On Proofs and Rule of Multiplication in Fuzzy Attribute Logic
The paper develops fuzzy attribute logic, i.e. a logic for reasoning about formulas of the form \(A\Rightarrow B\) where A and B are fuzzy sets of attributes. A formula \(A\Rightarrow B\) represents a dependency which is true in a data table with fuzzy attributes iff each object having all attributes from A has also all attributes from B, membership degrees in A and B playing a role of thresholds. We study axiomatic systems of fuzzy attribute logic which result by adding a single deduction rule, called a rule of multiplication, to an ordinary system of deduction rules complete w.r.t. bivalent semantics, i.e. to well-known Armstrong axioms. In this paper, we concentrate on the rule of multiplication and its role in fuzzy attribute logic. We show some advantageous properties of the rule of multiplication. In addition, we show that these properties enable us to reduce selected problems concerning proofs in fuzzy attribute logic to the corresponding problems in the ordinary case. As an example, we discuss the problem of normalization of proofs and present, in the setting of fuzzy attribute logic, a counterpart to a well-known theorem from database theory saying that each proof can be transformed to a so-called RAP-sequence.
KeywordsFuzzy Logic Association Rule Fuzzy Rule Residuated Lattice Formal Concept Analysis
Unable to display preview. Download preview PDF.
- 1.Armstrong, W.W.: Dependency structures in data base relationships. In: IFIP Congress, Geneva, Switzerland, pp. 580–583 (1974)Google Scholar
- 3.Belohlavek, R., Vychodil, V.: Fuzzy attribute logic: attribute implications, their validity, entailment, and non-redundant basis. In: IFSA 2005, vol. I, pp. 622–627. Springer, Heidelberg (2005)Google Scholar
- 4.Belohlavek, R., Vychodil, V.: Fuzzy attribute logic: syntactic entailment and completeness. In: Proc. JCIS 2005, Salt Lake City, Utah, pp. 78–81 (2005)Google Scholar
- 5.Belohlavek, R., Vychodil, V.: Axiomatizations of fuzzy attribute logic. In: Prasad, B. (ed.) IICAI 2005, Proc. 2nd Indian International Conference on Artificial Intelligence, Pune, India, Dec 20–22, 2005, pp. 2178–2193 (2005)Google Scholar
- 8.Belohlavek, R., Vychodil, V.: Computing non-redundant bases of if-then rules from data tables with graded attributes. In: Proc. IEEE GrC 2006, 2006 IEEE International Conference on Granular Computing, Atlanta, GA, May 10–12, 2006, pp. 205–210 (2006)Google Scholar
- 9.Belohlavek, R., Vychodil, V.: Properties of models of fuzzy attribute implications. In: SCIS & ISIS 2006, Int. Conf. Soft Computing and Intelligent Systems & Int. Symposium on Intelligent Systems, Tokyo, Japan, Sep. 20–24, 2006, pp. 291–296 (2006)Google Scholar
- 10.Belohlavek, R., Vychodil, V.: Pavelka-style fuzzy logic for attribute implications. In: Proc. JCIS 2006, Kaohsiung, Taiwan, ROC, pp. 1156–1159 (2006)Google Scholar
- 11.Belohlavek, R., Vychodil, V.: Codd’s relational model of data and fuzzy logic: comparisons, observations, and some new results. In: Mohammadian, M. (ed.) Proc. CIMCA 2006, Sydney, Australia (2006)Google Scholar