Skip to main content
SpringerLink
Log in

Symbolic Execution and Thresholding for Efficiently Tuning Fuzzy Logic Programs

  • Conference paper
  • First Online:
Logic-Based Program Synthesis and Transformation (LOPSTR 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10184))

Abstract

Fuzzy logic programming is a growing declarative paradigm aiming to integrate fuzzy logic into logic programming. One of the most difficult tasks when specifying a fuzzy logic program is determining the right weights for each rule, as well as the most appropriate fuzzy connectives and operators. In this paper, we introduce a symbolic extension of fuzzy logic programs in which some of these parameters can be left unknown, so that the user can easily see the impact of their possible values. Furthermore, given a number of test cases, the most appropriate values for these parameters can be automatically computed. Finally, we show some benchmarks that illustrate the usefulness of our approach.

This work has been partially supported by the EU (FEDER), the State Research Agency (AEI) and the Spanish Ministerio de Economía y Competitividad under grants TIN2013-45732-C4-2-P, TIN2013-44742-C4-1-R, TIN2016-76843-C4-1-R, TIN2016-76843-C4-2-R (AEI/FEDER, UE) and by the Generalitat Valenciana under grant PROMETEO-II/2015/013 (SmartLogic).

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    For instance, we have typically several adjoint pairs: Łukasiewicz logic \( \langle \& _\mathtt {L},{\leftarrow }_\mathtt {L} \rangle \), Gödel logic \( \langle \& _\mathtt {G},{\leftarrow }_\mathtt {G} \rangle \) and product logic \( \langle \& _\mathtt {P},{\leftarrow }_\mathtt {P} \rangle \), which might be used for modeling pessimist, optimist and realistic scenarios, respectively.

  2. 2.

    A complete lattice is a (partially) ordered set \(\langle L,\preceq \rangle \) such that every subset S of L has infimum and supremum elements. It is bounded if it has bottom and top elements, denoted by \(\bot \) and \(\top \), respectively. L is said to be the carrier set of the lattice, and \(\preceq \) its ordering relation.

  3. 3.

    For convenience, in the following sections, we do not distinguish between the connective \(\varsigma \) and its truth function \([\![ \varsigma ]\!]\).

  4. 4.

    Here, we assume that A in \(\mathcal {Q}[A]\) is the selected atom. Furthermore, as it is common practice, mgu(E) denotes the most general unifier of the set of equations E [14].

  5. 5.

    For simplicity, we consider that facts (H with v) are seen as rules of the form \((H{\leftarrow }_i \top ~with~v)\) for some implication \({\leftarrow }_i\). Furthermore, in this case, we directly derive the state \(\langle (\mathcal {Q}[A/v])\theta ;\sigma \theta \rangle \) since \( v\, \& _i \top = v\) for all &\(_i\).

  6. 6.

    It is important to note that, at execution time, each implication symbol belonging to a concrete adjoint pair is replaced by its adjoint conjunction (see again our repertoire of adjoint pairs in Fig. 1 in the preliminaries section).

  7. 7.

    Each cell refers to the average of 100 executions using a desktop computer equipped with an i3-2310 M CPU @ 2.10 GHz and 4,00 GB RAM.

  8. 8.

    Instead of focusing on satisfiability, (i.e., proving the existence of at least one model) as usually done in a SAT/SMT setting, in [1, 6] we have faced the problem of finding the whole set of models for a given fuzzy formula by re-using a previous method based on fuzzy logic programming where the formula is conceived as a goal whose derivation tree, provided by the FLOPER tool, contains in its leaves all the models of the original formula, together with other interpretations.

References

  1. Almendros-Jiménez, J.M., Bofill, M., Luna-Tedesqui, A., Moreno, G., Vázquez, C., Villaret, M.: Fuzzy XPath for the automatic search of fuzzy formulae models. In: Beierle, C., Dekhtyar, A. (eds.) SUM 2015. LNCS (LNAI), vol. 9310, pp. 385–398. Springer, Cham (2015). doi:10.1007/978-3-319-23540-0_26

    Chapter  Google Scholar 

  2. Almendros-Jiménez, J.M., Luna, A., Moreno, G.: Fuzzy XPath through fuzzy logic programming. New Gener. Comput. 33(2), 173–209 (2015)

    Article  Google Scholar 

  3. Ansótegui, C., Bofill, M., Manyà, F., Villaret, M.: Building automated theorem provers for infinitely-valued logics with satisfiability modulo theory solvers. In: Proceeding of ISMVL 2012, pp. 25–30 (2012)

    Google Scholar 

  4. Baldwin, J.F., Martin, T.P., Pilsworth, B.W.: Fril- Fuzzy and Evidential Reasoning in Artificial Intelligence. Wiley, New York (1995)

    Google Scholar 

  5. Barrett, C.W., Sebastiani, R., Seshia, S.A., Tinelli, C.: Satisfiability modulo theories. In: Handbook of Satisfiability, Frontiers in Artificial Intelligence and Applications, 185, pp. 825–885. IOS Press (2009)

    Google Scholar 

  6. Bofill, M., Moreno, G., Vázquez, C., Villaret, M.: Automatic proving of fuzzy formulae with fuzzy logic programming and SMT. In: Fredlund, L.A. (ed.) Programming and Computer Languages 2013, vol. 64, p. 19. ECEASST (2013)

    Google Scholar 

  7. Ishizuka, M., Kanai, N.: Prolog-ELF incorporating fuzzy logic. In: Proceeding of the IJCAI 1985, pp. 701–703. Morgan Kaufmann (1985)

    Google Scholar 

  8. Julián, P., Medina, J., Moreno, G., Ojeda-Aciego, M.: Efficient thresholded tabulation for fuzzy query answering. In: Bouchon-Meunier, B., Magdalena, L., Ojeda-Aciego, M., Verdegay, J.L., Yager, R.R. (eds.) Foundations of Reasoning under Uncertainty. STUDFUZZ, vol. 249, pp. 125–149. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  9. Julián, P., Moreno, G., Penabad, J.: Operational/interpretive unfolding of multi-adjoint logic programs. J. Univ. Comput. Sci. 12(11), 1679–1699 (2006)

    Google Scholar 

  10. Julián, P., Moreno, G., Penabad, J.: An improved reductant calculus using fuzzy partial evaluation techniques. Fuzzy Sets Syst. 160, 162–181 (2009). http://dx.doi.org/10.1016/j.fss.2008.05.006

    Article  MathSciNet  MATH  Google Scholar 

  11. Julián-Iranzo, P., Moreno, G., Penabad, J., Vázquez, C.: A fuzzy logic programming environment for managing similarity and truth degrees. In: EPTCS, vol. 173, pp. 71–86 (2015). http://dx.doi.org/10.4204/EPTCS.173.6

  12. Julián-Iranzo, P., Moreno, G., Penabad, J., Vázquez, C.: A declarative semantics for a fuzzy logic language managing similarities and truth degrees. In: Alferes, J.J.J., Bertossi, L., Governatori, G., Fodor, P., Roman, D. (eds.) RuleML 2016. LNCS, vol. 9718, pp. 68–82. Springer, Cham (2016). doi:10.1007/978-3-319-42019-6_5

    Chapter  Google Scholar 

  13. Kifer, M., Subrahmanian, V.S.: Theory of generalized annotated logic programming and its applications. J. Logic Program. 12, 335–367 (1992)

    Article  MathSciNet  Google Scholar 

  14. Lassez, J.L., Maher, M.J., Marriott, K.: Unification revisited. In: Foundations of Deductive Databases and Logic Programming, pp. 587–625. Morgan Kaufmann, Los Altos, CA (1988)

    Google Scholar 

  15. Lee, R.: Fuzzy logic and the resolution principle. J. ACM 19(1), 119–129 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  16. Li, D., Liu, D.: A Fuzzy Prolog Database System. Wiley, New York (1990)

    Google Scholar 

  17. Lloyd, J.W.: Foundations of Logic Programming. Springer-Verlag, Berlin (1987)

    Book  MATH  Google Scholar 

  18. Medina, J., Ojeda-Aciego, M., Vojtáš, P.: Similarity-based Unification: a multi-adjoint approach. Fuzzy Sets Syst. 146, 43–62 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  19. Morcillo, P.J., Moreno, G., Penabad, J., Vázquez, C.: A practical management of fuzzy truth-degrees using FLOPER. In: Dean, M., Hall, J., Rotolo, A., Tabet, S. (eds.) RuleML 2010. LNCS, vol. 6403, pp. 20–34. Springer, Heidelberg (2010). doi:10.1007/978-3-642-16289-3_4

    Chapter  Google Scholar 

  20. Moreno, G., Vázquez, C.: Fuzzy logic programming in action with FLOPER. J. Softw. Eng. Appl. 7, 237–298 (2014)

    Article  Google Scholar 

  21. Nguyen, H.T., Walker, E.A.: A First Course in Fuzzy Logic. Chapman & Hall, Boca Ratón (2006)

    MATH  Google Scholar 

  22. Rodríguez-Artalejo, M., Romero-Díaz, C.A.: Quantitative logic programming revisited. In: Garrigue, J., Hermenegildo, M.V. (eds.) FLOPS 2008. LNCS, vol. 4989, pp. 272–288. Springer, Heidelberg (2008). doi:10.1007/978-3-540-78969-7_20

    Chapter  Google Scholar 

  23. Straccia, U.: Managing uncertainty and vagueness in description logics, logic programs and description logic programs. In: Baroglio, C., Bonatti, P.A., Małuszyński, J., Marchiori, M., Polleres, A., Schaffert, S. (eds.) Reasoning Web. LNCS, vol. 5224, pp. 54–103. Springer, Heidelberg (2008). doi:10.1007/978-3-540-85658-0_2

    Chapter  Google Scholar 

  24. Vidal, A., Bou, F., Godo, L.: An SMT-based solver for continuous t-norm based logics. In: Hüllermeier, E., Link, S., Fober, T., Seeger, B. (eds.) SUM 2012. LNCS (LNAI), vol. 7520, pp. 633–640. Springer, Heidelberg (2012). doi:10.1007/978-3-642-33362-0_53

    Chapter  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computing Systems, UCLM, 02071, Albacete, Spain

    Ginés Moreno & José A. Riaza

  2. Department of Mathematics, UCLM, 02071, Albacete, Spain

    Jaime Penabad

  3. MiST, DSIC, Universitat Politècnica de València, Valencia, Spain

    Germán Vidal

Authors
  1. Ginés Moreno
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jaime Penabad
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. José A. Riaza
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Germán Vidal
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ginés Moreno .

Editor information

Editors and Affiliations

  1. IMDEA Software Institute and Universidad Politécnica de Madrid, Madrid, Spain

    Manuel V Hermenegildo

  2. IMDEA Software Institute and CSIC, Madrid, Spain

    Pedro Lopez-Garcia

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this paper

Cite this paper

Moreno, G., Penabad, J., Riaza, J.A., Vidal, G. (2017). Symbolic Execution and Thresholding for Efficiently Tuning Fuzzy Logic Programs. In: Hermenegildo, M., Lopez-Garcia, P. (eds) Logic-Based Program Synthesis and Transformation. LOPSTR 2016. Lecture Notes in Computer Science(), vol 10184. Springer, Cham. https://doi.org/10.1007/978-3-319-63139-4_8

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-63139-4_8

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-63138-7

  • Online ISBN: 978-3-319-63139-4

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation