On the Definability of Simulability and Bisimilarity by Finite Epistemic Models

Abstract

We explore when finite epistemic models are definable up to simulability or bisimulation, either over the basic multi-agent epistemic language $\mathsf L$ or over its extension $\mathsf L^C$ with common knowledge operators. Our negative results are that: simulability is not definable in general in $\mathsf L^C$, and finite epistemic states (i.e., pointed models) are not definable up to bisimulation in $\mathsf L$. Our positive results are that: finite epistemic states are definable up to bisimulation by model validity of $\mathsf L$-formulas, and there is a class of epistemic models we call well-multifounded for which simulability is definable over $\mathsf L$. From our method it also follows that finite epistemic models (i.e., not-pointed models) are definable up to bisimulation using model validity in $\mathsf L$. Our results may prove useful for the logical specification of multi-agent systems, as it provides justification for the ubiquitous but often unjustified claims of the form ‘suppose action a can only be performed in state s’: we show when such preconditions exist. An application are characteristic formulae for interpreted systems. They have a special form wherein factual knowledge, positive knowledge, and ignorance can be separated.