{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,23]],"date-time":"2026-07-23T20:16:20Z","timestamp":1784837780542,"version":"3.55.0"},"publisher-location":"Cham","reference-count":33,"publisher":"Springer International Publishing","isbn-type":[{"value":"9783319415277","type":"print"},{"value":"9783319415284","type":"electronic"}],"license":[{"start":{"date-parts":[[2016,1,1]],"date-time":"2016-01-01T00:00:00Z","timestamp":1451606400000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2016,1,1]],"date-time":"2016-01-01T00:00:00Z","timestamp":1451606400000},"content-version":"vor","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2016]]},"DOI":"10.1007\/978-3-319-41528-4_3","type":"book-chapter","created":{"date-parts":[[2016,7,12]],"date-time":"2016-07-12T09:34:06Z","timestamp":1468316046000},"page":"43-61","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":36,"title":["Synthesizing Probabilistic Invariants via Doob\u2019s Decomposition"],"prefix":"10.1007","author":[{"given":"Gilles","family":"Barthe","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Thomas","family":"Espitau","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Luis Mar\u00eda","family":"Ferrer Fioriti","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Justin","family":"Hsu","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2016,7,13]]},"reference":[{"key":"3_CR1","unstructured":"Allen, J.R., Kennedy, K., Porterfield, C., Warren, J.: Conversion of control dependence to data dependence. In: ACM Symposium on Principles of Programming Languages (POPL), Austin, Texas, pp. 177\u2013189. ACM, New York (1983). ISBN 0-89791-090-7"},{"key":"3_CR2","doi-asserted-by":"crossref","unstructured":"Ammarguellat, Z., Harrison III, W.L.: Automatic recognition of induction variables and recurrence relations by abstract interpretation. In: ACM SIGPLAN Conference on Programming Language Design and Implementation (PLDI), White Plains, New York, pp. 283\u2013295 (1990)","DOI":"10.1145\/93542.93583"},{"key":"3_CR3","unstructured":"Baier, C., Katoen, J.: Principles of Model Checking. MIT Press, Cambridge (2008). ISBN 978-0-262-02649-9"},{"key":"3_CR4","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"71","DOI":"10.1007\/978-3-642-22792-9_5","volume-title":"Advances in Cryptology \u2013 CRYPTO 2011","author":"G Barthe","year":"2011","unstructured":"Barthe, G., Gr\u00e9goire, B., Heraud, S., B\u00e9guelin, S.Z.: Computer-aided security proofs for the working cryptographer. In: Rogaway, P. (ed.) CRYPTO 2011. LNCS, vol. 6841, pp. 71\u201390. Springer, Heidelberg (2011)"},{"key":"3_CR5","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"323","DOI":"10.1007\/978-3-540-32033-3_24","volume-title":"Term Rewriting and Applications","author":"O Bournez","year":"2005","unstructured":"Bournez, O., Garnier, F.: Proving positive almost-sure termination. In: Giesl, J. (ed.) RTA 2005. LNCS, vol. 3467, pp. 323\u2013337. Springer, Heidelberg (2005)"},{"key":"3_CR6","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"511","DOI":"10.1007\/978-3-642-39799-8_34","volume-title":"Computer Aided Verification","author":"A Chakarov","year":"2013","unstructured":"Chakarov, A., Sankaranarayanan, S.: Probabilistic program analysis with martingales. In: Sharygina, N., Veith, H. (eds.) CAV 2013. LNCS, vol. 8044, pp. 511\u2013526. Springer, Heidelberg (2013)"},{"key":"3_CR7","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"crossref","first-page":"85","DOI":"10.1007\/978-3-319-10936-7_6","volume-title":"Static Analysis","author":"A Chakarov","year":"2014","unstructured":"Chakarov, A., Sankaranarayanan, S.: Expectation invariants for probabilistic program loops as fixed points. In: M\u00fcller-Olm, M., Seidl, H. (eds.) Static Analysis. LNCS, vol. 8723, pp. 85\u2013100. Springer, Heidelberg (2014)"},{"key":"3_CR8","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"260","DOI":"10.1007\/978-3-662-49674-9_15","volume-title":"Tools and Algorithms for the Construction and Analysis of Systems","author":"A Chakarov","year":"2016","unstructured":"Chakarov, A., Voronin, Y.-L., Sankaranarayanan, S.: Deductive proofs of almost sure persistence and recurrence properties. In: Chechik, M., Raskin, J.-F. (eds.) TACAS 2016. LNCS, vol. 9636, pp. 260\u2013279. Springer, Heidelberg (2016). doi: 10.1007\/978-3-662-49674-9_15"},{"key":"3_CR9","doi-asserted-by":"crossref","unstructured":"Chatterjee, K., Fu, H., Novotn\u00fd, P., Hasheminezhad, R.: Algorithmic analysis of qualitative and quantitative termination problems for affine probabilistic programs. In: ACM SIGPLAN\u2013SIGACT Symposium on Principles of Programming Languages (POPL), St. Petersburg, Florida, pp. 327\u2013342 (2016)","DOI":"10.1145\/2914770.2837639"},{"key":"3_CR10","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"658","DOI":"10.1007\/978-3-319-21690-4_44","volume-title":"Computer Aided Verification","author":"Y-F Chen","year":"2015","unstructured":"Chen, Y.-F., Hong, C.-D., Wang, B.-Y., Zhang, L.: Counterexample-guided polynomial loop invariant generation by lagrange interpolation. In: Kroening, D., P\u0103s\u0103reanu, C.S. (eds.) CAV 2015. LNCS, vol. 9206, pp. 658\u2013674. Springer, Heidelberg (2015)"},{"key":"3_CR11","doi-asserted-by":"crossref","unstructured":"Ciesinski, F., Baier, C.: LiQuor: a tool for qualitative and quantitative linear time analysis of reactive systems. In: Third International Conference on the Quantitative Evaluation of Systems (QEST), Riverside, California, pp. 131\u2013132 (2006)","DOI":"10.1109\/QEST.2006.25"},{"key":"3_CR12","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"420","DOI":"10.1007\/978-3-540-45069-6_39","volume-title":"Computer Aided Verification","author":"MA Col\u00f3n","year":"2003","unstructured":"Col\u00f3n, M.A., Sankaranarayanan, S., Sipma, H.B.: Linear invariant generation using non-linear constraint solving. In: Hunt Jr., W.A., Somenzi, F. (eds.) CAV 2003. LNCS, vol. 2725, pp. 420\u2013432. Springer, Heidelberg (2003)"},{"key":"3_CR13","doi-asserted-by":"crossref","unstructured":"Cousot, P., Halbwachs, N.: Automatic discovery of linear restraints among variables of a program. In: ACM Symposium on Principles of Programming Languages (POPL), Tucson, Arizona, pp. 84\u201396 (1978)","DOI":"10.1145\/512760.512770"},{"key":"3_CR14","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"337","DOI":"10.1007\/978-3-540-78800-3_24","volume-title":"Tools and Algorithms for the Construction and Analysis of Systems","author":"L de Moura","year":"2008","unstructured":"de Moura, L., Bj\u00f8rner, N.S.: Z3: an efficient SMT solver. In: Ramakrishnan, C.R., Rehof, J. (eds.) TACAS 2008. LNCS, vol. 4963, pp. 337\u2013340. Springer, Heidelberg (2008)"},{"issue":"8","key":"3_CR15","doi-asserted-by":"publisher","first-page":"453","DOI":"10.1145\/360933.360975","volume":"18","author":"EW Dijkstra","year":"1975","unstructured":"Dijkstra, E.W.: Guarded commands, nondeterminacy and formal derivation of programs. Commun. ACM 18(8), 453\u2013457 (1975)","journal-title":"Commun. ACM"},{"key":"3_CR16","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"280","DOI":"10.1007\/978-3-662-49674-9_16","volume-title":"Tools and Algorithms for the Construction and Analysis of Systems","author":"R Dimitrova","year":"2016","unstructured":"Dimitrova, R., Ferrer Fioriti, L.M., Hermanns, H., Majumdar, R.: Probabilistic $${{\\rm {CTL}}^{*}}$$: the deductive way. In: Chechik, M., Raskin, J.-F. (eds.) TACAS 2016. LNCS, vol. 9636, pp. 280\u2013296. Springer, Heidelberg (2016). doi: 10.1007\/978-3-662-49674-9_16"},{"key":"3_CR17","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"123","DOI":"10.1007\/978-3-642-31424-7_14","volume-title":"Computer Aided Verification","author":"J Esparza","year":"2012","unstructured":"Esparza, J., Gaiser, A., Kiefer, S.: Proving termination of probabilistic programs using patterns. In: Madhusudan, P., Seshia, S.A. (eds.) CAV 2012. LNCS, vol. 7358, pp. 123\u2013138. Springer, Heidelberg (2012)"},{"key":"3_CR18","doi-asserted-by":"crossref","unstructured":"Ferrer Fioriti, L.M., Hermanns, H.: Probabilistic termination: soundness, completeness, and compositionality. In: ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages (POPL), Mumbai, India, pp. 489\u2013501 (2015)","DOI":"10.1145\/2775051.2677001"},{"issue":"3","key":"3_CR19","doi-asserted-by":"publisher","first-page":"356","DOI":"10.1145\/2166.357214","volume":"5","author":"S Hart","year":"1983","unstructured":"Hart, S., Sharir, M., Pnueli, A.: Termination of probabilistic concurrent program. ACM Trans. Program. Lang. Syst. 5(3), 356\u2013380 (1983)","journal-title":"ACM Trans. Program. Lang. Syst."},{"issue":"3\u20134","key":"3_CR20","doi-asserted-by":"publisher","first-page":"225","DOI":"10.1145\/2110170.2110185","volume":"45","author":"D Joyner","year":"2012","unstructured":"Joyner, D., \u010cert\u00edk, O., Meurer, A., Granger, B.E.: Open source computer algebra systems: SymPy. ACM Commun. Comput. Algebra 45(3\u20134), 225\u2013234 (2012)","journal-title":"ACM Commun. Comput. Algebra"},{"key":"3_CR21","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"364","DOI":"10.1007\/978-3-662-49498-1_15","volume-title":"Programming Languages and Systems","author":"BL Kaminski","year":"2016","unstructured":"Kaminski, B.L., Katoen, J.-P., Matheja, C., Olmedo, F.: Weakest precondition reasoning for expected run\u2013times of probabilistic programs. In: Thiemann, P. (ed.) ESOP 2016. LNCS, vol. 9632, pp. 364\u2013389. Springer, Heidelberg (2016). doi: 10.1007\/978-3-662-49498-1_15"},{"key":"3_CR22","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"390","DOI":"10.1007\/978-3-642-15769-1_24","volume-title":"Static Analysis","author":"J-P Katoen","year":"2010","unstructured":"Katoen, J.-P., McIver, A.K., Meinicke, L.A., Morgan, C.C.: Linear-invariant generation for probabilistic programs: automated support for proof-based methods. In: Cousot, R., Martel, M. (eds.) SAS 2010. LNCS, vol. 6337, pp. 390\u2013406. Springer, Heidelberg (2010)"},{"issue":"2","key":"3_CR23","doi-asserted-by":"publisher","first-page":"90","DOI":"10.1016\/j.peva.2010.04.001","volume":"68","author":"J Katoen","year":"2011","unstructured":"Katoen, J., Zapreev, I.S., Hahn, E.M., Hermanns, H., Jansen, D.N.: The ins and outs of the probabilistic model checker MRMC. Perform. Eval. 68(2), 90\u2013104 (2011)","journal-title":"Perform. Eval."},{"key":"3_CR24","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"249","DOI":"10.1007\/978-3-540-78800-3_18","volume-title":"Tools and Algorithms for the Construction and Analysis of Systems","author":"L Kov\u00e1cs","year":"2008","unstructured":"Kov\u00e1cs, L.: Reasoning algebraically about P-solvable loops. In: Ramakrishnan, C.R., Rehof, J. (eds.) TACAS 2008. LNCS, vol. 4963, pp. 249\u2013264. Springer, Heidelberg (2008)"},{"issue":"3","key":"3_CR25","doi-asserted-by":"publisher","first-page":"328","DOI":"10.1016\/0022-0000(81)90036-2","volume":"22","author":"D Kozen","year":"1981","unstructured":"Kozen, D.: Semantics of probabilistic programs. J. Comput. Syst. Sci. 22(3), 328\u2013350 (1981)","journal-title":"J. Comput. Syst. Sci."},{"key":"3_CR26","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"585","DOI":"10.1007\/978-3-642-22110-1_47","volume-title":"Computer Aided Verification","author":"M Kwiatkowska","year":"2011","unstructured":"Kwiatkowska, M., Norman, G., Parker, D.: PRISM 4.0: verification of probabilistic real-time systems. In: Gopalakrishnan, G., Qadeer, S. (eds.) CAV 2011. LNCS, vol. 6806, pp. 585\u2013591. Springer, Heidelberg (2011)"},{"key":"3_CR27","series-title":"Monographs in Computer Science","volume-title":"Abstraction, Refinement and Proof for Probabilistic Systems","author":"A McIver","year":"2005","unstructured":"McIver, A., Morgan, C.: Abstraction, Refinement and Proof for Probabilistic Systems. Monographs in Computer Science. Springer, New York (2005)"},{"issue":"1","key":"3_CR28","doi-asserted-by":"publisher","first-page":"31","DOI":"10.1007\/s10990-006-8609-1","volume":"19","author":"A Min\u00e9","year":"2006","unstructured":"Min\u00e9, A.: The octagon abstract domain. High.-Order Symb. Comput. 19(1), 31\u2013100 (2006)","journal-title":"High.-Order Symb. Comput."},{"key":"3_CR29","series-title":"Lecture Notes in Computer Science","doi-asserted-by":"publisher","first-page":"111","DOI":"10.1007\/3-540-47764-0_7","volume-title":"Static Analysis","author":"D Monniaux","year":"2001","unstructured":"Monniaux, D.: An abstract analysis of the probabilistic termination of programs. In: Cousot, P. (ed.) SAS 2001. LNCS, vol. 2126, pp. 111\u2013126. Springer, Heidelberg (2001)"},{"key":"3_CR30","doi-asserted-by":"crossref","unstructured":"Motwani, R., Raghavan, P.: Randomized Algorithms. Cambridge University Press, Cambridge (1995). ISBN 0-521-47465-5","DOI":"10.1017\/CBO9780511814075"},{"issue":"2","key":"3_CR31","doi-asserted-by":"publisher","first-page":"107","DOI":"10.1016\/0167-6423(87)90029-3","volume":"9","author":"HR Nielson","year":"1987","unstructured":"Nielson, H.R.: A hoare-like proof system for analysing the computation time of programs. Sci. Comput. Program. 9(2), 107\u2013136 (1987)","journal-title":"Sci. Comput. Program."},{"issue":"4","key":"3_CR32","doi-asserted-by":"publisher","first-page":"443","DOI":"10.1016\/j.jsc.2007.01.002","volume":"42","author":"E Rodr\u00edguez-Carbonell","year":"2007","unstructured":"Rodr\u00edguez-Carbonell, E., Kapur, D.: Generating all polynomial invariants in simple loops. J. Symb. Comput. 42(4), 443\u2013476 (2007)","journal-title":"J. Symb. Comput."},{"key":"3_CR33","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511813658","volume-title":"Probability with Martingales","author":"D Williams","year":"1991","unstructured":"Williams, D.: Probability with Martingales. Cambridge University Press, Cambridge (1991)"}],"container-title":["Lecture Notes in Computer Science","Computer Aided Verification"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/978-3-319-41528-4_3","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,6,4]],"date-time":"2025-06-04T00:41:22Z","timestamp":1748997682000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/978-3-319-41528-4_3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016]]},"ISBN":["9783319415277","9783319415284"],"references-count":33,"URL":"https:\/\/doi.org\/10.1007\/978-3-319-41528-4_3","relation":{},"ISSN":["0302-9743","1611-3349"],"issn-type":[{"value":"0302-9743","type":"print"},{"value":"1611-3349","type":"electronic"}],"subject":[],"published":{"date-parts":[[2016]]},"assertion":[{"value":"13 July 2016","order":1,"name":"first_online","label":"First Online","group":{"name":"ChapterHistory","label":"Chapter History"}},{"value":"CAV","order":1,"name":"conference_acronym","label":"Conference Acronym","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"International Conference on Computer Aided Verification","order":2,"name":"conference_name","label":"Conference Name","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"Toronto","order":3,"name":"conference_city","label":"Conference City","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"Canada","order":4,"name":"conference_country","label":"Conference Country","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"2016","order":5,"name":"conference_year","label":"Conference Year","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"17 July 2016","order":7,"name":"conference_start_date","label":"Conference Start Date","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"23 July 2016","order":8,"name":"conference_end_date","label":"Conference End Date","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"28","order":9,"name":"conference_number","label":"Conference Number","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"cav2016","order":10,"name":"conference_id","label":"Conference ID","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"This content has been made available to all.","name":"free","label":"Free to read"}]}}