic and dynamic epistemic logic. The purpose of this paper is to show how greater interaction between formal semantics and dynamic epistemic logic in this area can be of mutual benefit. In one direction, we show how concepts and tools from modal logic and dynamic epistemic logic can be used to give a simple, complete axiomatization of Yalcin's [16] semantic consequence relation for a language with epistemic modals and indicative conditionals. In the other direction, the formal semantics for indicative conditionals due to Kolodny and MacFarlane [9] gives rise to a new dynamic operator that is very natural from the point of view of dynamic epistemic logic, allowing succinct expression of dependence (as in dependence logic) or supervenience statements. We prove decidability for the logic with epistemic modals and Kolodny and MacFarlane's indicative conditional via a full and faithful computable translation from their logic to the modal logic K45.

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Related topics

  • Modal Logic
  • Philosophy
  • Epistemic Logic
  • Doxastic Logic
  • Belief Revision
  • Dynamic Epistemic Logic
  • Inductive Inference

    • Cited by

    Truth-Tracking by Belief Revision
    Nina Gierasimczuk

    Studia Logica

    We study the learning power of iterated belief revision methods. Successful learning is understood as convergence to correct, i.e., true, beliefs. We focus on the issue of universality: whether or not a particular belief revision method is able to learn everything that in principle is learnable. We provide a general framework for interpreting belief revision policies as learning methods. We focus on three popular cases: conditioning, lexicographic revision, and minimal revision. Our main result is that conditioning and lexicographic revision can drive a universal learning mechanism, provided that the observations include only and all true data, and provided that a non-standard, i.e., non-well-founded prior plausibility relation is allowed. We show that a standard, i.e., well-founded belief revision setting is in general too narrow to guarantee universality of any learning method based on belief revision. We also show that minimal revision is not universal. Finally, we consider situations in which observational errors (false observations) may occur. Given a fairness condition, which says that only finitely many errors occur, and that every error is eventually corrected, we show that lexicographic revision is still universal in this setting, while the other two methods are not.

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    Learning to act: qualitative learning of deterministic action models
    Nina Gierasimczuk

    Journal of Logic and Computation

    In this article we study learnability of fully observable, universally applicable action models of dynamic epistemic logic. We introduce a framework for actions seen as sets of transitions between propositional states and we relate them to their dynamic epistemic logic representations as action models. We introduce and discuss a wide range of properties of actions and action models and relate them via correspondence results. We check two basic learnability criteria for action models: finite identifiability (conclusively inferring the appropriate action model in finite time) and identifiability in the limit (inconclusive convergence to the right action model). We show that deterministic actions are finitely identifiable, while arbitrary (nondeterministic) actions require more learning power-they are identifiable in the limit. We then move on to a particular learning method, i.e. learning via update, which proceeds via restriction of a space of events within a learning-specific action model. We show how this method can be adapted to learn conditional and unconditional deterministic action models. We propose update learning mechanisms for the afore mentioned classes of actions and analyse their computational complexity. Finally, we study a parametrized learning method which makes use of the upper bound on the number of propositions relevant for a given learning scenario. We conclude with describing related work and numerous directions of further work.

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