Abstract
This work proposes a declarative semantics based on a fuzzy variant of the classical notion of least Herbrand model for the so-called FASILL language (acronym of “Fuzzy Aggregators and Similarity Into a Logic Language”) which has been recently designed and implemented in our research group for coping with implicit/explicit truth degree annotations, a great variety of connectives and unification by similarity.
Work supported by the EU (FEDER), and the Spanish MINECO Ministry (Ministerio de Economía y Competitividad) under grant TIN2013-45732-C4-2-P.
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Notes
- 1.
Two different programming environments for \(\mathsf{Bousi}{\sim }\mathsf{Prolog}\) are available at http://dectau.uclm.es/bousi/.
- 2.
The tool is freely accessible from the Web site http://dectau.uclm.es/floper/.
- 3.
This convention is quite standard and even used in a pure logic language like Prolog, where the reserved words true and fail -which directly resemble the pair of elements conforming the fixed lattice of truth degrees associated to any Prolog program- can be freely used on goals and clause bodies.
- 4.
Here, the connectives \(\varsigma \) are binary operations but we usually generalize them with an arbitrary number of arguments, that is, with truth function \(\mathbf {\dot{\varsigma }} : L^n\rightarrow L\).
- 5.
Note that, in the antecedent of this implication we use the order for pairs, (which is defined as \((x_1,y_1)\le (x_2,y_2)\) if, and only if, \(x_1\le x_2 \text{ and } y_1\le y_2\)), while in the consequent the usual order on the interval [0, 1] is considered. Similarly, it is possible to extend the usual order on [0, 1], for n-ary connectives.
- 6.
For convenience, \({\mathcal R}(x,y)\), also denoted \(x{\mathcal R}y\), refers to both the syntactic expression (that symbolizes that the elements \(x,y\in {\mathcal U}\) are related by \({\mathcal R}\)) and the membership degree \(\mu _{\mathcal R}(x,y)\), i.e., the affinity degree of the pair \((x,y)\in {\mathcal U}\times {\mathcal U}\) with the verbal predicate (or fuzzy predicate) \({\mathcal R}\).
- 7.
Note that elegant(taxi) and vanguardist(metro) are 1-ary predicates, whereas that taxi, metro are terms with arity 0, i.e. constants.
- 8.
Note that, X-MALP programs do not rely on adjoint pairs.
- 9.
Sometimes we will abbreviate writing “fuzzy model” or simply “model”.
- 10.
Sometimes we will abbreviate writing “least fuzzy model” or simply “least model”.
- 11.
The last version of the FLOPER system which copes with similarity relations can be freely downloaded from http://dectau.uclm.es/floper/?q=sim and it can be tested on-line through http://dectau.uclm.es/floper/?q=sim/test.
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Julián-Iranzo, P., Moreno, G., Penabad, J., Vázquez, C. (2016). A Declarative Semantics for a Fuzzy Logic Language Managing Similarities and Truth Degrees. In: Alferes, J., Bertossi, L., Governatori, G., Fodor, P., Roman, D. (eds) Rule Technologies. Research, Tools, and Applications. RuleML 2016. Lecture Notes in Computer Science(), vol 9718. Springer, Cham. https://doi.org/10.1007/978-3-319-42019-6_5
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