When Do Introspection Axioms Matter for Multi-Agent Epistemic Reasoning?

Abstract

The early literature on epistemic logic in philosophy focused on reasoning about the knowledge or belief of a single agent, especially on controversies about "introspection axioms" such as the 4 and 5 axioms. By contrast, the later literature on epistemic logic in computer science and game theory has focused on multi-agent epistemic reasoning, with the single-agent 4 and 5 axioms largely taken for granted. In the relevant multi-agent scenarios, it is often important to reason about what agent A believes about what agent B believes about what agent A believes; but it is rarely important to reason just about what agent A believes about what agent A believes. This raises the question of the extent to which single-agent introspection axioms actually matter for multi-agent epistemic reasoning. In this paper, we formalize and answer this question. To formalize the question, we first define a set of multi-agent formulas that we call agent-alternating formulas, including formulas like 2 a 2 b 2 a p but not formulas like 2 a 2 a p. We then prove, for the case of belief, that if one starts with multi-agent K or KD, then adding both the 4 and 5 axioms (or adding the B axiom) does not allow the derivation of any new agent-alternating formulas-in this sense, introspection axioms do not matter. By contrast, we show that such conservativity results fail for knowledge and multi-agent KT, though they hold with respect to a smaller class of agent-nonrepeating formulas.

Towards Epistemic-Doxastic Planning with Observation and Revision

“…This is due to the restriction that epistemic atoms are repetition-free. The same restriction was exploited under the denomination 'agentalternating formulas' by Ding, Holliday, and Zhang (2019) in order to reduce reasoning with introspection to reasoning without introspection (though on 'knowledge-that' and 'belief-that' operators). Finally, we have shown how our formalism can be used for epistemic planning with the same complexity as classical planning.…”