Best Approximation of Ruspini Partitions in Gödel Logic

Abstract

A Ruspini partition is a finite family of fuzzy sets {f1,. .. , fn}, fi : [0, 1] → [0, 1], such that n i=1 fi(x) = 1 for all x ∈ [0, 1]. We analyze such partitions in the language of Gödel logic. Our main result identifies the precise degree to which the Ruspini condition is expressible in this language, and yields inter alia a constructive procedure to axiomatize a given Ruspini partition by a theory in Gödel logic.

An analysis of Ruspini partitions in Gödel logic

“…Acknowledgement. This paper is a revised and extended version of [6]. We are grateful to the three anonymous referees for several suggestions that have greatly improved the presentation of our results.…”