L. Aceto, A. Ingólfsdóttir, C. Prisacariu, and J. Sack, Compositional Reasoning for Multi-modal Logics, Lecture Notes in Computer Science, vol.7734, pp.1-15, 2013.
DOI : 10.1007/978-3-642-35722-0_1

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.244.9370

R. Alenda, N. Olivetti, and G. L. Pozzato, Nested Sequent Calculi for Conditional Logics, pp.14-27
DOI : 10.2307/2273315

URL : https://hal.archives-ouvertes.fr/hal-01479542

C. Areces, R. Fervari, and G. Hoffmann, Swap logic, Logic Journal of IGPL, vol.22, issue.2, pp.309-332, 2014.
DOI : 10.1093/jigpal/jzt030

C. Areces, R. Fervari, and G. Hoffmann, Relation-changing modal operators: Fig. 1., Logic Journal of IGPL, vol.23, issue.4, pp.601-627, 2015.
DOI : 10.1093/jigpal/jzv020

G. Aucher, A Combined System for Update Logic and Belief Revision, PRIMA, pp.1-17, 2004.
DOI : 10.1007/978-3-540-32128-6_1

URL : https://hal.archives-ouvertes.fr/inria-00537233

G. Aucher, Interpreting an action from what we perceive and what we expect, Journal of Applied Non-Classical Logics, vol.12, issue.4, pp.9-38, 2007.
DOI : 10.3166/jancl.17.9-38

URL : https://hal.archives-ouvertes.fr/inria-00556151

G. Aucher, Perspectives on belief and change, 2008.
URL : https://hal.archives-ouvertes.fr/tel-00556089

G. Aucher, DEL-sequents for progression, Journal of Applied Non-Classical Logics, vol.36, issue.6, pp.289-321, 2011.
DOI : 10.3166/jancl.21.289-321

URL : https://hal.archives-ouvertes.fr/hal-00674150

G. Aucher, DEL-sequents for regression and epistemic planning, Journal of Applied Non-Classical Logics, vol.17, issue.4, pp.337-367, 2012.
DOI : 10.1080/11663081.2012.736703

URL : https://hal.archives-ouvertes.fr/hal-00856471

G. Aucher, DEL as a substructural logic, Outstanding Contributions: Johan F. A. K. van Benthem on Logical and Informational Dynamics Trends in Logic, 2014.

G. Aucher, When conditional logic and belief revision meet substructural logics, Proceedings of the International Workshop on Defeasible and Ampliative Reasoning, DARe 2015, co-located with the 24th International Joint Conference on Artificial Intelligence Buenos Aires volume 1423 of CEUR Workshop Proceedings. CEUR-WS.org, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01193289

G. Aucher, Displaying updates in logic Journal of Logic and Computation, Special Issue on Substructural Logics and Information Dynam- ics(doi:10, p.2016, 1093.

G. Aucher, P. Balbiani, L. F. , D. Cerro, and A. Herzig, Global and Local Graph Modifiers, Methods for Modalities 5 (M4M-5), 2007.
DOI : 10.1016/j.entcs.2009.02.042

URL : https://hal.archives-ouvertes.fr/inria-00556034

G. Aucher and A. Herzig, Exploring the power of converse events. Dynamic Formal Epistemology, pp.51-74, 2011.
URL : https://hal.archives-ouvertes.fr/hal-00557002

G. Aucher, B. Maubert, and F. Schwarzentruber, Generalized DELsequents, pp.54-66
URL : https://hal.archives-ouvertes.fr/hal-00856472

P. Balbiani, R. Echahed, and A. Herzig, A Dynamic Logic for Termgraph Rewriting, Lecture Notes in Computer Science, vol.6372, pp.59-74, 2010.
DOI : 10.1007/978-3-642-15928-2_5

URL : https://hal.archives-ouvertes.fr/hal-00940868

P. Balbiani, A. Herzig, and N. Troquard, Dynamic Logic of Propositional Assignments: A Well-Behaved Variant of PDL, 2013 28th Annual ACM/IEEE Symposium on Logic in Computer Science, pp.143-152, 2013.
DOI : 10.1109/LICS.2013.20

URL : https://hal.archives-ouvertes.fr/hal-01240721

A. Baltag, B. Coecke, and M. Sadrzadeh, Algebra and Sequent Calculus for Epistemic Actions, Electronic Notes in Theoretical Computer Science, vol.126, pp.27-52, 2005.
DOI : 10.1016/j.entcs.2004.11.012

URL : http://doi.org/10.1016/j.entcs.2004.11.012

A. Baltag, B. Coecke, and M. Sadrzadeh, Epistemic Actions as Resources, Journal of Logic and Computation, vol.17, issue.3, pp.555-585, 2007.
DOI : 10.1093/logcom/exm015

URL : http://arxiv.org/abs/math/0608166

A. Baltag and L. S. Moss, Logics for Epistemic Programs, Synthese, vol.139, issue.2, pp.165-224, 2004.
DOI : 10.1023/B:SYNT.0000024912.56773.5e

A. Baltag, L. S. Moss, and S. Solecki, The Logic of Public Announcements, Common Knowledge, and Private Suspicions, pp.43-56, 1998.
DOI : 10.1007/978-3-319-20451-2_38

A. Baltag, L. S. Moss, and S. Solecki, The Logic of Public Announcements, Common Knowledge, and Private Suspicions, 1999.
DOI : 10.1007/978-3-319-20451-2_38

A. Baltag and S. Smets, Conditional Doxastic Models: A Qualitative Approach to Dynamic Belief Revision, Electronic Notes in Theoretical Computer Science, vol.165, pp.5-21, 2006.
DOI : 10.1016/j.entcs.2006.05.034

URL : http://doi.org/10.1016/j.entcs.2006.05.034

A. Baltag and S. Smets, Probabilistic dynamic belief revision, Synthese, vol.24, issue.11, pp.179-202, 2008.
DOI : 10.1007/s11229-008-9369-8

A. Baltag and S. Smets, A Qualitative Theory of Dynamic Interactive Belief Revision, Texts in Logic and Games, pp.9-58, 2008.
DOI : 10.1007/978-3-319-20451-2_39

J. Barwise and J. Perry, Situations and Attitudes, The Journal of Philosophy, vol.78, issue.11, 1983.
DOI : 10.2307/2026578

J. Barwise, Constraints, channels, and the flow of information. Situation theory and its applications, pp.3-27, 1993.

J. Barwise, D. Gabbay, and C. Hartonas, Information flow and the lambek calculus In Logic, Language and Computation: The Forthcoming from CSLI, Moraga Proceedings, pp.49-64, 1994.

J. Barwise, D. Gabbay, and C. Hartonas, On the Logic of Information Flow, Logic Journal of IGPL, vol.3, issue.1, pp.7-49, 1995.
DOI : 10.1093/jigpal/3.1.7

J. Beall, R. Brady, J. Michael-dunn, E. Hazen, . Mares et al., On the Ternary Relation and Conditionality, Journal of Philosophical Logic, vol.49, issue.3, pp.595-612, 2012.
DOI : 10.1007/s10992-011-9191-5

P. Blackburn, Y. Maarten-de-rijke, and . Venema, Modal Logic, volume 53 of Cambridge Tracts in Computer Science, 2001.

A. Bochman, Dynamic causal calculus, KR, 2014.

P. John and . Burgess, Quick completeness proofs for some logics of conditionals, Notre Dame Journal of Formal Logic, vol.22, issue.1, pp.76-84, 1981.

A. Church, Application of recursive arithmetic to the problem of circuit synthesis, Summaries of the Summer Institute of Symbolic Logic, pp.3-50, 1957.

J. , M. Dunn, and G. Restall, Relevance logic. Handbook of philosophical logic, pp.1-128, 2002.
DOI : 10.1007/978-94-017-0460-1_1

R. Fagin, J. Halpern, Y. Moses, and M. Vardi, Reasoning about knowledge, 1995.

R. Fervari, Relation-Changing Modal Logics, 2014.

M. Fitting, Modal proof theory Handbook of Modal Logic, pp.85-138, 2007.

T. French, P. Wiebe-van-der-hoek, B. Iliev, and . Kooi, On the succinctness of some modal logics, Artificial Intelligence, vol.197, issue.0, pp.56-85, 2013.
DOI : 10.1016/j.artint.2013.02.003

URL : https://hal.archives-ouvertes.fr/hal-01273572

S. Frittella, G. Greco, A. Kurz, A. Palmigiano, and V. Sikimi´csikimi´c, Multi-type display calculus for dynamic epistemic logic, Journal of Logic and Computation, vol.26, issue.6, 2014.
DOI : 10.1093/logcom/exu068

URL : https://hal.archives-ouvertes.fr/hal-01509398

S. Frittella, G. Greco, and A. Kurz, Alessandra Palmigiano, and Vlasta Sikimi´cSikimi´c. A proof-theoretic semantic analysis of dynamic epistemic logic, Journal of Logic and Computation, 2014.

M. Dov, C. J. Gabbay, J. A. Hogger, . Robinson, D. Siekmann et al., Handbook of logic in artificial intelligence and logic programming, volume Nonmonotonic reasoning and uncertain reasoning, 1998.

P. Gärdenfors, Knowledge in Flux (Modeling the Dynamics of Epistemic States), 1988.

P. Gärdenfors, Belief revision and nonmonotonic logic: Two sides of the same coin?, pp.52-54, 1991.
DOI : 10.1007/BFb0018432

G. Gentzen, Untersuchungen ???ber das logische Schlie???en. I, Mathematische Zeitschrift, vol.39, issue.1, pp.176-210, 1935.
DOI : 10.1007/BF01201353

L. Giordano, V. Gliozzi, N. Olivetti, G. L. Pozzato, and C. B. Schwind, Non-classical logics for knowledge representation and reasoning, Intelligenza Artificiale, vol.5, issue.1, pp.127-131, 2011.

L. Giordano and C. Schwind, Conditional logic of actions and causation, Artificial Intelligence, vol.157, issue.1-2, pp.239-279, 2004.
DOI : 10.1016/j.artint.2004.04.009

URL : https://hal.archives-ouvertes.fr/halshs-00261167

P. Girard, J. Seligman, and F. Liu, General dynamic dynamic logic, Advances in Modal Logic, pp.239-260, 2012.

J. Halpern and Y. Moses, Knowledge and common knowledge in a distributed environment, Journal of the ACM, vol.37, issue.3, pp.549-587, 1990.
DOI : 10.1145/79147.79161

URL : http://arxiv.org/abs/cs/0006009

J. Hintikka, Knowledge and Belief, An Introduction to the Logic of the Two Notions, 1962.

C. A. Hoare, An axiomatic basis for computer programming, Communications of the ACM, vol.12, issue.10, pp.567-580, 1969.
DOI : 10.1007/978-3-642-59412-0_23

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.116.2392

T. Hoshi, Epistemic dynamics and protocol information, 2009.

B. Kooi, Expressivity and completeness for public update logics via reduction axioms, Journal of Applied Non-Classical Logics, vol.1, issue.2, pp.231-253, 2007.
DOI : 10.1017/CBO9780511569852

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.495.7754

B. Kooi and B. Renne, Arrow update logic. The Review of Symbolic Logic, pp.536-559, 2011.

O. Kupferman and M. Y. Vardi, Abstract, Bulletin of Symbolic Logic, vol.48, issue.02, pp.245-263, 1999.
DOI : 10.1145/357084.357090

N. Kurtonina, Frames and Labels. A Modal Analysis of Categorial Deduction, 1995.

J. Lambek, The Mathematics of Sentence Structure, The American Mathematical Monthly, vol.65, issue.3, pp.154-170, 1958.
DOI : 10.2307/2310058

J. Lambek, On the calculus of syntactic types, Structure of language and its mathematical aspects, pp.166-178, 1961.
DOI : 10.1090/psapm/012/9972

F. Liu, Changing for the Better: Preference Dynamics and Agent Diversity, 2008.

D. Makinson, Bridges from classical to nonmonotonic logic. King's College, 2005.
DOI : 10.1093/jigpal/11.1.69

D. Makinson and P. Gärdenfors, Relations between the logic of theory change and nonmonotonic logic The Logic of Theory Change, Lecture Notes in Computer Science, vol.465, pp.185-205

D. Edwin and . Mares, Relevant logic and the theory of information, Synthese, vol.109, issue.3, pp.345-360, 1996.

D. Edwin, R. K. Mares, and . Meyer, The Blackwell guide to philosophical logic, chapter Relevant Logics, 2001.

M. Martinez and S. Sequoiah-grayson, Logic and information The Stanford Encyclopedia of Philosophy, 2014.

E. Mendelson, Introduction to mathematical logic, 1997.
DOI : 10.1007/978-1-4615-7288-6

M. Moortgat, Categorial type logics, Handbook of logic and language, pp.95-180, 2010.
DOI : 10.1016/b978-044481714-3/50005-9

R. Moot and C. Retoré, A logic for categorial grammars: Lambek's syntactic calculus In The Logic of Categorial Grammars, Lecture Notes in Computer Science, vol.6850, pp.23-63, 2012.
DOI : 10.1007/978-3-642-31555-8

R. Muskens, J. Van-benthem, and A. Visser, Handbook of logic and language , chapter Dynamics, pp.607-670, 2011.

S. Negri and N. Olivetti, A Sequent Calculus for Preferential Conditional Logic Based on Neighbourhood Semantics, Nivelle [73], pp.115-134
DOI : 10.1007/978-3-319-24312-2_9

S. Negri and G. Sbardolini, Proof analysis for Lewis counterfactuals. The Review of Symbolic Logic, pp.1-32

D. Nute, B. Charles, and . Cross, Conditional logic, Handbook of philosophical logic, pp.1-98, 2001.
DOI : 10.1007/978-94-017-0456-4_1

N. Olivetti and G. L. Pozzato, A Standard Internal Calculus for Lewis??? Counterfactual Logics, Nivelle [73], pp.270-286
DOI : 10.1007/978-3-319-24312-2_19

N. Olivetti, G. L. Pozzato, and C. Schwind, A sequent calculus and a theorem prover for standard conditional logics, ACM Transactions on Computational Logic, vol.8, issue.4, 2007.
DOI : 10.1145/1276920.1276924

URL : https://hal.archives-ouvertes.fr/halshs-00261169

H. Ono, Proof-Theoretic Methods in Nonclassical Logic --an Introduction, Theories of Types and Proofs, pp.207-254, 1998.
DOI : 10.2969/msjmemoirs/00201C060

URL : http://projecteuclid.org/download/pdf_1/euclid.msjm/1389985704

J. Perry and D. Israel, What is information? Information, Language, and Cognition, vol.1, 1990.

J. Plaza, Logics of public communcations, Proceedings of the 4th International Symposium on Methodologies for Intelligent Systems, pp.201-216, 1989.

F. Poggiolesi, Gentzen calculi for modal propositional logic, 2010.
DOI : 10.1007/978-90-481-9670-8

URL : https://hal.archives-ouvertes.fr/halshs-00775804

F. Poggiolesi and G. Restall, Interpreting and Applying Proof Theories for Modal Logic, New Waves in Philosophical Logic. Palgrave Macmillan, 2012.
DOI : 10.1057/9781137003720_4

V. Pratt, Semantical considerations on floyd-hoare logic, Proceedings of the 17th IEEE Symposium on the Foundations of Computer Science, pp.109-121, 1976.
DOI : 10.1109/sfcs.1976.27

P. Frank and . Ramsey, General propositions and causality, 1929.

R. Reiter, Knowledge in Action: Logical Foundations for Specifying and Implementing Dynamical Systems, 2001.

G. Restall, Information flow and relevant logics In Logic, Language and Computation: The, Moraga Proceedings. CSLI, pp.463-477, 1994.

G. Restall, An Introduction to Substructural Logics, Routledge, 2000.
DOI : 10.4324/9780203252642

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.183.8564

G. Restall, Relevant and substructural logics. Handbook of the History of Logic, pp.289-398, 2006.
DOI : 10.1016/s1874-5857(06)80030-0

URL : http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.22.6179

R. Routley, K. Robert, and . Meyer, The semantics of entailment?II, Journal of Philosophical Logic, vol.16, issue.1, pp.53-73, 1972.
DOI : 10.1007/BF00649991

R. Routley, K. Robert, and . Meyer, The semantics of entailment ? III, Journal of Philosophical Logic, vol.16, issue.2, pp.192-208, 1972.
DOI : 10.1007/BF00650498

R. Routley, K. Robert, and . Meyer, The Semantics of Entailment, Studies in Logic and the Foundations of Mathematics, pp.199-243, 1973.
DOI : 10.1016/S0049-237X(08)71541-6

R. Routley, V. Plumwood, K. Robert, and . Meyer, Relevant logics and their rivals, 1982.

M. Shanahan, Solving the Frame Problem, 1997.

A. Urquhart, Completeness of weak implication, Theoria, vol.59, issue.3, pp.274-282, 1971.
DOI : 10.1111/j.1755-2567.1971.tb00072.x

A. Urquhart, A general theory of implication, Journal of Symbolic Logic, vol.37, issue.443, p.270, 1972.

A. Urquhart, Semantics for relevant logics, The Journal of Symbolic Logic, vol.32, issue.01, pp.159-169, 1972.
DOI : 10.1007/BF00818100

J. Van-benthem, General dynamics. Theoretical Linguistics, pp.159-202, 1991.

J. Van-benthem, Exploring logical dynamics. CSLI publications Stanford, 1996.

J. Van-benthem, Meaning: the Dynamic Turn, chapter Structural Properties of Dynamic Reasoning, pp.15-31, 2003.

J. Van-benthem, Dynamic logic for belief revision, Journal of Applied Non-Classical Logics, vol.147, issue.11, pp.129-155, 2007.
DOI : 10.1007/BF00248150

J. Van-benthem, Inference in action, pp.3-16, 2007.

J. Van-benthem, Logical dynamics meets logical pluralism? The Australasian Journal of Logic, pp.182-209, 2008.

J. Van-benthem, Modal logic for open minds. CSLI publications, 2010.

J. Van-benthem, Logical Dynamics of Information and Interaction, 2011.
DOI : 10.1017/CBO9780511974533

J. Van-benthem, McCarthy variations in a modal key, Artificial Intelligence, vol.175, issue.1, pp.428-439, 2011.
DOI : 10.1016/j.artint.2010.04.014

J. Van-benthem, J. Gerbrandy, T. Hoshi, and E. Pacuit, Merging frameworks for interaction, Proceedings of the 11th conference on Theoretical aspects of rationality and knowledge , TARK '07, pp.491-526, 2009.
DOI : 10.1145/1324249.1324262

J. Van-benthem, J. Gerbrandy, and B. Kooi, Dynamic Update with Probabilities, Studia Logica, vol.204, issue.11, pp.67-96, 2009.
DOI : 10.1007/s11225-009-9209-y

J. Van-benthem and B. Kooi, Reduction axioms for epistemic actions, AiML-2004: Advances in Modal Logic, number UMCS-04-9-1 in Technical Report Series, pp.197-211, 2004.

R. Van-der-meyden and M. Y. Vardi, Synthesis from knowledge-based specifications, Lecture Notes in Computer Science, vol.1466, pp.34-49, 1998.
DOI : 10.1007/BFb0055614

H. Van and D. , Prolegomena to dynamic logic for belief revision, Synthese, vol.147, pp.229-275, 2005.

. Hans-van-ditmarsch, B. Wiebe-van-der-hoek, and . Kooi, Dynamic Epistemic Logic, volume 337 of Synthese library, 2007.

P. Hans, S. Van-ditmarsch, R. Ghosh, Y. Verbrugge, and . Wang, Hidden protocols: Modifying our expectations in an evolving world, Artif. Intell, vol.208, pp.18-40, 2014.

P. Hans, A. Van-ditmarsch, T. Herzig, and . Lima, From situation calculus to dynamic epistemic logic, Journal of Logic and Computation, vol.21, issue.2, pp.179-204, 2009.

J. Van and E. , Reducing dynamic epistemic logic to PDL by program transformation, 2004.

J. Van-eijck, F. Sietsma, and Y. Wang, Composing models, Journal of Applied Non-Classical Logics, vol.34, issue.2, pp.397-425, 2011.
DOI : 10.3166/jancl.21.397-425

URL : https://hal.archives-ouvertes.fr/inria-00535882

Y. Wang, Epistemic Modelling and Protocol Dynamics, 2010.

H. Wansing, Sequent Calculi for Normal Modal Propositional Logics, Journal of Logic and Computation, vol.4, issue.2, pp.125-142, 1994.
DOI : 10.1093/logcom/4.2.125

@. An and U. *. , C ? )?consistent set of S P * is a set ? of L (P, C ? , P)?structures of S P * such that there are no X 1

L. Del, C ? )?structures of S A * . Then, like in [12], we can prove the same preliminary results and the same Cut Lemma and Lindenbaum Lemma. For the sake of completeness, we recall them below, Similar definitions hold for the