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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).

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