Abstract
The European Erasmus+ project ARC – Automated Reasoning in the Class aims at improving the academic education in disciplines related to Computational Logic by using Automated Reasoning tools. We present the technical aspects of the tools as well as our education experiments, which took place mostly in virtual lectures due to the COVID pandemics. Our education goals are: to support the virtual interaction between teacher and students in the absence of the blackboard, to explain the basic Computational Logic algorithms, to study their implementation in certain programming environments, to reveal the main relationships between logic and programming, and to develop the proof skills of the students. For the introductory lectures we use some programs in C and in Mathematica in order to illustrate normal forms, resolution, and DPLL (Davis-Putnam-Logemann-Loveland) with its Chaff version, as well as an implementation of sequent calculus in the Theorema system. Furthermore we developed special tools for SAT (propositional satisfiability), some based on the original methods from the partners, including complex tools for SMT (Satisfiability Modulo Theories) that allow the illustration of various solving approaches. An SMT related approach is natural-style proving in Elementary Analysis, for which we developed and interesting set of practical heuristics. For more advanced lectures on rewrite systems we use the Coq programming and proving environment, in order on one hand to demonstrate programming in functional style and on the other hand to prove properties of programs. Other advanced approaches used in some lectures are the deduction based synthesis of algorithms and the techniques for program transformation.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Ábrahám, E., Nalbach, J., Kremer, G.: Embedding the virtual substitution method in the model constructing satisfiability calculus framework. In: proceedings of SC-square’17. CEUR Workshop Proceedings, vol. 1974. CEUR-WS.org (2017). urn:nbn:de:0074–1974-4, http://ceur-ws.org/Vol-1974/EAb.pdf
Arias, E.J.G., Pin, B., Jouvelot, P.: jsCoq: Towards hybrid theorem proving interfaces. In: Proceedings of the 12th Workshop on User Interfaces for Theorem Provers, UITP. EPTCS, vol. 239, pp. 15–27 (2016)
Barrett, C., Fontaine, P., Tinelli, C.: The satisfiability modulo theories library (SMT-LIB). www.SMT-LIB.org (2016)
Barrett, C., Kroening, D., Melham, T.: Problem solving for the 21st century: Efficient solvers for satisfiability modulo theories. Knowledge Transfer Report 3, London Mathematical Society and Smith Institute for Industrial Mathematics and System Engineering (2014)
Barrett, C., Sebastiani, R., Seshia, S.A., Tinelli, C.: Satisfiability modulo theories. In: Handbook of Satisfiability, Frontiers in Artificial Intelligence and Applications, chap. 26, vol. 185, pp. 825–885. IOS Press (2009)
Biere, A., Heule, M., van Maaren, H., Walsh, T.: Handbook of Satisfiability, Frontiers in Artificial Intelligence and Applications, vol. 185. IOS Press (2009)
Buchberger, B., Jebelean, T., Kutsia, T., Maletzky, A., Windsteiger, W.: Theorema 2.0: computer-assisted natural-style mathematics. J. Formalized Reasoning 9(1), 149–185 (2016). https://doi.org/10.6092/issn.1972-5787/4568
Buss, S.R.: An introduction to proof theory. In: Handbook of Proof Theory, pp. 31–35. Elsevier (1998)
CArL: Project homepage. https://github.com/smtrat/carl
Collins, G.E.: Quantifier elimination for real closed fields by cylindrical algebraic decompostion. In: Brakhage, H. (ed.) GI-Fachtagung 1975. LNCS, vol. 33, pp. 134–183. Springer, Heidelberg (1975). https://doi.org/10.1007/3-540-07407-4_17
Corzilius, F.: Integrating virtual substitution into strategic SMT solving. Ph.D. thesis, RWTH Aachen University, Germany (2016). http://publications.rwth-aachen.de/record/688379
Corzilius, F., Kremer, G., Junges, S., Schupp, S., Ábrahám, E.: SMT-RAT: an open source C++ toolbox for strategic and parallel SMT solving. In: Heule, M., Weaver, S. (eds.) SAT 2015. LNCS, vol. 9340, pp. 360–368. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-24318-4_26
Corzilius, F., Loup, U., Junges, S., Ábrahám, E.: SMT-RAT: an SMT-compliant nonlinear real arithmetic toolbox. In: Cimatti, A., Sebastiani, R. (eds.) SAT 2012. LNCS, vol. 7317, pp. 442–448. Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-31612-8_35
Davis, M., Logemann, G., Loveland, D.: A machine program for theorem-proving. Commun. ACM 5(7), 394–397 (1962). https://doi.org/10.1145/368273.368557
http://www.domagoj-babic.com/uploads/ResearchProjects/Spear/dimacs-cnf.pdf
Doig, A.G., Land, B.H., Doig, A.G.: An automatic method for solving discrete programming problems. Econometrica 28, 497–520 (1960)
Dolmans, D.H.J.M., De Grave, W., Wolfhagen, I.H.A.P., Van Der Vleuten, C.P.M.: Problem-based learning: future challenges for educational practice and research. Mediac Educ. 39(7), 732–741 (2005). https://doi.org/10.1111/j.1365-2929.2005.02205.x
Dramnesc, I., Jebelean, T.: Synthesis of sorting algorithms using multisets in Theorema. J. Logical Algebraic Methods Program. 119, 100635 (2020). https://doi.org/10.1016/j.jlamp.2020.100635
Dramnesc, I., Jebelean, T., Stratulat, S.: Mechanical synthesis of sorting algorithms for binary trees by logic and combinatorial techniques. J. Symb. Comput. 90, 3–41 (2019). https://doi.org/10.1016/j.jsc.2018.04.002
Dramnesc, I., Jebelean, T.: Implementation of deletion algorithms on lists and binary trees in Theorema. RISC Report Series 20–04, Research Institute for Symbolic Computation, Johannes Kepler University Linz (2020)
Fontaine, P., Ogawa, M., Sturm, T., Vu, X.T.: Subtropical satisfiability. In: Dixon, C., Finger, M. (eds.) FroCoS 2017. LNCS (LNAI), vol. 10483, pp. 189–206. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-66167-4_11
Friedgut, E.: Appendix by Jean bourgain: sharp thresholds of graph properties, and the k-sat problem. J. Am. Math. Soc. 12, 1017–1054 (1999)
Gao, S., Ganai, M., Ivančić, F., Gupta, A., Sankaranarayanan, S., Clarke, E.M.: Integrating ICP and LRA solvers for deciding nonlinear real arithmetic problems. In: Proceedings of FMCAD’10, pp. 81–90. IEEE (2010)
Herbort, S., Ratz, D.: Improving the efficiency of a nonlinear-system-solver using a componentwise Newton method. Technical report 2/1997, Inst. für Angewandte Mathematik, University of Karlsruhe (1997)
CEA-EDF-INRIA summer school, INRIA Paris-Rocquencourt, Antenne Parisienne, 2011. https://fzn.fr/teaching/coq/ecole11/
Jebelean, T.: A Heuristic Prover for Elementary Analysis in Theorema. In: Kamareddine, F., Sacerdoti Coen, C. (eds.) CICM 2021. LNCS (LNAI), vol. 12833, pp. 130–134. Springer, Cham (2021). https://doi.org/10.1007/978-3-030-81097-9_10
Junges, S., Loup, U., Corzilius, F., Ábrahám, E.: On Gröbner bases in the context of satisfiability-modulo-theories solving over the real numbers. In: Muntean, T., Poulakis, D., Rolland, R. (eds.) CAI 2013. LNCS, vol. 8080, pp. 186–198. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-40663-8_18
Knuth, D.E.: The Art of Computer Programming: Satisfiability, Volume 4, Fascicle 6. Addison-Wesley Professional (2015)
Kremer, G., Abraham, E.: Fully incremental cylindrical algebraic decomposition. J. Symb. Comput. 100, 11–37 (2020). https://doi.org/10.1016/j.jsc.2019.07.018
Kremer, G., Corzilius, F., Ábrahám, E.: A generalised branch-and-bound approach and its application in SAT modulo nonlinear integer arithmetic. In: Gerdt, V.P., Koepf, W., Seiler, W.M., Vorozhtsov, E.V. (eds.) CASC 2016. LNCS, vol. 9890, pp. 315–335. Springer, Cham (2016). https://doi.org/10.1007/978-3-319-45641-6_21
Kroening, D., Strichman, O.: Decision Procedures: An Algorithmic Point of View. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-74105-3
Kusper, G., Balla, T., Biró, C., Tajti, T., Yang, Z.G., Baják, I.: Generating minimal unsatisfiable SAT instances from strong digraphs. In: SYNASC 2020, pp. 84–92. IEEE Computer Society Press (2020). https://doi.org/10.1109/SYNASC51798.2020.00024
Kusper, G., Biró, C., Iszály, G.B.: SAT solving by CSFLOC, the next generation of full-length clause counting algorithms. In: 2018 IEEE International Conference on Future IoT Technologies, pp. 1–9. IEEE Computer Society Press (2018). https://doi.org/10.1109/FIOT.2018.8325589
Moskewicz, M.W., Madigan, C.F., Zhao, Y., Zhang, L., Malik, S.: Chaff: Engineering an efficient SAT solver. In: Proceedings of the 38th Annual Design Automation Conference, pp. 530–535. Association for Computing Machinery (2001). https://doi.org/10.1145/378239.379017
de Moura, L., Bjørner, N.: Z3: an efficient SMT solver. In: Ramakrishnan, C.R., Rehof, J. (eds.) TACAS 2008. LNCS, vol. 4963, pp. 337–340. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-78800-3_24
The Coq development team: The Coq Reference Manual. INRIA (2020)
Weispfenning, V.: A new approach to quantifier elimination for real algebra. In: Caviness, B.F., Johnson, J.R. (eds.) Quantifier Elimination and Cylindrical Algebraic Decomposition. Texts and Monographs in Symbolic ComputationTexts and Monographs in Symbolic Computation, pp. 376–392. Springer, Vienna (1998). https://doi.org/10.1007/978-3-7091-9459-1_20
Weispfenning, V.: Quantifier elimination for real algebra - the quadratic case and beyond. Appl. Algebra Eng. Commun. Comput. 8(2), 85–101 (1997). https://doi.org/10.1007/s002000050055
Windsteiger, W.: Theorema 2.0: a system for mathematical theory exploration. In: Hong, H., Yap, C. (eds.) ICMS 2014. LNCS, vol. 8592, pp. 49–52. Springer, Heidelberg (2014). https://doi.org/10.1007/978-3-662-44199-2_9
Wolfram Research Inc.: Mathematica, Version 13.0.0. https://www.wolfram.com/mathematica
https://github.com/Z3Prover/doc/tree/master/programmingz3/code
Acknowledgements
This work is co-funded by the Erasmus+ Programme of the European Union, project ARC: Automated Reasoning in the Class, 2019-1-RO01-KA203-063943.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this paper
Cite this paper
Drămnesc, I., Ábrahám, E., Jebelean, T., Kusper, G., Stratulat, S. (2022). Experiments with Automated Reasoning in the Class. In: Buzzard, K., Kutsia, T. (eds) Intelligent Computer Mathematics. CICM 2022. Lecture Notes in Computer Science(), vol 13467. Springer, Cham. https://doi.org/10.1007/978-3-031-16681-5_20
Download citation
DOI: https://doi.org/10.1007/978-3-031-16681-5_20
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-16680-8
Online ISBN: 978-3-031-16681-5
eBook Packages: Computer ScienceComputer Science (R0)