Beyond multi-adjoint logic programming

^†Extended version of a previous work entitled ‘Relaxing the Role of Adjoint Pairs in Multi-adjoint Logic Programming’ and presented in the ‘13th International Conference on Mathematical Methods in Science and Engineering, CMMSE 2013 [46]’.

Abstract

In ‘multi-adjoint logic programming’, MALP in brief, each fuzzy logic program is associated with its own ‘multi-adjoint lattice’ for modelling truth degrees beyond the simpler case of true and false, where a large set of fuzzy connectives can be defined. On this wide repertoire, it is crucial to connect each implication symbol with a proper conjunction thus conforming constructs of the form (←_i, &_i) called ‘adjoint pairs’, whose use directly affects both declarative and operational semantics of the MALP framework. In this work, we firstly show how the strong dependence of adjoint pairs can be largely weakened for an interesting ‘sub-class’ of MALP programs. Then, we reason in a similar way till conceiving a ‘super-class’ of fuzzy logic programs beyond MALP, which definitively drops out the need for using adjoint pairs, since the new semantics behaviour relies on much more relaxed lattices than multi-adjoint ones.

Keywords:

• multi-adjoint logic programming
• adjoint pairs
• fuzzy logic programming

2010 AMS Subject Classifications:

• 06B23
• 68N17
• 90C70
• 97P50

Acknowledgements

This work was supported by the EU (FEDER), and the Spanish MINECO Ministry (Ministerio de Economía y Competitividad) under grant TIN2013-45732-C4-2-P.

Notes

^† Extended version of a previous work entitled ‘Relaxing the Role of Adjoint Pairs in Multi-adjoint Logic Programming’ and presented in the ‘13th International Conference on Mathematical Methods in Science and Engineering, CMMSE 2013 [46]’.

1. Visit too http://dectau.uclm.es/floper/ and http://dectau.uclm.es/fuzzyXPath/.

2. Then, it is a bounded lattice, that is, it has bottom and top elements, denoted by ⊥ and ⊤, respectively.

3. This condition is the most important feature of the framework.

4. Sometimes we will say only least fuzzy model or least model.

5. The non-reputational behaviour of MALP has a decisive influence in this sense.

6. This concept does not make use of adjoint pairs and weights of program rules.

7. By definition, given (L,≤), the subset X⊂L is a directed set if for all $\{x_1,\dots ,x_n\}\subset X,$ it is verified that $sup\{x_1,\dots ,x_n\}\in X$.

8. The existence of this atom is guaranteed by the definition of the Herbrand universe and Herbrand base of $\mathcal{P}$.