Nothing Special   »   [go: up one dir, main page]

skip to main content
10.1007/978-3-031-43369-6_2guideproceedingsArticle/Chapter ViewAbstractPublication PagesConference Proceedingsacm-pubtype
Article

Recurrence-Driven Summations in Automated Deduction

Published: 20 September 2023 Publication History

Abstract

Many problems in mathematics and computer science involve summations. We present a procedure that automatically proves equations involving finite summations, inspired by the theory of holonomic sequences. The procedure is designed to be interleaved with the activities of a higher-order automatic theorem prover. It performs an induction and automatically solves the induction step, leaving the base cases to the theorem prover.

References

[1]
Abramov, S.A., Bronstein, M., Petkovsek, M., Schneider, C.: On rational and hypergeometric solutions of linear ordinary difference equations in *-field extensions. J. Symb. Comput. 107, 23–66 (2021)
[2]
Barbosa H, Reynolds A, El Ouraoui D, Tinelli C, and Barrett C Fontaine P Extending SMT solvers to higher-order logic Automated Deduction – CADE 27 2019 Cham Springer 35-54
[3]
Bhayat A and Reger G Peltier N and Sofronie-Stokkermans V A combinator-based superposition calculus for higher-order logic Automated Reasoning 2020 Cham Springer 278-296
[4]
Blanchette JC, Kaliszyk C, Paulson LC, and Urban J Hammering towards QED J. Formaliz. Reason. 2016 9 1 101-148
[5]
Bueso J, Gómez-Torrecillas J, and Verschoren A Bueso J, Gómez-Torrecillas J, and Verschoren A Gröbner bases for modules Algorithmic Methods in Non-Commutative Algebra: Applications to Quantum Groups 2003 Dordrecht Springer 169-202
[6]
Chyzak, F., Kauers, M., Salvy, B.: A non-holonomic systems approach to special function identities. In: Johnson, J.R., Park, H., Kaltofen, E. (eds.) Symbolic and Algebraic Computation, International Symposium, ISSAC 2009, Seoul, Republic of Korea, 29–31 July 2009, pp. 111–118. ACM (2009)
[7]
Cox DA, Little J, and O’Shea D Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra 2015 4 Cham Springer
[8]
Fajardo W, Gallego C, Lezama O, Reyes A, Suárez H, and Venegas H Gallego C, Lezama O, Reyes A, Suárez H, and Venegas H Gröbner bases of modules Skew PBW Extensions 2020 Cham Springer 261-286
[9]
Gosper RW Decision procedure for indefinite hypergeometric summation Proc. Natl. Acad. Sci. USA 1978 75 1 40-42
[10]
Kandri-Rody A and Weispfenning V Non-commutative Gröbner bases in algebras of solvable type J. Symb. Comput. 1990 9 1 1-26
[11]
Koutschan C Advanced applications of the holonomic systems approach ACM Comm. Comput. Algebra 2009 43 3/4 119
[12]
Ludwig M and Waldmann U Dershowitz N and Voronkov A An extension of the Knuth-Bendix ordering with LPO-Like properties Logic for Programming, Artificial Intelligence, and Reasoning 2007 Heidelberg Springer 348-362
[13]
Maletzky, A., Immler, F.: Gröbner bases of modules and Faugère’s f algorithm in Isabelle/HOL. CoRR abs/1805.00304 (2018)
[14]
Nummelin, V., Blanchette, J., Dahmen, S.R.: Automated deduction with recurrence-driven summations (technical report). Technical report (2023). https://lean-forward.github.io/pubs/sums_report.pdf
[15]
Schneider C Symbolic summation assists combinatorics Sem. Lothar. Combin. 2007 56 1-36
[16]
Steen A and Benzmüller C Extensional higher-order paramodulation in Leo-III J. Autom. Reason. 2021 65 6 775-807
[17]
Vukmirović P, Bentkamp A, Blanchette J, Cruanes S, Nummelin V, and Tourret S Making higher-order superposition work J. Autom. Reason. 2022 66 4 541-564
[18]
Vukmirović P, Blanchette J, and Schulz S Sankaranarayanan S and Sharygina N Extending a high-performance prover to higher-order logic TACAS 2023 2023 Cham Springer 111-129
[19]
Wilf HS and Zeilberger D Rational functions certify combinatorial identities J. Am. Math. Soc. 1990 3 1 147-158
[20]
Zeilberger D A holonomic systems approach to special functions identities J. Comput. Appl. Math. 1990 32 3 321-368

Recommendations

Comments

Please enable JavaScript to view thecomments powered by Disqus.

Information & Contributors

Information

Published In

cover image Guide Proceedings
Frontiers of Combining Systems: 14th International Symposium, FroCoS 2023, Prague, Czech Republic, September 20–22, 2023, Proceedings
Sep 2023
281 pages
ISBN:978-3-031-43368-9
DOI:10.1007/978-3-031-43369-6
  • Editors:
  • Uli Sattler,
  • Martin Suda
Open Access This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

Publisher

Springer-Verlag

Berlin, Heidelberg

Publication History

Published: 20 September 2023

Qualifiers

  • Article

Contributors

Other Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

  • 0
    Total Citations
  • 0
    Total Downloads
  • Downloads (Last 12 months)0
  • Downloads (Last 6 weeks)0
Reflects downloads up to 26 Sep 2024

Other Metrics

Citations

View Options

View options

Get Access

Login options

Media

Figures

Other

Tables

Share

Share

Share this Publication link

Share on social media