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Logics of knowledge and action: critical analysis and challenges

Autonomous Agents and Multi-Agent Systems volume 29, pages 719–753 (2015)Cite this article

Abstract

We overview the most prominent logics of knowledge and action that were proposed and studied in the multiagent systems literature. We classify them according to these two dimensions, knowledge and action, and moreover introduce a distinction between individual knowledge and group knowledge, and between a nonstrategic an a strategic interpretation of action operators. For each of the logics in our classification we highlight problematic properties. They indicate weaknesses in the design of these logics and call into question their suitability to represent knowledge and reason about it. This leads to a list of research challenges.

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Fig. 1

Notes

  1. The set of agent names cannot be infinite if we want the set of all groups of agents to be enumerable—for example, if \(\mathbb {I}\) were allowed to be the set of natural numbers then the set of subsets of \(\mathbb {I}\) would be uncountable—, and enumerability is necessary to obtain finite axiomatisations. Another way out is to only consider finite subsets of an infinite set of agents. We refer to [48] for non-finite axiomatisations of logics of group knowledge.

  2. We have chosen the notation \({\langle \![ J ]\!\rangle }\)—the nesting of a modal diamond and a modal box—in order to signal a \(\forall \exists \) quantification. It is however not standard in the literature, where one can find \({\langle J \rangle }\) in Pauly’s Coalition Logic \(\mathtt {CL}\) and \({\langle \!\langle J \rangle \!\rangle }\) in Alternating-time Temporal Logic (\(\mathtt {ATL}\)).

  3. We here give a simplified version; the original formulation is not about the above extension \(\mathtt {PDL^{\!\forall }}\) of \(\mathtt {PDL}\), but about logic \(\mathtt {ES}\) [70]. The latter is a variant of the Situation Calculus where situation terms are suppressed (and which is therefore closer to modal logics) and which moreover comes with sensing actions and epistemic operators.

  4. Actually we give an equivalent axiomatisation that better matches the other axiomatisations in this paper: our \(\mathtt {RE({\langle \![ J ]\!\rangle })}\) and \(\mathtt {M({\langle \![ J ]\!\rangle })}\) do not appear in the original presentation, while \({\langle \![ \emptyset ]\!\rangle }\mathsf {G}\top \) and the inference rules \(\frac{\textstyle \varphi }{\textstyle ~ {\langle \![ \emptyset ]\!\rangle } \mathsf {G}\varphi ~}\) and \(\frac{\textstyle \varphi \rightarrow \psi }{\textstyle ~ {\langle \![ \emptyset ]\!\rangle } {\mathsf {X}}\varphi \rightarrow {\langle \![ \emptyset ]\!\rangle } {\mathsf {X}}\psi ~ }\) are missing in ours. The first can be proved from \(\mathtt {N({\langle \![ J ]\!\rangle })}\) via \(\mathtt {RE({\langle \![ J ]\!\rangle })}\) and the Greatest Fixpoint Axiom \(\mathtt {GFP(\mathsf {G})}\). The second can be derived from the first via \(\mathtt {RE({\langle \![ J ]\!\rangle })}\). The third can be derived from \(\mathtt {M({\langle \![ J ]\!\rangle })}\) via \(\mathtt {RE({\langle \![ J ]\!\rangle })}\).

  5. Thanks are due to Nicholas Asher for a confirmation of that point in conversation.

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Acknowledgments

The ideas in this paper were presented in an invited talk at the 2013 International Joint Conference on Artificial Intelligence (IJCAI 2013) that was entitled “Logics for multiagent systems: a critical overview”. It is based on joint work with several colleagues that I would like to thank here for our smooth and fruitful collaboration: Philippe Balbiani (U. Toulouse, CNRS), Jan Broersen (U. Utrecht), Hans van Ditmarsch (U. Nancy, CNRS), Benoit Gaudou (U. Toulouse), Tiago de Lima (U. Artois, CNRS), Emiliano Lorini (U. Toulouse, CNRS), Faustine Maffre (U. Toulouse), Frédédric Moisan (U. Toulouse), François Schwarzentruber (U. Rennes, ENS), Nicolas Troquard (CNR, Trento), Dirk Walther (U. Politecnica de Madrid/U. Dresden). The overview presented in this paper is part of the efforts of the SINTELNET network (www.sintelnet.eu) to reexamine the logical foundations of concepts that are commonly used in artificial intelligence and the social sciences.

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  1. CNRS and University of Toulouse, IRIT, 118 Route de Narbonne, 31062 , Toulouse Cedex 9, France

    Andreas Herzig

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Herzig, A. Logics of knowledge and action: critical analysis and challenges. Auton Agent Multi-Agent Syst 29, 719–753 (2015). https://doi.org/10.1007/s10458-014-9267-z

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Keywords

  • Logic of action
  • Logic of knowledge
  • Common knowledge
  • Frame problem
  • Uniform strategy