# Public announcement logic with distributed knowledge: expressivity, completeness and complexity

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## Abstract

While dynamic epistemic logics with common knowledge have been extensively studied, dynamic epistemic logics with *distributed* knowledge have so far received far less attention. In this paper we study extensions of public announcement logic (\(\mathcal{PAL }\)) with distributed knowledge, in particular their expressivity, axiomatisations and complexity. \(\mathcal{PAL }\) extended only with distributed knowledge is not more expressive than standard epistemic logic with distributed knowledge. Our focus is therefore on \(\mathcal{PACD }\), the result of adding both common and distributed knowledge to \(\mathcal{PAL }\), which is more expressive than each of its component logics. We introduce an axiomatisation of \(\mathcal{PACD }\), which is not surprising: it is the combination of well-known axioms. The completeness proof, however, is not trivial, and requires novel combinations and extensions of techniques for dealing with \(S5\) knowledge, distributed knowledge, common knowledge and public announcements at the same time. We furthermore show that \(\mathcal{PACD }\) is decidable, more precisely that it is \(\textsc {exptime}\)-complete. This result also carries over to \(\mathcal{S 5\mathcal CD }\) with common and distributed knowledge operators for all coalitions (and not only the grand coalition). Finally, we propose a notion of a trans-bisimulation to generalise certain results and give deeper insight into the proofs.

## Keywords

Public announcement logic Distributed knowledge Expressivity Completeness Decidability Computational complexity Unravelling Folding Trans-bisimulation## Notes

### Acknowledgments

We thank the attendees at LORI-III in Guangzhou, the anonymous LORI reviewers and Synthese reviewers, and Pål Grønås Drange for helpful remarks and suggestions. Yì Wáng gratefully acknowledges funding support from the Major Project of National Social Science Foundation of China (No. 11&ZD088). Thomas Ågothes is also affiliated with Southwest University, China, and Bergen University College, Norway.

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