Computing sets of graded attribute implications with witnessed non-redundancy

In this paper we extend our previous results on sets of graded attribute implications with witnessed non-redundancy. We assume finite residuated lattices as structures of truth degrees and use arbitrary idempotent truth-stressing linguistic hedges as parameters which influence the semantics of graded attribute implications. In this setting, we introduce algorithm which transforms any set of graded attribute implications into an equivalent non-redundant set of graded attribute implications with saturated consequents whose non-redundancy is witnessed by antecedents of the formulas. As a consequence, we solve the open problem regarding the existence of general systems of pseudo-intents which appear in formal concept analysis of object-attribute data with graded attributes and linguistic hedges. Furthermore, we show a polynomial-time procedure for determining bases given by general systems of pseudo-intents from sets of graded attribute implications which are complete in data

On minimal sets of graded attribute implications

We explore the structure of non-redundant and minimal sets consisting of graded if-then rules. The rules serve as graded attribute implications in object-attribute incidence data and as similarity-based functional dependencies in a similarity-based generalization of the relational model of data. Based on our observations, we derive a polynomial-time algorithm which transforms a given finite set of rules into an equivalent one which has the least size in terms of the number of rules

On sets of graded attribute implications with witnessed non-redundancy

We study properties of particular non-redundant sets of if-then rules describing dependencies between graded attributes. We introduce notions of saturation and witnessed non-redundancy of sets of graded attribute implications are show that bases of graded attribute implications given by systems of pseudo-intents correspond to non-redundant sets of graded attribute implications with saturated consequents where the non-redundancy is witnessed by antecedents of the contained graded attribute implications. We introduce an algorithm which transforms any complete set of graded attribute implications parameterized by globalization into a base given by pseudo-intents. Experimental evaluation is provided to compare the method of obtaining bases for general parameterizations by hedges with earlier graph-based approaches