Logics of knowledge and action: critical analysis and challenges

Andreas Herzig
2014 Autonomous Agents and Multi-Agent Systems  
OATAO is an open access repository that collects the work of Toulouse researchers and makes it freely available over the web where possible. This is an author-deposited version published in : http://oatao.univ-toulouse.fr/ Eprints ID : 15176 To link to 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
more » ... duce 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. Table 8 Axiomatisation of epistemic logics with common knowledge EL CK EL Some axiomatics of epistemic logic . In game theory, the hypothesis of common knowledge of rationality was proved to be fundamental for equilibria [9] . In theoretical computer science it is used to study communication in distributed systems [35] . The language has common knowledge operators CK J , one per group J ⊆ I. The formula CK J ϕ is read "it is common knowledge in group J that ϕ". As we have said in Sect. 2.1, common knowledge of ϕ, CK J ϕ, is typically explained in terms of the modal operator of shared knowledge EK J as being the infinite conjunction EK J ϕ ∧ EK J EK J ϕ ∧ EK J EK J EK J ϕ ∧ · · · . However, due to this infiniteness CK J cannot be syntactically reduced to EK J and requires a proper axiomatisation. Most of the axiomatisations in the literature are built on top of the epistemic logic S5. Given that we have stressed the inappropriateness of the latter as a logic of knowledge in Sect. 5.2, the axiomatics we give in Table 8 is based on a non-specified normal modal logic of individual knowledge EL that may be any modal logic between K and S5. It is obtained by adding two axioms to the logic of individual knowledge EL: the Fixpoint Axiom FP(CK J ) and the Greatest Fixpoint Axiom GFP(CK J ). Both involve shared knowledge EK J ϕ, which as we have mentioned in Sect. 2.1 abbreviates i∈J K i ϕ. Axiom GFP(CK J ) can be replaced by the following inference rule: It can be proved that CK J is a normal modal operator. We observe that the formula is a theorem of EL CK if EL is a normal modal logic. It says that if it is common knowledge that ϕ 1 and ϕ 2 have the same truth value and that each agent i knows the truth value of 'his' ϕ i then it is common knowledge that if both are true then everybody knows this. Here is a proof.
doi:10.1007/s10458-014-9267-z fatcat:fa7rshcicratvd2n7zuvfpkyii