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Efficient representation of piecewise linear functions into Łukasiewicz logic modulo satisfiability

Published online by Cambridge University Press:  17 May 2022

Sandro Preto Open the ORCID record for Sandro Preto [Opens in a new window]  and
Marcelo Finger
Sandro Preto*
Affiliation:
Institute of Mathematics and Statistics, University of São Paulo, São Paulo, Brazil
Marcelo Finger
Affiliation:
Institute of Mathematics and Statistics, University of São Paulo, São Paulo, Brazil
*
*Corresponding author. Email: spreto@ime.usp.br
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Abstract

This work concerns the representation of a class of continuous functions into Logic, so that one may automatically reason about properties of these functions using logical tools. Rational McNaughton functions may be implicitly represented by logical formulas in Łukasiewicz Infinitely-valued Logic by constraining the set of allowed valuations; such a restriction contemplates only those valuations that satisfy specific formulas. This work investigates two approaches to such depiction, called representation modulo satisfiability. Furthermore, a polynomial-time algorithm that builds this representation is presented, producing a pair of formulas consisting of the representative formula and the constraining one, given as input a rational McNaughton function in a suitable encoding. An implementation of the algorithm is discussed.

Keywords

Function representationŁukasiewicz Infinitely-valued Logicrational McNaughton functionspiecewise linear functions
Type
Special Issue: LSFA’19 and LSFA’20
Information
Mathematical Structures in Computer Science , Volume 32 , Special Issue 9: LSFA 2019 and 2020 , October 2022 , pp. 1119 - 1144
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Copyright
© The Author(s), 2022. Published by Cambridge University Press

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Footnotes

†

This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior – Brasil (CAPES) – Finance Code 001. This work was carried out at the Center for Artificial Intelligence (C4AI-USP), with support by the São Paulo Research Foundation (FAPESP) (grant #2019/07665-4) and by the IBM Corporation. Sandro Preto was partly supported by the São Paulo Research Foundation (FAPESP) (grant #2021/03117-2). Marcelo Finger was partly supported by the São Paulo Research Foundation (FAPESP) (grants #2015/21880-4, #2014/12236-1); and the National Council for Scientific and Technological Development (CNPq) (grant PQ 303609/2018-4).

References

Aguzzoli, S. (1998). The complexity of McNaughton functions of one variable. Advances in Applied Mathematics 21 (1) 58–77.CrossRefGoogle Scholar
Aguzzoli, S. and Mundici, D. (2001). Weierstrass approximations by Łukasiewicz formulas with one quantified variable. In: Proceedings 31st IEEE International Symposium on Multiple-Valued Logic, IEEE, 361–366.CrossRefGoogle Scholar
Aguzzoli, S. and Mundici, D. (2003). Weierstrass Approximation Theorem and Łukasiewicz Formulas with one Quantified Variable, Physica-Verlag HD, Heidelberg, 315–335.CrossRefGoogle Scholar
Amato, P., Di Nola, A. and Gerla, B. (2002). Neural networks and rational Łukasiewicz logic. In: 2002 Annual Meeting of the North American Fuzzy Information Processing Society Proceedings. NAFIPS-FLINT 2002 (Cat. No. 02TH8622), IEEE, 506–510.Google Scholar
Amato, P. and Porto, M. (2000). An algorithm for the automatic generation of a logical formula representing a control law. Neural Network World 10 (5) 777–786.Google Scholar
Ansótegui, C., Bofill, M., Manyà, F. and Villaret, M. (2012). Building automated theorem provers for infinitely-valued logics with satisfiability modulo theory solvers. In: 2012 IEEE 42nd International Symposium on Multiple-Valued Logic, 25–30.CrossRefGoogle Scholar
Barrett, C., Fontaine, P. and Tinelli, C. (2016). The satisfiability modulo theories library (SMT-LIB). www.SMT-LIB.org.Google Scholar
Bertsimas, D. and Tsitsiklis, J. N. (1997). Introduction to Linear Optimization, Athena Scientific, Belmont, MA.Google Scholar
Bofill, M., Manya, F., Vidal, A. and Villaret, M. (2015). Finding hard instances of satisfiability in Łukasiewicz logics. In: 2015 IEEE International Symposium on Multiple-Valued Logic (ISMVL), IEEE, 30–35.CrossRefGoogle Scholar
Cignoli, R., D’Ottaviano, I. and Mundici, D. (2000). Algebraic Foundations of Many-Valued Reasoning , Trends in Logic, Springer, Netherlands.Google Scholar
Di Nola, A. and Leuştean, I. (2011). Riesz MV-algebras and their logic. In: Proceedings of the 7th Conference of the European Society for Fuzzy Logic and Technology (EUSFLAT-11), Atlantis Press, 140–145.Google Scholar
Di Nola, A. and Leuştean, I. (2014). Łukasiewicz logic and Riesz spaces. Soft Computing 18 2349–2363.CrossRefGoogle Scholar
Dutertre, B. (2014). Yices 2.2. In: Biere, A. and Bloem, R. (eds.) Computer-Aided Verification (CAV’2014), Lecture Notes in Computer Science, vol. 8559, Springer, 737–744.CrossRefGoogle Scholar
Esteva, F., Godo, L. and Montagna, F. (2001). The $\unicode{x0141}\Pi$ and $\unicode{x0141}\Pi\frac{1}{2}$ logics: Two complete fuzzy systems joining Łukasiewicz and product logics. Archive for Mathematical Logic 40 (1) 39–67.CrossRefGoogle Scholar
Finger, M. and Preto, S. (2018). Probably half true: Probabilistic satisfiability over Łukasiewicz infinitely-valued logic. In: Galmiche, D., Schulz, S. and Sebastiani, R. (eds.) Automated Reasoning, Cham, Springer International Publishing, 194–210.CrossRefGoogle Scholar
Finger, M. and Preto, S. (2020). Probably partially true: Satisfiability for Łukasiewicz infinitely-valued probabilistic logic and related topics. Journal of Automated Reasoning 64 (7) 1269–1286.CrossRefGoogle Scholar
Gamrath, G., Anderson, D., Bestuzheva, K., Chen, W.-K., Eifler, L., Gasse, M., Gemander, P., Gleixner, A., Gottwald, L., Halbig, K., Hendel, G., Hojny, C., Koch, T., Bodic, P. L., Maher, S. J., Matter, F., Miltenberger, M., Mühmer, E., Müller, B., Pfetsch, M., Schlösser, F., Serrano, F., Shinano, Y., Tawfik, C., Vigerske, S., Wegscheider, F., Weninger, D. and Witzig, J. (2020). The SCIP optimization suite 7.0. Technical report, Optimization Online.Google Scholar
Gerla, B. (2001). Rational Łukasiewicz logic and DMV-algebras. Neural Network World 11 (6) 579–594.Google Scholar
Leshno, M., Lin, V. Y., Pinkus, A. and Schocken, S. (1993). Multilayer feedforward networks with a nonpolynomial activation function can approximate any function. Neural Networks 6 (6) 861–867.CrossRefGoogle Scholar
McNaughton, R. (1951). A theorem about infinite-valued sentential logic. Journal of Symbolic Logic 16 1–13.CrossRefGoogle Scholar
Mundici, D. (1987). Satisfiability in many-valued sentential logic is NP-complete. Theoretical Computer Science 52 (1–2) 145–153.CrossRefGoogle Scholar
Mundici, D. (1994). A constructive proof of McNaughton’s theorem in infinite-valued logic. Journal of Symbolic Logic 59 (2) 596–602.CrossRefGoogle Scholar
Preto, S. and Finger, M. (2020). An efficient algorithm for representing piecewise linear functions into logic. Electronic Notes in Theoretical Computer Science 351 167–186. Proceedings of LSFA 2020, the 15th International Workshop on Logical and Semantic Frameworks, with Applications (LSFA 2020).CrossRefGoogle Scholar
Tarela, J., Alonso, E. and Martínez, M. (1990). A representation method for PWL functions oriented to parallel processing. Mathematical and Computer Modelling 13 (10) 75–83.CrossRefGoogle Scholar
Tarela, J. and Martínez, M. (1999). Region configurations for realizability of lattice piecewise-linear models. Mathematical and Computer Modelling 30 (11–12) 17–27.CrossRefGoogle Scholar
Xu, J. and Wang, S. (2019). Lattice piecewise affine representations on convex projection regions. In: 2019 IEEE 58th Conference on Decision and Control (CDC), 7240–7245.CrossRefGoogle Scholar