Cut Sets as Recognizable Tree Languages Björn Borchardt and Andreas Maletti 1 Department of Computer Science, Dresden University of Technology, D-01062 Dresden, Germany Branimir e²elja and Andreja Tepav£evi¢ ∗,2 Department of Mathematics and Informatics, University of Novi Sad, 21000 Novi Sad, Serbia and Montenegro Heiko Vogler Department of Computer Science, Dresden University of Technology, D-01062 Dresden, Germany Abstract A tree series over a semiring with partially ordered carrier set can be considered as a fuzzy set. We investigate conditions under which it can also be understood as a fuzzied recognizable tree language. In this sense, sucient conditions are presented which, when imposed, ensure that every cut set, i.e., the pre-image of a prime lter of the carrier set, is a recognizable tree language. Moreover, such conditions are also presented for cut sets of recognizable tree series. 1 Introduction There are two sources for the investigations in this paper, namely (i) fuzzy sets and (ii) tree series and recognizable tree series, in particular. Both sources ∗ Corresponding author. Address: Department of Mathematics and Informatics, Trg Dositeja Obradovi¢a 4, 21000 Novi Sad, Serbia and Montenegro Email addresses: {borchard,maletti}@tcs.inf.tu-dresden.de (Björn Borchardt and Andreas Maletti),
[email protected] (Branimir e²elja and Andreja Tepav£evi¢),
[email protected] (Heiko Vogler). 1 Financially supported by the German Research Foundation (DFG, GK 334/3). 2 Financially supported by the Herbert Quandt Foundation and by the Serbian Ministry of Science, grant number 1227. Preprint submitted to Fuzzy Sets and Systems 6 October 2006 are derivatives of the concept of characteristic functions, where as usual, given a set S every characteristic function χ : S → {0, 1} on S identies the subset { s ∈ S | χ(s) = 1 } of S.