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View all- Ciobâcă ŞLucanu DBuruiană A(2023)Operationally-based Program Equivalence Proofs using LCTRSsJournal of Logical and Algebraic Methods in Programming10.1016/j.jlamp.2023.100894(100894)Online publication date: Jul-2023
Proving inductive validity of constrained inequalities
Pages 50 - 61
Abstract
Rewriting induction (RI) frameworks consist of inference rules to prove equations to be inductive theorems of a given term rewriting system, i.e., to be inductively valid w.r.t. reduction of the given system. To prove inductive validity of inequalities within such frameworks, one may reduce inequalities to equations. However, it is often hard to prove inductive validity of such reduced equations within the existing RI frameworks due to their indirect handling of inequalities. In this paper, we adapt the notion of inductive theorems to inequalities and propose an RI framework for directly proving inductive validity of inequalities of constrained term rewriting systems. Within the framework, we handle inequalities that may contain function symbols defined in a given rewriting system but not necessarily interpreted by the built-in semantics. Direct handling of inequalities facilitates the utilization of transitivity of magnitude relations via inequalities obtained as induction hypotheses. Our approach succeeds in proving inductive validity of some inequalities that are hard to verify within the existing RI frameworks for equations.
References
[1]
T. Aoto. Soundness of rewriting induction based on an abstract principle. IPSJ Transactions on Programming, 49(SIG 1 (PRO 35)):28--38, 2008.
[2]
T. Arts and J. Giesl. Termination of term rewriting using dependency pairs. Theoretical Computer Science, 236(1-2):133--178, 2000.
[3]
F. Baader and T. Nipkow. Term Rewriting and All That. Cambridge University Press, 1998.
[4]
A. Bouhoula. Automated theorem proving by test set induction. Journal of Symbolic Computation, 23(1):47--77, 1997.
[5]
A. Bouhoula and F. Jacquemard. Automated induction with constrained tree automata. In A. Armando, P. Baumgartner, and G. Dowek, editors, Proceedings of the 4th International Joint Conference on Automated Reasoning, volume 5195 of Lecture Notes in Computer Science, pages 539--554. Springer, 2008.
[6]
A. Bouhoula and F. Jacquemard. Sufficient completeness verification for conditional and constrained term rewriting systems. Journal of Applied Logic, 10(1):127--143, 2012.
[7]
A. Bouhoula and M. Rusinowitch. Implicit induction in conditional theories. Journal of Automated Reasoning, 14(2):189--235, 1995.
[8]
R. S. Boyer and J. S. Moore. A Computational Logic. Academic Press, 1979.
[9]
A. Bundy. The automation of proof by mathematical induction. In Handbook of Automated Reasoning, volume I, pages 845--911. Elsevier Science, 2001.
[10]
A. Bundy, F. van Harmelen, A. Smaill, and A. Ireland. Extensions to the rippling-out tactic for guiding inductive proofs. In M. E. Stickel, editor, Proceedings of the 10th International Conference on Automated Deduction, volume 449 of Lecture Notes in Artificial Intelligence, pages 132--146. Springer, 1990.
[11]
S. Falke and D. Kapur. Dependency pairs for rewriting with built-in numbers and semantic data structures. In A. Voronkov, editor, Proceedings of the 19th International Conference on Rewriting Techniques and Applications, volume 5117 of Lecture Notes in Computer Science, pages 94--109. Springer, 2008.
[12]
S. Falke and D. Kapur. Rewriting induction + linear arithmetic = decision procedure. In B. Gramlich, D. Miller, and U. Sattler, editors, Proceedings of the 6th International Joint Conference on Automated Reasoning, volume 7364 of Lecture Notes in Computer Science, pages 241--255. Springer, 2012.
[13]
Y. Furuichi, N. Nishida, M. Sakai, K. Kusakari, and T. Sakabe. Approach to procedural-program verification based on implicit induction of constrained term rewriting systems. IPSJ Transactions on Programming, 1(2):100--121, 2008. In Japanese.3
[14]
G. Huet and J.-M. Hullot. Proof by induction in equational theories with constructors. Journal of Computer and System Science, 25(2):239--266, 1982.
[15]
M. Huth and M. Ryan. Logic in Computer Science: Modeling and Reasoning about Systems. Cambridge University Press, 2000.
[16]
H. Koike and Y. Toyama. Inductionless induction and rewriting induction. Computer Software, 17(6):1--12, 2000. In Japanese.
[17]
C. Kop. Termination of LCTRSs. In Proceedings of the 13th International Workshop on Termination, pages 59--63, 2013.
[18]
C. Kop and N. Nishida. Term rewriting with logical constraints. In P. Fontaine, C. Ringeissen, and R. A. Schmidt, editors, Proceedings of the 9th International Symposium on Frontiers of Combining Systems, volume 8152 of Lecture Notes in Artificial Intelligence, pages 343--358. Springer, 2013.
[19]
C. Kop and N. Nishida. Automatic constrained rewriting induction towards verifying procedural programs. In Proceedings of the 12th Asian Symposium on Programming Languages and Systems, volume 8858 of Lecture Notes in Computer Science, pages 334--353. Springer, 2014.
[20]
D. R. Musser. On proving inductive properties of abstract data types. In Proceedings of 7th ACM Symposium on Principles of Programming Languages, pages 154--162, 1980.
[21]
N. Nakabayashi, N. Nishida, K. Kusakari, T. Sakabe, and M. Sakai. Lemma generation method in rewriting induction for constrained term rewriting systems. Computer Software, 28(1):173--189, 2010. In Japanese.3
[22]
N. Nishida, F. Nomura, K. Kurahashi, and M. Sakai. Constrained tree automata and their closure properties. In Proceedings of the 1st International Workshop on Trends in Tree Automata and Tree Transducers, pages 24--34, 2012.
[23]
N. Nishida, M. Sakai, and Y. Nakano. On constructing constrained tree automata recognizing ground instances of constrained terms. In Proceedings of the 2nd International Workshop on Trends in Tree Automata and Tree Transducers, volume 134 of Electronic Proceedings in Theoretical Computer Science, pages 1--10. 2013.
[24]
U. S. Reddy. Term rewriting induction. In Proceedings of the 10th International Conference on Automated Deduction, volume 449 of Lecture Notes in Computer Science, pages 162--177. Springer, 1990.
[25]
T. Sakata, N. Nishida, and T. Sakabe. On proving termination of constrained term rewrite systems by eliminating edges from dependency graphs. In H. Kuchen, editor, Proceedings of the 20th International Workshop on Functional and Constraint Logic Programming, volume 6816 of Lecture Notes in Computer Science, pages 138--155. Springer, 2011.
[26]
T. Sakata, N. Nishida, T. Sakabe, M. Sakai, and K. Kusakari. Rewriting induction for constrained term rewriting systems. IPSJ Transactions on Programming, 2(2):80--96, 2009. In Japanese.3
Index Terms
- Proving inductive validity of constrained inequalities
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Published: 05 September 2016
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View all- Ciobâcă ŞLucanu DBuruiană A(2023)Operationally-based Program Equivalence Proofs using LCTRSsJournal of Logical and Algebraic Methods in Programming10.1016/j.jlamp.2023.100894(100894)Online publication date: Jul-2023
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