{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,18]],"date-time":"2026-04-18T10:48:30Z","timestamp":1776509310639,"version":"3.51.2"},"publisher-location":"Cham","reference-count":40,"publisher":"Springer Nature Switzerland","isbn-type":[{"value":"9783032070203","type":"print"},{"value":"9783032070210","type":"electronic"}],"license":[{"start":{"date-parts":[[2026,1,1]],"date-time":"2026-01-01T00:00:00Z","timestamp":1767225600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2026,1,1]],"date-time":"2026-01-01T00:00:00Z","timestamp":1767225600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2026]]},"DOI":"10.1007\/978-3-032-07021-0_7","type":"book-chapter","created":{"date-parts":[[2025,10,7]],"date-time":"2025-10-07T13:40:36Z","timestamp":1759844436000},"page":"113-132","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Graded Quantitative Narrowing"],"prefix":"10.1007","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0089-3905","authenticated-orcid":false,"given":"Mauricio","family":"Ayala-Rinc\u00f3n","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0852-9086","authenticated-orcid":false,"given":"Thaynara Arielly","family":"de Lima","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0009-0003-5931-9673","authenticated-orcid":false,"given":"Georg","family":"Ehling","sequence":"additional","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4084-7380","authenticated-orcid":false,"given":"Temur","family":"Kutsia","sequence":"additional","affiliation":[]}],"member":"297","reference":[{"key":"7_CR1","doi-asserted-by":"publisher","unstructured":"Ahrens, E., Kassing, J., Giesl, J., Katoen, J.: Weighted rewriting: Semiring semantics for abstract reduction systems. In: Fern\u00e1ndez, M. (ed.) 10th International Conference on Formal Structures for Computation and Deduction, FSCD 2025, July 14-20, 2025, Birmingham, UK. LIPIcs, vol.\u00a0337, pp. 6:1\u20136:21. Schloss Dagstuhl - Leibniz-Zentrum f\u00fcr Informatik (2025). https:\/\/doi.org\/10.4230\/LIPICS.FSCD.2025.6","DOI":"10.4230\/LIPICS.FSCD.2025.6"},{"key":"7_CR2","doi-asserted-by":"publisher","unstructured":"A\u00eft-Kaci, H., Pasi, G.: Fuzzy lattice operations on first-order terms over signatures with similar constructors: A constraint-based approach. Fuzzy Sets Syst. 391, 1\u201346 (2020). https:\/\/doi.org\/10.1016\/J.FSS.2019.03.019","DOI":"10.1016\/J.FSS.2019.03.019"},{"key":"7_CR3","doi-asserted-by":"publisher","unstructured":"Alpuente, M., Escobar, S., Iborra, J.: Termination of narrowing revisited. Theor. Comput. Sci. 410(46), 4608\u20134625 (2009). https:\/\/doi.org\/10.1016\/J.TCS.2009.07.037","DOI":"10.1016\/J.TCS.2009.07.037"},{"key":"7_CR4","doi-asserted-by":"publisher","unstructured":"Alpuente, M., Escobar, S., Iborra, J.: Modular termination of basic narrowing and equational unification. Log. J. IGPL 19(6), 731\u2013762 (2011). https:\/\/doi.org\/10.1093\/JIGPAL\/JZQ009","DOI":"10.1093\/JIGPAL\/JZQ009"},{"key":"7_CR5","doi-asserted-by":"publisher","unstructured":"Antoy, S., Hanus, M.: Functional logic programming. Commun. ACM 53(4), 74\u201385 (2010). https:\/\/doi.org\/10.1145\/1721654.1721675","DOI":"10.1145\/1721654.1721675"},{"key":"7_CR6","doi-asserted-by":"crossref","unstructured":"Avanzini, M., Yamada, A.: Weighted rewriting (2025), https:\/\/akihisayamada.github.io\/AY2025.pdf, to appear in FroCoS 2025.","DOI":"10.1007\/978-3-032-04167-8_11"},{"key":"7_CR7","doi-asserted-by":"publisher","unstructured":"Ayala-Rinc\u00f3n, M., de\u00a0Carvalho\u00a0Segundo, W., Fern\u00e1ndez, M., Silva, G.F., Nantes-Sobrinho, D.: Formalising nominal C-unification generalised with protected variables. Math. Struct. Comput. Sci. 31(3), 286\u2013311 (2021). https:\/\/doi.org\/10.1017\/S0960129521000050","DOI":"10.1017\/S0960129521000050"},{"key":"7_CR8","doi-asserted-by":"publisher","unstructured":"Ayala-Rinc\u00f3n, M., Fern\u00e1ndez, M., Silva, G.F., Kutsia, T., Nantes-Sobrinho, D.: Nominal AC-Matching. In: Dubois, C., Kerber, M. (eds.) Intelligent Computer Mathematics - 16th International Conference, CICM 2023, Cambridge, UK, September 5-8, 2023, Proceedings. Lecture Notes in Computer Science, vol. 14101, pp. 53\u201368. Springer (2023). https:\/\/doi.org\/10.1007\/978-3-031-42753-4_4","DOI":"10.1007\/978-3-031-42753-4_4"},{"key":"7_CR9","doi-asserted-by":"publisher","unstructured":"Ayala-Rinc\u00f3n, M., Fern\u00e1ndez, M., Silva, G.F., Kutsia, T., Nantes-Sobrinho, D.: Certified First-Order AC-Unification and Applications. J. Autom. Reason. 68(4), \u00a025 (2024). https:\/\/doi.org\/10.1007\/S10817-024-09714-5","DOI":"10.1007\/S10817-024-09714-5"},{"key":"7_CR10","doi-asserted-by":"publisher","unstructured":"Ayala-Rinc\u00f3n, M., de\u00a0Lima, T.A., Ehling, G., Kutsia, T.: Graded quantitative narrowing. CoRR abs\/2507.19630 (2025). https:\/\/doi.org\/10.48550\/ARXIV.2507.19630","DOI":"10.48550\/ARXIV.2507.19630"},{"key":"7_CR11","doi-asserted-by":"publisher","unstructured":"Baader, F., Nipkow, T.: Term rewriting and all that. Cambridge University Press (1998). https:\/\/doi.org\/10.1017\/CBO9781139172752","DOI":"10.1017\/CBO9781139172752"},{"key":"7_CR12","doi-asserted-by":"publisher","unstructured":"Baader, F., Snyder, W.: Chapter 8 - Unification Theory. In: Robinson, A., Voronkov, A. (eds.) Handbook of Automated Reasoning, p. 445\u2013533. North-Holland, Amsterdam (2001). https:\/\/doi.org\/10.1016\/B978-044450813-3\/50010-2","DOI":"10.1016\/B978-044450813-3\/50010-2"},{"key":"7_CR13","doi-asserted-by":"publisher","unstructured":"Bacci, G., Mardare, R., Panangaden, P., Plotkin, G.D.: Quantitative Equational Reasoning. In: Barthe, G., Katoen, J.P., Silva, A. (eds.) Foundations of Probabilistic Programming, p. 333\u2013360. Cambridge University Press (2020). https:\/\/doi.org\/10.1017\/9781108770750.011","DOI":"10.1017\/9781108770750.011"},{"key":"7_CR14","doi-asserted-by":"publisher","unstructured":"Bae, K., Escobar, S., Meseguer, J.: Abstract logical model checking of infinite-state systems using narrowing. In: van Raamsdonk, F. (ed.) 24th International Conference on Rewriting Techniques and Applications, RTA 2013, June 24-26, 2013, Eindhoven, The Netherlands. LIPIcs, vol.\u00a021, pp. 81\u201396. Schloss Dagstuhl - Leibniz-Zentrum f\u00fcr Informatik (2013). https:\/\/doi.org\/10.4230\/LIPICS.RTA.2013.81","DOI":"10.4230\/LIPICS.RTA.2013.81"},{"key":"7_CR15","doi-asserted-by":"publisher","unstructured":"Cholewa, A., Escobar, S., Meseguer, J.: Constrained narrowing for conditional equational theories modulo axioms. Sci. Comput. Program. 112, 24\u201357 (2015). https:\/\/doi.org\/10.1016\/J.SCICO.2015.06.001","DOI":"10.1016\/J.SCICO.2015.06.001"},{"key":"7_CR16","doi-asserted-by":"publisher","unstructured":"Clavel, M., Dur\u00e1n, F., Eker, S., Escobar, S., Lincoln, P., Mart\u00ed-Oliet, N., Talcott, C.L.: Two decades of Maude. In: Mart\u00ed-Oliet, N., \u00d6lveczky, P.C., Talcott, C.L. (eds.) Logic, Rewriting, and Concurrency - Essays dedicated to Jos\u00e9 Meseguer on the Occasion of His 65th Birthday. Lecture Notes in Computer Science, vol.\u00a09200, pp. 232\u2013254. Springer (2015). https:\/\/doi.org\/10.1007\/978-3-319-23165-5_11","DOI":"10.1007\/978-3-319-23165-5_11"},{"key":"7_CR17","unstructured":"Dershowitz, N., Sivakumar, G.: Goal-directed equation solving. In: Shrobe, H.E., Mitchell, T.M., Smith, R.G. (eds.) Proceedings of the 7th National Conference on Artificial Intelligence, St. Paul, MN, USA, August 21-26, 1988. pp. 166\u2013170. AAAI Press \/ The MIT Press (1988)"},{"key":"7_CR18","doi-asserted-by":"publisher","unstructured":"Dundua, B., Kutsia, T., Marin, M., Pau, C.: Constraint solving over multiple similarity relations. In: Ariola, Z.M. (ed.) 5th International Conference on Formal Structures for Computation and Deduction, FSCD 2020, June 29-July 6, 2020, Paris, France (Virtual Conference). LIPIcs, vol.\u00a0167, pp. 30:1\u201330:19. Schloss Dagstuhl - Leibniz-Zentrum f\u00fcr Informatik (2020). https:\/\/doi.org\/10.4230\/LIPICS.FSCD.2020.30","DOI":"10.4230\/LIPICS.FSCD.2020.30"},{"key":"7_CR19","doi-asserted-by":"publisher","unstructured":"Ehling, G., Kutsia, T.: Solving quantitative equations. In: Benzm\u00fcller, C., Heule, M.J.H., Schmidt, R.A. (eds.) Automated Reasoning - 12th International Joint Conference, IJCAR 2024, Nancy, France, July 3-6, 2024, Proceedings, Part II. Lecture Notes in Computer Science, vol. 14740, pp. 381\u2013400. Springer (2024). https:\/\/doi.org\/10.1007\/978-3-031-63501-4_20","DOI":"10.1007\/978-3-031-63501-4_20"},{"key":"7_CR20","doi-asserted-by":"publisher","unstructured":"Escobar, S., Meseguer, J., Thati, P.: Narrowing and Rewriting Logic: from foundations to applications. In: L\u00f3pez-Fraguas, F.J. (ed.) Proceedings of the 15th Workshop on Functional and (Constraint) Logic Programming, WFLP 2006, Madrid, Spain, November 16-17, 2006. Electronic Notes in Theoretical Computer Science, vol.\u00a0177, pp. 5\u201333. Elsevier (2006). https:\/\/doi.org\/10.1016\/J.ENTCS.2007.01.004","DOI":"10.1016\/J.ENTCS.2007.01.004"},{"key":"7_CR21","doi-asserted-by":"publisher","unstructured":"Escobar, S., Sasse, R., Meseguer, J.: Folding variant narrowing and optimal variant termination. J. Log. Algebraic Methods Program. 81(7-8), 898\u2013928 (2012). https:\/\/doi.org\/10.1016\/J.JLAP.2012.01.002","DOI":"10.1016\/J.JLAP.2012.01.002"},{"key":"7_CR22","unstructured":"Fay, M.: First-order unification in an equational theory. In: 4th Workshop on Automated Deduction, Austin, TX. pp. 161\u2013167 (1979)"},{"key":"7_CR23","doi-asserted-by":"publisher","unstructured":"Formato, F., Gerla, G., Sessa, M.I.: Similarity-based unification. Fundam. Informaticae 41(4), 393\u2013414 (2000). https:\/\/doi.org\/10.3233\/FI-2000-41402","DOI":"10.3233\/FI-2000-41402"},{"key":"7_CR24","doi-asserted-by":"publisher","unstructured":"Gavazzo, F., Di Florio, C.: Quantitative and Metric Rewriting: Abstract, Non-Expansive, and Graded Systems. CoRR abs\/2206.13610 (2022). https:\/\/doi.org\/10.48550\/ARXIV.2206.13610","DOI":"10.48550\/ARXIV.2206.13610"},{"key":"7_CR25","doi-asserted-by":"publisher","unstructured":"Gavazzo, F., Di Florio, C.: Elements of quantitative rewriting. Proc. ACM Program. Lang. 7(POPL), 1832\u20131863 (2023). https:\/\/doi.org\/10.1145\/3571256","DOI":"10.1145\/3571256"},{"key":"7_CR26","unstructured":"Hanus, M.: Curry: An integrated functional logic language (vers. 0.9.0). Available at http:\/\/www.curry-lang.org (2016)"},{"key":"7_CR27","doi-asserted-by":"publisher","unstructured":"Hullot, J.: Canonical Forms and Unification. In: Bibel, W., Kowalski, R.A. (eds.) 5th Conference on Automated Deduction, Les Arcs, France, July 8-11, 1980, Proceedings. Lecture Notes in Computer Science, vol.\u00a087, p. 318\u2013334. Springer (1980). https:\/\/doi.org\/10.1007\/3-540-10009-1_25","DOI":"10.1007\/3-540-10009-1_25"},{"key":"7_CR28","doi-asserted-by":"publisher","unstructured":"Juli\u00e1n-Iranzo, P., Rubio-Manzano, C.: Proximity-based unification theory. Fuzzy Sets and Systems 262, 21\u201343 (2015). https:\/\/doi.org\/10.1016\/j.fss.2014.07.006","DOI":"10.1016\/j.fss.2014.07.006"},{"key":"7_CR29","doi-asserted-by":"publisher","unstructured":"Lawvere, F.W.: Metric spaces, generalized logic, and closed categories. Rendiconti del Seminario Matematico e Fisico di Milano 43, 135\u2013166 (1973). https:\/\/doi.org\/10.1007\/BF02924844","DOI":"10.1007\/BF02924844"},{"key":"7_CR30","doi-asserted-by":"publisher","unstructured":"L\u00f3pez-Rueda, R., Escobar, S., Sapi\u00f1a, J.: An efficient canonical narrowing implementation with irreducibility and SMT constraints for generic symbolic protocol analysis. J. Log. Algebraic Methods Program. 135, 100895 (2023). https:\/\/doi.org\/10.1016\/J.JLAMP.2023.100895","DOI":"10.1016\/J.JLAMP.2023.100895"},{"key":"7_CR31","doi-asserted-by":"publisher","unstructured":"Mardare, R., Panangaden, P., Plotkin, G.D.: Quantitative algebraic reasoning. In: Proceedings of the 31st Annual ACM\/IEEE Symposium on Logic in Computer Science. p. 700\u2013709. LICS \u201916, Association for Computing Machinery, New York, NY, USA (2016). https:\/\/doi.org\/10.1145\/2933575.2934518","DOI":"10.1145\/2933575.2934518"},{"key":"7_CR32","doi-asserted-by":"publisher","unstructured":"Mardare, R., Panangaden, P., Plotkin, G.D.: On the axiomatizability of quantitative algebras. In: 32nd Annual ACM\/IEEE Symposium on Logic in Computer Science, LICS 2017, Reykjavik, Iceland, June 20-23, 2017. pp. 1\u201312. IEEE Computer Society (2017). https:\/\/doi.org\/10.1109\/LICS.2017.8005102","DOI":"10.1109\/LICS.2017.8005102"},{"key":"7_CR33","doi-asserted-by":"publisher","unstructured":"Middeldorp, A., Hamoen, E.: Counterexamples to completeness results for basic narrowing (extended abstract). In: Kirchner, H., Levi, G. (eds.) Algebraic and Logic Programming, Third International Conference, Volterra, Italy, September 2-4, 1992, Proceedings. Lecture Notes in Computer Science, vol.\u00a0632, pp. 244\u2013258. Springer (1992). https:\/\/doi.org\/10.1007\/BFB0013830","DOI":"10.1007\/BFB0013830"},{"key":"7_CR34","doi-asserted-by":"publisher","unstructured":"Middeldorp, A., Hamoen, E.: Completeness results for basic narrowing. Appl. Algebra Eng. Commun. Comput. 5, 213\u2013253 (1994). https:\/\/doi.org\/10.1007\/BF01190830","DOI":"10.1007\/BF01190830"},{"key":"7_CR35","doi-asserted-by":"publisher","unstructured":"Middeldorp, A., Hamoen, E.: Completeness Results for Basic Narrowing. Appl. Algebra Eng. Commun. Comput. 5, 213\u2013253 (1994). https:\/\/doi.org\/10.1007\/BF01190830","DOI":"10.1007\/BF01190830"},{"key":"7_CR36","unstructured":"Pau, C.: Symbolic Techniques for Approximate Reasoning. Ph.D. thesis, RISC, Johannes Kepler University Linz (2022), https:\/\/epub.jku.at\/obvulihs\/content\/titleinfo\/7818355"},{"key":"7_CR37","doi-asserted-by":"publisher","unstructured":"Pau, C., Kutsia, T.: Proximity-Based Unification and Matching for Fully Fuzzy Signatures. In: 30th IEEE International Conference on Fuzzy Systems, FUZZ-IEEE 2021, Luxembourg, July 11-14, 2021. p. 1\u20136. IEEE (2021). https:\/\/doi.org\/10.1109\/FUZZ45933.2021.9494438","DOI":"10.1109\/FUZZ45933.2021.9494438"},{"key":"7_CR38","doi-asserted-by":"publisher","unstructured":"Sessa, M.I.: Approximate reasoning by similarity-based SLD resolution. Theor. Comput. Sci. 275(1-2), 389\u2013426 (2002). https:\/\/doi.org\/10.1016\/S0304-3975(01)00188-8","DOI":"10.1016\/S0304-3975(01)00188-8"},{"key":"7_CR39","unstructured":"Terese: Term rewriting systems, Cambridge tracts in theoretical computer science, vol.\u00a055. Cambridge University Press (2003)"},{"key":"7_CR40","doi-asserted-by":"publisher","unstructured":"Tran, D.D., Do, C.M., Escobar, S., Ogata, K.: Hybrid post-quantum transport layer security formal analysis in Maude-NPA and its parallel version. PeerJ Comput. Sci. 9, e1556 (2023). https:\/\/doi.org\/10.7717\/PEERJ-CS.1556","DOI":"10.7717\/PEERJ-CS.1556"}],"container-title":["Lecture Notes in Computer Science","Intelligent Computer Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/978-3-032-07021-0_7","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,18]],"date-time":"2026-04-18T10:22:35Z","timestamp":1776507755000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/978-3-032-07021-0_7"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026]]},"ISBN":["9783032070203","9783032070210"],"references-count":40,"URL":"https:\/\/doi.org\/10.1007\/978-3-032-07021-0_7","relation":{},"ISSN":["0302-9743","1611-3349"],"issn-type":[{"value":"0302-9743","type":"print"},{"value":"1611-3349","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026]]},"assertion":[{"value":"The authors have no competing interests to declare that are relevant to the content of this article.","order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Disclosure of Interests"}},{"value":"CICM","order":1,"name":"conference_acronym","label":"Conference Acronym","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"International Conference on Intelligent Computer Mathematics","order":2,"name":"conference_name","label":"Conference Name","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"Brasilia","order":3,"name":"conference_city","label":"Conference City","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"Brazil","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":"6 October 2025","order":7,"name":"conference_start_date","label":"Conference Start Date","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"11 October 2025","order":8,"name":"conference_end_date","label":"Conference End Date","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"18","order":9,"name":"conference_number","label":"Conference Number","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"mkm2025","order":10,"name":"conference_id","label":"Conference ID","group":{"name":"ConferenceInfo","label":"Conference Information"}},{"value":"https:\/\/cicm-conference.org\/2025\/cicm.php","order":11,"name":"conference_url","label":"Conference URL","group":{"name":"ConferenceInfo","label":"Conference Information"}}]}}