By introducing element equivalence class that proposes dynamic characteristic into Pawlak Z rough sets theory, the first author of this paper improved Pawlak Z rough sets and put forward S-rough sets (singular rough sets). S-rough sets are defined by element equivalence class that proposes dynamic characteristic.
S-rough sets have dynamic characteristic. By introducing the function equivalence class (law equivalence class) that proposes dynamic characteristic into S-rough sets, the first author improved S-rough sets and put forward function S-rough sets (function singular rough sets). Function S-rough sets have dynamic characteristic and law characteristic, and a function is a law. By using function S-rough sets, this paper presents law identification, law identification theorem, and law identification criterion and applications. Function S-rough sets are a new research direction of rough sets theory, and it is also a new tool to the research of system law identification.function S-rough sets, system law, law identification, identification criterion, identification theorem Information system with multiple outputs or single output needs to identify its outputs law state in its work time interval 1 [ , ] j k T t t , so that we can affirm that the output law of information system on T j 1 [ , ] k t t is law stabilization or law turbulence, 1, 2, , j . The outputs 1 2 , , , m u u u in the information system with m outputs on 1 [ , ] j k T t t compose function equivalence class [1, 2] (law equivalence class) 1 2 [ ] { , , , } j T m u u u u with respect to 1 [ , ] j k T t t , [ ]j i T u u is a function, and a function is a law. The outputs 1 2 ( ), ( ), , ( ) m u t u t u t at j t T compose element equivalence class (law state value at t ) 1 2 [ ] { ( ), ( ), , ( )} t m x u t u t u t 1 2