On minimal sets of graded attribute implications

Reasoning with various types of if-then rules is crucial in many disciplines ranging from theoretical computer science to applications. Among the most widely used rules are those taking from of implications between conjunctions of attributes. Such rules are utilized in database systems (as functional dependencies or inclusion dependencies [23]), logic programming (as particular definite clauses representing programs [22]), and data mining (as attribute implications [14] or association rules [1, 33]). One of the most important problems regarding the rules is to find for a given set T of rules a set of rules which is equivalent to T and minimal in terms of its size. In relational database theory [23], the problem is referred to as finding minimal covers of T.