A lightweight epistemic logic and its application to planning

Abstract

We study multiagent epistemic planning with a simple epistemic logic whose language is a restriction of that of standard epistemic logic. Its formulas are boolean combinations of observability atoms: sequences of ‘knowing whether’ operators followed by propositional variables. This compares favourably with other restricted languages where formulas are boolean combinations of epistemic literals: sequences of ‘knowing that’ epistemic operators and negations followed by propositional variables; or in other terms: epistemic formulas without conjunctions or disjunctions. The reason is that our language enables a richer theory of mind: we can express statements such as “I don't know whether p, but I know that you know whether p” which are important in communication and more generally in interaction and which cannot be expressed with epistemic literals. Going beyond previous work, we also introduce a ‘common knowledge whether’ operator. We show that satisfiability is nevertheless NP-complete. We then define simple epistemic planning tasks as generalisations of classical planning tasks: action descriptions have sets of observability atoms as add- and delete-lists, initial states are sets of observability atoms, and goals are boolean combinations of observability atoms. We show that simple epistemic planning tasks can be polynomially translated into classical planning tasks. It follows that checking solvability of simple epistemic planning tasks is PSpace-complete. We present some application examples such as the gossip problem and some experimental results and clarify the relationship with Dynamic Epistemic Logic-based planning.