One of the most basic fuzzy set systems use min and max for and and or of pairs of membership grades which each must be between 0 and 1. Other operators are used in other fuzzy systems.
Three well-studied triangular norm (t_norm) systems are Kleene-Dienes, Reichenbach, and Lukasiewicz. The t_norm operator for each system is its "and" operator NB. t_norms Kleene-Dienes, Reichenbach, Lukasiewicz KD =: <. reich =: * lukas =: (0 >. 1 -~ +)"0 NB. sample and tables NB. KD /~ 0 0.25 0.5 0.75 1 NB. reich /~ 0 0.25 0.5 0.75 1 NB. lukas /~ 0 0.25 0.5 0.75 1 NB. With the deMorgan "not", the t_conorm for each system can be defined as follows using the adverb S . not =: -. S =: 1 : 'n...@u& not' NB. t_conorm S =: & not(not@) NB. t_conorm NB. sample or tables NB. KD S/~ 0 0.25 0.5 0.75 1 NB. reich S/~ 0 0.25 0.5 0.75 1 NB. lukas S/~ 0 0.25 0.5 0.75 1 NB. next shows that t_conorm for KD is >. NB. (>./-:KD S/)~ 0 0.25 0.5 0.75 1 NB. KD t_conorm is >. NB. and next 2 show similar equivalences NB. ((+- *)"0/-:reich S/)~ 0 0.25 0.5 0.75 1 NB. reich t_conorm is (+-*)"0 NB. ((1<.+)"0/-:lukas S/)~ 0 0.25 0.5 0.75 1 NB. lukas t_conorm is (1<.+)"0 NB. material implication for these systems can be generated as follows I =: 1 : 'n...@[ u S ]' NB. I is the implication operator NB. KD I"0/~0 0.25 0.5 0.75 1 NB. reich I"0/~0 0.25 0.5 0.75 1 NB. lukas I"0/~0 0.25 0.5 0.75 1 NB. next shows the equivalent implication operator for KD is -...@[>.]"0 NB. (( -...@[>.])-:KD I"0)/~0 0.25 0.5 0.75 1 NB. (( -...@[+ *)-:reich I"0)/~0 0.25 0.5 0.75 1 NB. ((1<....@[+ ])-:lukas I"0)/~0 0.25 0.5 0.75 1 On Wed, Mar 4, 2009 at 4:28 PM, Raul Miller <[email protected]> wrote: > On Wed, Mar 4, 2009 at 4:07 PM, Roger Hui <[email protected]> wrote: >> It is by definition that 0=x*.y if x=0 or y=0. >> Then the advantages are that *. is the same >> as "and" for booleans and that the identity >> (x*.y)=x(*%+.)y holds for x=0 or y=0 . > > Is it worth noting that min and max (<. and >.) > could likewise serve as and and or, without > needing any special case treatment? > > -- > Raul > ---------------------------------------------------------------------- > For information about J forums see http://www.jsoftware.com/forums.htm > -- (B=) <-----my sig Brian Schott ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
