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