Mathematical foundations of fuzzy systems

1. Mathematical foundations of fuzzy
systems
I.S. Soldatenko
Department of Information Technologies
Tver State University, Tver, Russia
A.V. Yazenin

2. Curriculum details
“Mathematical foundations of fuzzy systems” is a discipline of first year of
master program “Fundamental computer sciences and information
technologies”.
It has 324 hours (180 – first semester, 144 – second semester).

3. Theory of fuzzy sets

4. Theory of possibility
In 1978 Stephen Nahmias introduced a new theory using axiomatic approach
and basing on fuzzy sets – theory of possibility.
Let  – be a model set,  – its elements, P() – power set of .
Definition 1. A possibility measure : P()  E1 is a set function with the
following properties:
() = 0, () = 1,
(iIAi) = supiI (Ai), AiP(),I.
Definition 2. A triplet (,P(),) is called possibilistic space.
Definition 3. A possibilistic (fuzzy) variable is a real-valued function
possible values of which are characterized by possibility distribution µA(x):
µA(x) is the possibility that A can have x as a value.
,:)( 1
A
  .,)(:)( 1
 xxAxA 

5. Crisp vs fuzzy values
1
10987654321 1514131211
crisp
values
1
10987654321 1514131211
fuzzy
values

6. Model task
Here and i=1, …, m,
aij, bi – are crisp coefficients.
max,)(0 xf





 .
,,,1,0)(
N
i
Rx
mixf 
 

n
j jj xaxf 1 00 )( i
n
j jiji bxaxf   1
)(

7. Possibilistic optimization task
Here and i=1, …, m,
aij(), bi() – are possibilistic variables.
max,k
  ,),( 00   kxf
 





 .
,,,1,0),(
N
ii
Rx
mixf 
 

n
j jj xaxf 1 00 )(),(  )()(),( 1
 i
n
j jiji bxaxf   

8. Fuzzy systems and soft computing
Optimization, classification, data-mining, prognosis…