{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T07:24:36Z","timestamp":1760081076170,"version":"3.44.0"},"publisher-location":"Cham","reference-count":21,"publisher":"Springer Nature Switzerland","isbn-type":[{"value":"9783032060846","type":"print"},{"value":"9783032060853","type":"electronic"}],"license":[{"start":{"date-parts":[[2025,9,25]],"date-time":"2025-09-25T00:00:00Z","timestamp":1758758400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2025,9,25]],"date-time":"2025-09-25T00:00:00Z","timestamp":1758758400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2026]]},"abstract":"<jats:title>Abstract<\/jats:title>\n          <jats:p>Converse <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$\\textsf{PDL}$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>PDL<\/mml:mi>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> is the extension of propositional dynamic logic with a converse operation on programs. Our main result states that Converse <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$\\textsf{PDL}$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>PDL<\/mml:mi>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> enjoys the (local) Craig Interpolation Property, with respect to both atomic programs and propositional variables. As a corollary we establish the Beth Definability Property for the logic.<\/jats:p>\n          <jats:p>Our interpolation proof is based on an adaptation of Maehara\u2019s proof-theoretic method. For this purpose we introduce a sound and complete cyclic sequent system for this logic. This calculus features an analytic cut rule and uses a focus mechanism for recognising successful cycles.<\/jats:p>","DOI":"10.1007\/978-3-032-06085-3_14","type":"book-chapter","created":{"date-parts":[[2025,9,27]],"date-time":"2025-09-27T10:45:10Z","timestamp":1758969910000},"page":"258-277","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Interpolation for\u00a0Converse PDL"],"prefix":"10.1007","author":[{"given":"Johannes","family":"Kloibhofer","sequence":"first","affiliation":[]},{"given":"Valentina","family":"Trucco Dalmas","sequence":"additional","affiliation":[]},{"given":"Yde","family":"Venema","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2025,9,25]]},"reference":[{"key":"14_CR1","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"197","DOI":"10.1007\/978-3-030-98479-3_10","volume-title":"Language, Logic, and Computation","author":"B Afshari","year":"2022","unstructured":"Afshari, B., Leigh, G.E.: Lyndon interpolation for\u00a0modal $$\\mu $$-calculus. In: \u00d6zg\u00fcn, A., Zinova, Y. (eds.) TbiLLC 2019. LNCS, vol. 13206, pp. 197\u2013213. Springer, Cham (2022). https:\/\/doi.org\/10.1007\/978-3-030-98479-3_10"},{"key":"14_CR2","doi-asserted-by":"publisher","unstructured":"Baader, F., Lutz, C.: Description logics. In: Blackburn, P., van Benthem, J., Wolter, F. (eds.) Handbook of Modal Logic, Studies in logic and practical reasoning, vol.\u00a03, pp. 757\u2013820. North-Holland (2007). https:\/\/doi.org\/10.1016\/S1570-2464(07)80015-2","DOI":"10.1016\/S1570-2464(07)80015-2"},{"issue":"3","key":"14_CR3","doi-asserted-by":"publisher","first-page":"402","DOI":"10.1016\/0022-0000(82)90018-6","volume":"25","author":"M Ben-Ari","year":"1982","unstructured":"Ben-Ari, M., Halpern, J.Y., Pnueli, A.: Deterministic propositional dynamic logic: finite models, complexity, and completeness. J. Comput. Syst. Sci. 25(3), 402\u2013417 (1982). https:\/\/doi.org\/10.1016\/0022-0000(82)90018-6","journal-title":"J. Comput. Syst. Sci."},{"key":"14_CR4","unstructured":"Borzechowski, M.: Tableau-Kalk\u00fcl f\u00fcr PDL und Interpolation. Master\u2019s thesis, Department of Mathematics, Freie Universit\u00e4t Berlin (1988)"},{"key":"14_CR5","unstructured":"Borzechowski, M., Gattinger, M., Hansen, H.H., Ramanayake, R., Dalmas, V.T., Venema, Y.: Propositional dynamic logic has Craig interpolation: a tableau-based proof (2025). https:\/\/arxiv.org\/abs\/2503.13276"},{"key":"14_CR6","doi-asserted-by":"publisher","first-page":"347","DOI":"10.1613\/JAIR.4057","volume":"48","author":"B Cate","year":"2013","unstructured":"Cate, B., Franconi, E., Seylan, I.: Beth definability in expressive description logics. J. Artif. Intell. Res. 48, 347\u2013414 (2013). https:\/\/doi.org\/10.1613\/JAIR.4057","journal-title":"J. Artif. Intell. Res."},{"issue":"1","key":"14_CR7","doi-asserted-by":"publisher","first-page":"132","DOI":"10.1137\/S0097539793304741","volume":"29","author":"E Emerson","year":"1999","unstructured":"Emerson, E., Jutla, C.: The complexity of tree automata and logics of programs. SIAM J. Comput. 29(1), 132\u2013158 (1999)","journal-title":"SIAM J. Comput."},{"issue":"2","key":"14_CR8","doi-asserted-by":"publisher","first-page":"194","DOI":"10.1016\/0022-0000(79)90046-1","volume":"18","author":"MJ Fischer","year":"1979","unstructured":"Fischer, M.J., Ladner, R.E.: Propositional dynamic logic of regular programs. J. Comput. Syst. Sci. 18(2), 194\u2013211 (1979). https:\/\/doi.org\/10.1016\/0022-0000(79)90046-1","journal-title":"J. Comput. Syst. Sci."},{"key":"14_CR9","doi-asserted-by":"crossref","unstructured":"Gabbay, D.M., Maksimova, L.: Interpolation and Definability: Modal and Intuitionistic Logics. Oxford University Press (2005)","DOI":"10.1093\/acprof:oso\/9780198511748.001.0001"},{"key":"14_CR10","doi-asserted-by":"publisher","unstructured":"Jung, J.C., Lutz, C., Pulcini, H., Wolter, F.: Separating data examples by description logic concepts with restricted signatures. In: Proceedings of the 18th International Conference on Principles of Knowledge Representation and Reasoning, KR (2021). https:\/\/doi.org\/10.24963\/KR.2021\/37","DOI":"10.24963\/KR.2021\/37"},{"key":"14_CR11","unstructured":"Kloibhofer, J., Venema, Y.: Interpolation for the two-way modal $$\\mu $$-calculus. In: 40th Annual ACM\/IEEE Symposium on Logic in Computer Science (LICS 2025) (2025). to appear"},{"key":"14_CR12","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"413","DOI":"10.1007\/11538363_29","volume-title":"Computer Science Logic","author":"C Lutz","year":"2005","unstructured":"Lutz, C.: PDL with intersection and converse is decidable. In: Ong, L. (ed.) CSL 2005. LNCS, vol. 3634, pp. 413\u2013427. Springer, Heidelberg (2005). https:\/\/doi.org\/10.1007\/11538363_29"},{"key":"14_CR13","unstructured":"Marti, J., Venema, Y.: Focus-style proof systems and interpolation for the alternation-free $$\\mu $$-calculus. CoRR abs\/2103.01671 (2021). https:\/\/arxiv.org\/abs\/2103.01671"},{"key":"14_CR14","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/978-3-540-31980-1_1","volume-title":"Tools and Algorithms for the Construction and Analysis of Systems","author":"KL McMillan","year":"2005","unstructured":"McMillan, K.L.: Applications of craig interpolants in model checking. In: Halbwachs, N., Zuck, L.D. (eds.) TACAS 2005. LNCS, vol. 3440, pp. 1\u201312. Springer, Heidelberg (2005). https:\/\/doi.org\/10.1007\/978-3-540-31980-1_1"},{"key":"14_CR15","unstructured":"Mostowski, A.: Games with forbidden positions (1991), technical Report\u00a078, Instytut Matematyki, Uniwersytet Gda\u0144ski, Poland"},{"key":"14_CR16","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"403","DOI":"10.1007\/3-540-08921-7_88","volume-title":"Mathematical Foundations of Computer Science 1978","author":"R Parikh","year":"1978","unstructured":"Parikh, R.: The completeness of propositional dynamic logic. In: Winkowski, J. (ed.) MFCS 1978. LNCS, vol. 64, pp. 403\u2013415. Springer, Heidelberg (1978). https:\/\/doi.org\/10.1007\/3-540-08921-7_88"},{"key":"14_CR17","doi-asserted-by":"publisher","first-page":"231","DOI":"10.1016\/0022-0000(80)90061-6","volume":"20","author":"V Pratt","year":"1980","unstructured":"Pratt, V.: A near-optimal method for reasoning about action. J. Comput. Syst. Sci. 20, 231\u2013254 (1980). https:\/\/doi.org\/10.1016\/0022-0000(80)90061-6","journal-title":"J. Comput. Syst. Sci."},{"key":"14_CR18","doi-asserted-by":"publisher","unstructured":"Renardel de Lavalette, G.R.: Interpolation in computing science: the semantics of modularization. Synthese 164(3), 437\u2013450 (2008). https:\/\/doi.org\/10.1007\/S11229-008-9358-Y","DOI":"10.1007\/S11229-008-9358-Y"},{"key":"14_CR19","unstructured":"Segerberg, K.: A completeness theorem in the modal logic of programs. Not. Am. Math. Soc. 24, 522 (1977). http:\/\/eudml.org\/doc\/209235"},{"issue":"3\u20134","key":"14_CR20","doi-asserted-by":"publisher","first-page":"575","DOI":"10.1134\/s0001434614090326","volume":"96","author":"DS Shamkanov","year":"2014","unstructured":"Shamkanov, D.S.: Circular proofs for the G\u00f6del-L\u00f6b provability logic. Math. Notes 96(3\u20134), 575\u2013585 (2014). https:\/\/doi.org\/10.1134\/s0001434614090326","journal-title":"Math. Notes"},{"key":"14_CR21","unstructured":"Troquard, N., Balbiani, P.: Propositional dynamic logic. In: Zalta, E.N., Nodelman, U. (eds.) The Stanford Encyclopedia of Philosophy. Metaphysics Research Lab, Stanford University, Fall 2023 edn. (2023). https:\/\/plato.stanford.edu\/archives\/fall2023\/entries\/logic-dynamic\/"}],"container-title":["Lecture Notes in Computer Science","Automated Reasoning with Analytic Tableaux and Related Methods"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/978-3-032-06085-3_14","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,9,27]],"date-time":"2025-09-27T10:45:12Z","timestamp":1758969912000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/978-3-032-06085-3_14"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,9,25]]},"ISBN":["9783032060846","9783032060853"],"references-count":21,"URL":"https:\/\/doi.org\/10.1007\/978-3-032-06085-3_14","relation":{},"ISSN":["0302-9743","1611-3349"],"issn-type":[{"value":"0302-9743","type":"print"},{"value":"1611-3349","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,9,25]]},"assertion":[{"value":"25 September 2025","order":1,"name":"first_online","label":"First Online","group":{"name":"ChapterHistory","label":"Chapter History"}},{"value":"TABLEAUX","order":1,"name":"conference_acronym","label":"Conference Acronym","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"International Conference on Automated Reasoning with Analytic Tableaux and Related Methods","order":2,"name":"conference_name","label":"Conference Name","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"Reykjavik","order":3,"name":"conference_city","label":"Conference City","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"Iceland","order":4,"name":"conference_country","label":"Conference Country","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"2025","order":5,"name":"conference_year","label":"Conference Year","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"27 September 2025","order":7,"name":"conference_start_date","label":"Conference Start Date","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"29 September 2025","order":8,"name":"conference_end_date","label":"Conference End Date","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"34","order":9,"name":"conference_number","label":"Conference Number","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"tableaux2025","order":10,"name":"conference_id","label":"Conference ID","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"https:\/\/icetcs.github.io\/frocos-itp-tableaux25\/tableaux\/","order":11,"name":"conference_url","label":"Conference URL","group":{"name":"ConferenceInfo","label":"Conference Information"}}]}}