{"type":"ScholarlyArticle","title":"Primal Grammars Driven Automated Induction","authors":[{"name":"Adel Bouhoula","slug":"adel-bouhoula","given":"Adel","family":"Bouhoula","affiliation":"Arabian Gulf University , Bahrain"},{"name":"Miki Hermann","slug":"miki-hermann","given":"Miki","family":"Hermann","affiliation":"LIX , CNRS ,; École Polytechnique , Institut Polytechnique de Paris , France"}],"abstract":"Automated induction is a powerful method for the validation of\r\n  critical systems. However, the inductive proof process faces major\r\n  challenges: it is undecidable and diverges even with small examples.\r\n\r\n  Previous methods have proposed ad-hoc heuristics to speculate on\r\n  additional lemmas that hopefully stop the divergence. Although these\r\n  methods have succeeded in proving interesting theorems, they have\r\n  significant limitations: in particular, they often fail to find\r\n  appropriate lemmas, and the lemmas they provide may not be valid.\r\n\r\n  We present a new method that allows us to perform inductive proofs\r\n  in conditional theories. This method automatically detects\r\n  divergence in proof traces and derives primal grammars as well as\r\n  new lemmas that schematize the divergent sequence. This new\r\n  construction allows us to break the divergence and complete the\r\n  proof.\r\n\r\n  Our method is presented as a set of inference rules whose soundness\r\n  and refutational completeness have been formally proved. Unlike\r\n  previous methods, our method is fully automated and has no risk of\r\n  over-generalization. Moreover, our technique for capturing and\r\n  schematizing divergence represents the most general decidable\r\n  schematization, with respect to description power, among all known\r\n  schematizations.\r\n\r\n  Our method has been implemented in C++ and successfully proved\r\n  over fifty complex examples that fail with well-known theorem\r\n  provers (e.g., ACL2, Isabelle, PVS, SPIKE) and related methods for\r\n  handling divergence in proofs by induction.\r\n\r\n  Our method represents a significant contribution to the field of\r\n  automated reasoning as it can be integrated with existing automated\r\n  and interactive inductive proof systems to enhance their\r\n  performance. Moreover, it has the potential to substantially reduce\r\n  the time needed for the verification of critical systems.","doi":"10.24963/ijcai.2024/361","url":"https://ijcai.org/proceedings/2024/361","pdf":"https://ijcai.org/proceedings/2024/0361.pdf","markdown":"https://ijcai.org/proceedings/2024/361.md","bibtex":"https://ijcai.org/proceedings/2024/bibtex/361","volume":{"slug":"2024","title":"Proceedings of the Thirty-Third International Joint Conference on Artificial Intelligence","short_title":"IJCAI 2024","url":"https://ijcai.org/proceedings/2024/","year":2024,"location":"Jeju, Korea","dates":"3-9 August 2024","editors":"Kate Larson","isbn":"978-1-956792-04-1","publisher":"International Joint Conferences on Artificial Intelligence Organization","doi_prefix":"10.24963/ijcai.2024"},"section":"Knowledge Representation and Reasoning","track":"Main Track","number":361,"pages":{"first":3259,"last":3269},"keywords":["Knowledge Representation and Reasoning: KRR: Automated reasoning and theorem proving"],"videos":[],"references":[{"position":1,"raw":"[Aubin, 1976] Raymond Aubin. Mechanizing structural in- duction. PhD thesis, University of Edinburgh, UK, 1976.","title":"Mechanizing structural induction","authors":"Raymond Aubin","year":"1976","venue":null,"doi":null,"url":null,"internal_url":null},{"position":2,"raw":"[Baader and Nipkow, 1998] Franz Baader and Tobias Nip- kow. Term rewriting and all that. Cambridge University Press, 1998.","title":"Term rewriting and all that","authors":"Nipkow Baader, Franz Baader, Tobias Nipkow","year":"1998","venue":null,"doi":null,"url":null,"internal_url":null},{"position":3,"raw":"[Basin and Walsh, 1992] David A. Basin and Toby Walsh. Difference matching. In D. Kapur, editor, Proceedings 11th International Conference on Automated Deduction (CADE 1992), Saratoga Springs (New York, USA), volume 607 of Lecture Notes in Computer Science, pages 295- 309. Springer, June 1992.","title":"Difference matching","authors":"David A Basin, Toby Walsh","year":"1992","venue":"Proceedings 11th International Conference on Automated Deduction (CADE 1992)","doi":null,"url":null,"internal_url":null},{"position":4,"raw":"[Basin and Walsh, 1993] David A. Basin and Toby Walsh. Difference unification. In Ružena Bajcsy, editor, Proceed- ings 13th International Joint Conference on Artificial In- telligence (IJCAI 1993), Chambéry (France), pages 116- 122.","title":"Difference unification","authors":"David A Basin, Toby Walsh","year":null,"venue":"Proceedings 13th International Joint Conference on Artificial Intelligence (IJCAI 1993)","doi":null,"url":null,"internal_url":null},{"position":5,"raw":"Morgan Kaufmann, 1993.","title":null,"authors":"Morgan Kaufmann","year":"1993","venue":null,"doi":null,"url":null,"internal_url":null},{"position":6,"raw":"[Bouhoula and Jouannaud, 1997] Adel Bouhoula and Jean- Pierre Jouannaud. Automata-driven automated induction. In Proceedings 12th Annual IEEE Symposium on Logic in Computer Science (LICS 1997), Warsaw (Poland), pages 14-25. IEEE Computer Society, 1997.","title":"Automata-driven automated induction","authors":"Jouannaud Bouhoula, Adel Bouhoula, Jean-Pierre Jouannaud","year":"1997","venue":"Proceedings 12th Annual IEEE Symposium on Logic in Computer Science (LICS 1997)","doi":null,"url":null,"internal_url":null},{"position":7,"raw":"[Bouhoula and Jouannaud, 2001] Adel Bouhoula and Jean- Pierre Jouannaud. Automata-driven automated induction. Information and Computation, 169(1):1-22, 2001. A pre- liminary version was published in [Bouhoula and Jouan- naud, 1997].","title":"Automata-driven automated induction","authors":"Jouannaud Bouhoula, Adel Bouhoula, Jean-Pierre Jouannaud","year":"2001","venue":"Information and Computation","doi":null,"url":null,"internal_url":null},{"position":8,"raw":"[Bouhoula and Rusinowitch, 1993] Adel Bouhoula and Michaël Rusinowitch. Automatic case analysis in proof by induction. In Ružena Bajcsy, editor, Proceedings 13th International Joint Conference on Artificial Intelligence (IJCAI 1993), Chambéry (France), pages 88--94.","title":"Adel Bouhoula and Michaël Rusinowitch. Automatic case analysis in proof by induction","authors":"Rusinowitch Bouhoula","year":"1993","venue":"Proceedings 13th International Joint Conference on Artificial Intelligence (IJCAI 1993)","doi":null,"url":null,"internal_url":null},{"position":9,"raw":"Morgan Kaufmann, 1993.","title":null,"authors":"Morgan Kaufmann","year":"1993","venue":null,"doi":null,"url":null,"internal_url":null},{"position":10,"raw":"[Bouhoula et al., 1995] Adel Bouhoula, Emmanuel Kounalis, and Michaël Rusinowitch. Automated math- ematical induction. Journal of Logic and Computation, 5(5):631-668, 1995.","title":"Automated mathematical induction","authors":"Adel Bouhoula, Emmanuel Kounalis, Michaël Rusinowitch","year":"1995","venue":"Journal of Logic and Computation","doi":null,"url":null,"internal_url":null},{"position":11,"raw":"[Bouhoula et al., 2000] Adel Bouhoula, Jean-Pierre Jouan- naud, and José Meseguer. Specification and proof in mem- bership equational logic. Theoretical Computer Science, 236(1-2):35-132, 2000.","title":"Specification and proof in membership equational logic","authors":"Adel Bouhoula, Jean-Pierre Jouannaud, José Meseguer","year":"2000","venue":"Theoretical Computer Science","doi":null,"url":null,"internal_url":null},{"position":12,"raw":"[Bouhoula, 1997] Adel Bouhoula. Automated theorem prov- ing by test set induction. Journal of Symbolic Computa- tion, 23(1):47-77, 1997.","title":"Automated theorem proving by test set induction","authors":"Adel Bouhoula","year":"1997","venue":"Journal of Symbolic Computation","doi":null,"url":null,"internal_url":null},{"position":13,"raw":"[Bouhoula, 2000] Adel Bouhoula. Simultaneous checking of completeness and ground confluence. In Proceedings 15th IEEE International Conference on Automated Soft- ware Engineering, (ASE 2000), Grenoble (France), pages 143-151. IEEE Computer Society, 2000.","title":"Simultaneous checking of completeness and ground confluence","authors":"Adel Bouhoula","year":"2000","venue":"Proceedings 15th IEEE International Conference on Automated Software Engineering","doi":null,"url":null,"internal_url":null},{"position":14,"raw":"[Bouhoula, 2009] Adel Bouhoula. Simultaneous checking of completeness and ground confluence for algebraic spec- ifications. ACM Transactions on Computational Logic, 10(3):20:1-20:33, 2009. A preliminary version was pub- lished in [Bouhoula, 2000].","title":"Simultaneous checking of completeness and ground confluence for algebraic specifications","authors":"Adel Bouhoula","year":"2009","venue":"ACM Transactions on Computational Logic","doi":null,"url":null,"internal_url":null},{"position":15,"raw":"[Boyer and Moore, 1979] Robert S. Boyer and J Strother Moore. A computational logic handbook, volume 23 of Perspectives in computing. Academic Press, 1979.","title":"A computational logic handbook","authors":"Robert S Boyer, J Strother, Moore","year":"1979","venue":"Perspectives in computing","doi":null,"url":null,"internal_url":null},{"position":16,"raw":"[Bundy et al., 1993] Alan Bundy, Andrew Stevens, Frank van Harmelen, Andrew Ireland, and Alan Smaill. Rip- pling: A heuristic for guiding inductive proofs. Artificial Intelligence, 62(2):185-253, 1993.","title":"Rippling: A heuristic for guiding inductive proofs","authors":"Alan Bundy, Andrew Stevens, Frank Van Harmelen, Andrew Ireland, Alan Smaill","year":"1993","venue":"Artificial Intelligence","doi":null,"url":null,"internal_url":null},{"position":17,"raw":"[Chen et al., 1990] Hong Chen, Jieh Hsiang, and Hwa- Chung Kong. On finite representations of infinite se- quences of terms. In Stéphane Kaplan and Mitsuhiro Okada, editors, Proceedings 2nd International Work- shop on Conditional and Typed Rewriting Systems (CTRS 1990), Montreal (Canada), volume 516 of Lecture Notes in Computer Science, pages 100-114. Springer, 1990.","title":"On finite representations of infinite sequences of terms","authors":"Chen","year":"1990","venue":"Proceedings 2nd International Workshop on Conditional and Typed Rewriting Systems (CTRS 1990)","doi":null,"url":null,"internal_url":null},{"position":18,"raw":"[Comon, 1995] Hubert Comon. On unification of terms with integer exponents. Mathematical Systems Theory, 28(1):67-88, 1995.","title":"On unification of terms with integer exponents","authors":"Hubert Comon","year":"1995","venue":"Mathematical Systems Theory","doi":null,"url":null,"internal_url":null},{"position":19,"raw":"[de Moura and Ullrich, 2021] Leonardo de Moura and Se- bastian Ullrich. The Lean 4 theorem prover and program- ming language. In André Platzer and Geoff Sutcliffe, ed- itors, Proceedings 28th International Conference on Au- tomated Deduction (CADE 2021), Virtual Event, volume 12699 of Lecture Notes in Computer Science, pages 625- 635. Springer, 2021.","title":"Leonardo de Moura and Sebastian Ullrich. The Lean 4 theorem prover and programming language","authors":"De Moura, Ullrich","year":"2021","venue":"Proceedings 28th International Conference on Automated Deduction (CADE 2021), Virtual Event","doi":null,"url":null,"internal_url":null},{"position":20,"raw":"[Dershowitz and Jouannaud, 1991] Nachum Dershowitz and Jean-Pierre Jouannaud. Notations for rewting. Bulletin of EATCS, 43:162-174, 1991.","title":"Nachum Dershowitz and Jean-Pierre Jouannaud. Notations for rewting","authors":"Jouannaud Dershowitz","year":"1991","venue":"Bulletin of EATCS","doi":null,"url":null,"internal_url":null},{"position":21,"raw":"[Hajdú et al., 2020] Márton Hajdú, Petra Hozzová, Laura Kovács, Johannes Schoisswohl, and Andrei Voronkov. In- duction with generalization in superposition reasoning. In Christoph Benzmüller and Bruce R. Miller, editors, Proceedings 13th International Conference on Intelligent Computer Mathematics (CICM 2020), Bertinoro (Italy), volume 12236 of Lecture Notes in Computer Science, pages 123-137. Springer, 2020.","title":"Induction with generalization in superposition reasoning","authors":"Márton Hajdú, Petra Hozzová, Laura Kovács, Johannes Schoisswohl, Andrei Voronkov","year":"2020","venue":"Proceedings 13th International Conference on Intelligent Computer Mathematics (CICM 2020)","doi":null,"url":null,"internal_url":null},{"position":22,"raw":"[Hermann and Galbavý, 1997] Miki Hermann and Roman Galbavý. Unification of infinite sets of terms schema- tized by primal grammars. Theoretical Computer Science, 176(1-2):111-158, 1997.","title":"Unification of infinite sets of terms schematized by primal grammars","authors":"Galbavý Hermann, Miki Hermann, Roman Galbavý","year":"1997","venue":"Theoretical Computer Science","doi":null,"url":null,"internal_url":null},{"position":23,"raw":"[Hozzová et al., 2021] Petra Hozzová, Laura Kovács, and Andrei Voronkov. Integer induction in saturation. In André Platzer and Geoff Sutcliffe, editors, Proceedings 28th In- ternational Conference on Automated Deduction (CADE 2021), Virtual Event, volume 12699 of Lecture Notes in Computer Science, pages 361-377. Springer, 2021.","title":"Integer induction in saturation","authors":"Petra Hozzová, Laura Kovács, Andrei Voronkov","year":"2021","venue":"Proceedings 28th International Conference on Automated Deduction (CADE 2021), Virtual Event","doi":null,"url":null,"internal_url":null},{"position":24,"raw":"[Ireland and Bundy, 1996] Andrew Ireland and Alan Bundy. Extensions to a generalization critic for inductive proof. In Michael A. McRobbie and John K. Slaney, editors, Pro- ceedings 13th International Conference on Automated De- duction (CADE-13), New Brunswick (New Jersey, USA), volume 1104 of Lecture Notes in Computer Science, pages 47-61. Springer, 1996.","title":"Extensions to a generalization critic for inductive proof","authors":"Bundy Ireland, Andrew Ireland, Alan Bundy","year":"1996","venue":"Proceedings 13th International Conference on Automated Deduction (CADE-13)","doi":null,"url":null,"internal_url":null},{"position":25,"raw":"[Kapur and Subramaniam, 1996] Deepak Kapur and Ma- hadevan Subramaniam. Lemma discovery in automated induction. In Michael A. McRobbie and John K. Slaney, editors, Proceedings 13th International Conference on Au- tomated Deduction (CADE-13), New Brunswick (New Jer- sey, USA), volume 1104 of Lecture Notes in Computer Sci- ence, pages 538-552. Springer, 1996.","title":"Lemma discovery in automated induction","authors":"Subramaniam ; Hadevan Kapur, Subramaniam","year":"1996","venue":"Proceedings 13th International Conference on Automated Deduction (CADE-13)","doi":null,"url":null,"internal_url":null},{"position":26,"raw":"[Kapur and Zhang, 1995] Deepak Kapur and Hantao Zhang. An overview of rewrite rule laboratory (RRL). Computers & Mathematics with Applications, 29(2):91-114, 1995.","title":"An overview of rewrite rule laboratory (RRL)","authors":"Zhang Kapur, Deepak Kapur, Hantao Zhang","year":"1995","venue":"Computers & Mathematics with Applications","doi":null,"url":null,"internal_url":null},{"position":27,"raw":"[Kaufmann et al., 2000] Matt Kaufmann, Panagiotis Mano- lios, and J. Strother Moore, editors. Computer-Aided Rea- soning: ACL2 Case Studies. Kluwer Academic Publishers, June 2000.","title":"Computer-Aided Reasoning: ACL2 Case Studies","authors":null,"year":"2000","venue":null,"doi":null,"url":null,"internal_url":null},{"position":28,"raw":"[Nipkow et al., 2002] Tobias Nipkow, Lawrence C. Paulson, and Markus Wenzel. Isabelle/HOL -A Proof Assistant for Higher-Order Logic, volume 2283 of Lecture Notes in Computer Science. Springer, 2002.","title":"Isabelle/HOL -A Proof Assistant for Higher-Order Logic","authors":"Nipkow","year":"2002","venue":"Lecture Notes in Computer Science","doi":null,"url":null,"internal_url":null},{"position":29,"raw":"[Rushby et al., 1998] John M. Rushby, Sam Owre, and Natarajan Shankar. Subtypes for specifications: Predicate subtyping in PVS. IEEE Transactions on Software Engi- neering, 24(9):709-720, 1998.","title":"Subtypes for specifications: Predicate subtyping in PVS","authors":"John M Rushby, Sam Owre, Natarajan Shankar","year":"1998","venue":"IEEE Transactions on Software Engineering","doi":null,"url":null,"internal_url":null},{"position":30,"raw":"[Salzer, 1992] Gernot Salzer. The unification of infinite sets of terms and its applications. In Andrei Voronkov, editor, Proceedings 3rd International Conference on Logic Pro- gramming and Automated Reasoning (LPAR 1992), St. Pe- tersburg (Russia), volume 624 of Lecture Notes in Com- puter Science, pages 409-420. Springer, 1992.","title":"The unification of infinite sets of terms and its applications","authors":"Gernot Salzer","year":"1992","venue":"Proceedings 3rd International Conference on Logic Programming and Automated Reasoning (LPAR 1992), St. Petersburg","doi":null,"url":null,"internal_url":null},{"position":31,"raw":"[Urso and Kounalis, 2004] Pascal Urso and Emmanuel Kounalis. Sound generalizations in mathematical induc- tion. Theoretical Computer Science, 323(1-3):443-471, 2004.","title":"Pascal Urso and Emmanuel Kounalis. Sound generalizations in mathematical induction","authors":"Kounalis Urso","year":"2004","venue":"Theoretical Computer Science","doi":null,"url":null,"internal_url":null},{"position":32,"raw":"[Walsh, 1996] Toby Walsh. Divergence critic for inductive proofs. Journal of Artificial Intelligence Research, 4:209- 235, 1996.","title":"Divergence critic for inductive proofs","authors":"Toby Walsh","year":"1996","venue":"Journal of Artificial Intelligence Research","doi":null,"url":null,"internal_url":null}],"license_note":"Copyright International Joint Conferences on Artificial Intelligence."}