Abstract
The purpose of this paper is to present a fresh idea on how symbolic learning might be realized via analogical reasoning. For this, we introduce directed analogical proportions between logic programs of the form “P transforms into Q as R transforms into S” as a mechanism for deriving similar programs by analogy-making. The idea is to instantiate a fragment of a recently introduced abstract algebraic framework of analogical proportions in the domain of logic programming. Technically, we define proportions in terms of modularity where we derive abstract forms of concrete programs from a “known” source domain which can then be instantiated in an “unknown” target domain to obtain analogous programs. To this end, we introduce algebraic operations for syntactic logic program composition and concatenation. Interestingly, our work suggests a close relationship between modularity, generalization, and analogy which we believe should be explored further in the future. In a broader sense, this paper is a further step towards a mathematical theory of logic-based analogical reasoning and learning with potential applications to open AI-problems like commonsense reasoning and computational learning and creativity.
Article PDF
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.Avoid common mistakes on your manuscript.
Data availability statement
The manuscript has no data associated.
References
Antić, C.: Analogical proportions. Ann. Math. Artif. Intell. 90(6), 595–644 (2022). https://doi.org/10.1007/s10472-022-09798-y
Antić, C.: Analogical proportions II. https://hal.science/hal-04246079
Antić, C.: Boolean proportions. https://arxiv.org/pdf/2109.00388.pdf
Antić, C.: On syntactically similar logic programs and sequential decompositions. https://arxiv.org/pdf/2109.05300.pdf
Antić, C.: Sequential composition of answer set programs. https://arxiv.org/pdf/2104.12156.pdf
Antić, C.: Sequential composition of propositional logic programs. https://arxiv.org/pdf/2009.05774.pdf
Apt, K.R.: Logic programming. In: van Leeuwen, J. (ed.) Handbook of Theoretical Computer Science, vol. B, pp. 493–574. Elsevier, Amsterdam (1990)
Baader, F., Büttner, W.: Unification in commutative idempotent monoids. Theoret. Comput. Sci. 56, 345–352 (1988)
Baader, F., Ludmann, P.: The unification type of ACUI w.r.t. the unrestricted instantiation preorder is not finitary. In: 30th International workshop on unification, pp. 31–36 (2016)
Baader, F., Marantidis, P., Mottet, A., Okhotin, A.: Extensions of unification modulo ACUI. Math. Struct. Comput. Sci. 30(6), 597–626 (2020)
Barbot, N., Miclet, L., Prade, H.: Analogy between concepts. Artif. Intell. 275, 487–539 (2019)
Brewka, G., Eiter, T., Truszczynski, M.: Answer set programming at a glance. Commun. ACM 54(12), 92–103 (2011)
Brogi, A., Mancarella, P., Pedreschi, D., Turini, F.: Modular logic programming. ACM Trans. Program. Lang. Syst. 16(4), 1361–1398 (1999)
Buchberger, B.: Automated programming, symbolic computation, machine learning: my personal view. Ann. Math. Artif. Intell. 91, 569–589 (2023)
Bugliesi, M., Lamma, E., Mello, P.: Modularity in logic programming. J. Logic Program. 19–20(1), 443–502 (1994)
Chen, W., Kifer, M., Warren, D.S.: HiLog: a foundation for higher-order logic programming. J. Logic Program. 15(3), 187–230 (1993)
Falkenhainer, B., Forbus, K.D., Gentner, D.: The structure-mapping engine: algorithm and examples. Artif. Intell. 41(1), 1–63 (1989)
Gelfond, M., Lifschitz, V.: Classical negation in logic programs and disjunctive databases. N. Gener. Comput. 9(3–4), 365–385 (1991)
Gentner, D.: Structure-mapping: a theoretical framework for analogy. Cogn. Sci. 7(2), 155–170 (1983)
Goodfellow, I., Bengio, Y., Courville, A.: Deep Learning: Adaptive Computation and Machine Learning. MIT Press, Cambridge USA (2016)
Gowers, T.: The two cultures of mathematics. In: Arnold, W. I., Atiyah, M. F., Lax, P., Mazur, B. (Eds.), Mathematics: frontiers and perspectives, pp. 65–78. American Mathematical Society (2000)
Hofstadter, D., Mitchell, M.: The copycat project: a model of mental fluidity and analogy-making. In: Fluid Concepts and Creative Analogies. Computer Models of the Fundamental Mechanisms of Thought, vol. 5, pp. 205–267. Basic Books, New York (1995)
Hofstadter, D., Sander, E.: Surfaces and Essences. Basic Books, New York, Analogy as the Fuel and Fire of Thinking (2013)
Kowalski, R.: The relation between logic programming and logic specification. Philos. Trans. R. Soc. Lond. 312(1522), 345–361 (1984)
LeCun, Y., Bengio, Y., Hinton, G.: Deep learning. Nature 521, 436–444 (2015)
Lepage, Y.: Analogy and formal languages. Electron. Note Theor. Comput. Sci. 53, 180–191 (2001)
Miclet, L., Prade, H.: Handling analogical proportions in classical logic and fuzzy logics settings. In: Sossai, C., Chemello, G. (Eds.), ECSQARU 2009, LNAI 5590, pp. 638–650. Springer-Verlag, Berlin/Heidelberg (2009)
Miller, D., Nadathur, G.: Programming with Higher-Order Logic. Cambridge University Press, Cambridge (2012)
Muggleton, S.: Inductive logic programming. N. Gener. Comput. 8(4), 295–318 (1991)
Plotkin, G.D.: A note on inductive generalization. Mach. Intell. 5, 153–163 (1970)
Reynolds, J.C.: Transformational systems and the algebraic structure of atomic formulas. Mach. Intell. 5(1), 135–151 (1970)
Schmidt, M., Krumnack, U., Gust, H., Kühnberger, K.-U.: Heuristic-driven theory projection: an overview. In: Prade, H., Richard, G. (eds.) Computational Approaches to Analogical Reasoning: Current Trends. Studies in Computational Intelligence, vol. 548, pp. 163–194. Berlin/Heidelberg, Springer-Verlag (2014)
Stroppa, N., Yvon, F.: Formal models of analogical proportions. Technical Report D008, Telecom ParisTech - École Nationale Supérieure de Télécommunications, Télécom Paris (2006)
Acknowledgements
I would like to thank the reviewers for their thoughtful and constructive comments, and for their helpful suggestions to improve the presentation of the article. I would also like to thank Temur Kutsia for providing the references for unification and minimal complete set of unifiers.
Funding
Open access funding provided by TU Wien (TUW).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, 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 licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence 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. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Antić, C. Logic program proportions. Ann Math Artif Intell (2023). https://doi.org/10.1007/s10472-023-09904-8
Accepted:
Published:
DOI: https://doi.org/10.1007/s10472-023-09904-8