1. CPSC 386 Artificial Intelligence Ellen Walker Hiram College
Fuzzy Logic
CPSC 386 Artificial Intelligence
Ellen Walker
Hiram College

2. Why fuzzy? Imprecision better approximates human reasoning
It is warm today.
65 degrees F in April is warm
65 degrees F in August is cool
Imprecise calculation can be much faster (and “good enough”)
This is the idea behind heuristics
TSP (0.75%) 1,000,000 cities, 7 months
TSP (3.5%) 1,000,000 cities, 3.5 hours

3. Fuzzy vs. Crisp Sets Crisp set A Fuzzy set A
Each item x is an element of A (100%) or is not (0%)
Fuzzy set A
Each item x has degree of membership in the set A
An item can have non-zero membership in both A and ~A

4. Membership function
Horizontal axis: all potential members of the set (sorted, if possible)
Vertical axis: membership value from 0 (not at all) to 1 (totally)
hot
cold
warm
0.8
0.2
65F
Temperature

5. Linguistic Values
Cold, Warm and Hot are linguistic values, represented by overlapping membership functions
We can also define linguistic modifiers
Very A = membership in A squared
Somewhat hot = sqrt (membership in A)
Not A = 1-A
So, “not very A” = 1-A2

6. Fuzzy Logic Operations
AND
X cannot have more membership in the set (A and B) than it does in A
Therefore, membership in (A and B) = min (membership in A, membership in B) OR
X must have at least as much membership in (A or B) as in A
Therefore, membership in (A or B) = max (membership in A, membership in B)

7. Fuzzy Rules The basic fuzzy rule form
If x is A then y is B
First, determine membership of x in A
Then use that value to limit the degree to which y is B

8. Example: Adjusting a Bath/Shower
Rules
If temperature is cold, then add-hot-water = high
If temperature is warm, then add-hot-water = med
If temperature is cold, then add-cold-water = low
If temperature is warm, then add-cold-water = low

9. Example: Membership functions
cold
warm
hot
0.8
0.2
65F
Temperature
low
med
high
Water flow

10. Resulting Membership Functions
low
med
high
Center of mass
Cold water flow (.8 low, .2 low)
low
med
high
Center of mass
Hot water flow (.8 med, .2 high)

11. More complex left sides
If X is A and Y is B, then Z is C
Take AND (min) of A and B membership values to limit C
If X is A and Y is not B …
Take min (A, 1-B) membership values to limit C

12. Designing a Fuzzy Rule-based System
Develop linguistic variables for each input and output variable
Express a set of rules in terms of variables
One rule for each input,output combination
Test and adjust
Change rules
Change linguistic variables
Often done using neural net, genetic algorithm, etc.

13. Operating a Fuzzy Rule Based System
Fuzzify inputs
Numerical input -> set of linguistic memberships
Apply rules
Find the degree to which the left side applies
Limit the m.f. of the right side at that value
OR (union) m.f.s of all applicable rules
Defuzzify output
Find the center of mass of the resulting fuzzy set, translate that to a numerical output (if needed)

14. Rules as Fuzzy Graphs
If temperature is cold, then add-hot-water = high
If temperature is warm, then add-hot-water = med
If temperature is hot, then add-hot-water = low
hot
warm
cold
high
low
med

15. Advantages of Fuzzy Rules
Rules are expressed in (close to) natural language
Fewer rules are needed because of interpolation
No precise thresholds
Precise thresholds tend to make systems brittle

16. Fuzzy Clustering
If you have data that you want to classify, you are essentially specifying subsets of your data.
Crisp clustering = crisp sets. Each item in (at most) one cluster.
Fuzzy clustering = fuzzy sets. Items can belong to multiple clusters (with membership values between 0 and 1)
Typically, membership values add up to 1

17. K Means Clustering (Crisp)
Choose K random cluster centers
Repeat
For each data point (feature vector)
Find the closest cluster
Assign point to cluster
For each cluster
Recompute the center (avg. data pt)
Until clusters don’t change

18. Fuzzy C-Means Clustering
Same as fuzzy K-means except:
Point has (0-1) membership in cluster
Membership computed based on distance from center
Center computed based on weighted average of cluster members (weight by membership)
“Borderline points” belong to multiple clusters, but don’t pull the centers much
Overlapping clusters make sense