What Fuzzy Logic Is – AI with Fuzzy Logic (Senior 6 ICT) | Ganzaa

AI with Fuzzy Logic

Lessons

Senior 6 · ICT

AI with Fuzzy Logic

5 lessons

Lesson 1 of 5

What Fuzzy Logic Is

22 min

When does a day become hot?

Ask a class in Kigali "Is today hot?" At 31 °C everyone says yes, and at 19 °C everyone says no. At 27 °C the class splits: some say "a bit", some say "not really", and nobody is wrong. Now try to program it. A conventional program needs a cut-off: if temperature >= 28: hot = True. Then 27.9 °C is "not hot" and 28.0 °C is "hot", although nobody can feel a tenth of a degree. The word hot has no sharp edge, and forcing one onto it makes the program behave strangely near the edge.

Fuzzy logic deals with this by letting a statement be true to a degree. At 27 °C, "the day is hot" can be 0.5 true. This unit shows how computers reason with such degrees, and why the idea, more than sixty years old, still runs inside rice cookers, washing machines, camera autofocus and train controllers.

Definition

Crisp set (classical set): a set that each element either belongs to or does not; membership is 0 or 1. Example: the set of students in S6.

Fuzzy set: a set in which each element has a degree of membership, written $\mu(x)$ , any number from 0 (not at all a member) to 1 (fully a member). Example: the set of hot days, in which a 27 °C day has $\mu = 0.5$ .

Fuzzy logic: a way of reasoning in which statements have degrees of truth between 0 and 1, rather than only true or false. It was introduced by Lotfi A. Zadeh in the paper "Fuzzy Sets" (1965).

Crisp and fuzzy, side by side

The crisp definition is a step: at 27 °C a day is not hot at all; one degree later it is fully hot. The fuzzy definition is a ramp: membership rises by $\frac{1}{6}$ for every degree from 24 °C to 30 °C, so

$\mu_{\text{hot}}(27) = \frac{27 - 24}{30 - 24} = \frac{3}{6} = 0.5$

A fan controlled by the crisp rule switches fully on at 28 °C and fully off at 27.9 °C, and on a day that hovers around 28 °C it keeps clicking on and off. A fan controlled by the fuzzy value can run at half speed at 27 °C and speed up smoothly as the room warms. This smoothness is the main practical reason engineers use fuzzy logic in control.

Fuzzy is not probability

Both use numbers from 0 to 1, so they are easily confused, but they answer different questions.

	Degree of membership (fuzzy)	Probability
Question it answers	How far does this thing fit a vague description?	How likely is an uncertain event?
"0.7" means	The day is hot to degree 0.7, now, with nothing unknown	There is a 70% chance it will be hot; it will turn out fully hot or fully not
After you look	Still 0.7: the vagueness is in the word	Becomes 0 or 1: the uncertainty disappears
Must values add to 1?	No: a day can be 0.5 hot and 0.6 humid	Yes, over all possible outcomes

A water example makes the difference sharp. Bottle A is known to be 0.9 a member of "safe to drink": the water is slightly cloudy but harmless. Bottle B has a probability of 0.9 of being safe: nine bottles in ten from that source are clean, and one in ten is contaminated. Bottle A will never make you seriously ill; bottle B might.

Where fuzzy logic sits in AI

Artificial intelligence has two broad traditions. Knowledge-based (symbolic) AI writes human knowledge as rules that a program applies. Machine learning lets a program find patterns in data. Fuzzy logic belongs to the first tradition: an expert writes rules in words, such as "if the soil is dry and the day is hot, water for a long time", and fuzzy logic turns those words into numbers a machine can compute with. It does not learn by itself, although it can be combined with learning methods (for example, neuro-fuzzy systems tune their membership functions from data).

Year	Milestone
1965	Lotfi A. Zadeh, a professor at the University of California, Berkeley, publishes "Fuzzy Sets" in the journal Information and Control.
1975	Ebrahim Mamdani and Sedrak Assilian in London control a laboratory steam engine with fuzzy rules: the first fuzzy controller.
1987	The Namboku Line of the Sendai subway in Japan opens with fuzzy control of acceleration and braking, built by Hitachi.
Late 1980s onwards	Fuzzy control appears in consumer products: washing machines, rice cookers, air conditioners and camera autofocus.

Computerphile: Fuzzy Logic. Watch for the difference between a crisp and a fuzzy set, and for how the degrees of membership are drawn as graphs; compare them with the "hot day" graph above.

Worked example 1: crisp and fuzzy "full"

A school's water tank is "full" in a crisp program when the level is at least 90%. A fuzzy version says membership in "full" rises from 0 at 70% to 1 at 95%. Find both values at 80% and at 92%.

Crisp at 80%: below 90%, so 0. Crisp at 92%: at least 90%, so 1.
Fuzzy at 80%: $\mu = \frac{80 - 70}{95 - 70} = \frac{10}{25} = 0.4$ .
Fuzzy at 92%: $\mu = \frac{92 - 70}{95 - 70} = \frac{22}{25} = 0.88$ .

A pump controller using the fuzzy value can slow the pump as the tank nears full instead of slamming it off at exactly 90%.

Worked example 2: a membership curve from people

Ten students each give the temperature at which a Kigali day becomes "hot" for them (illustrative data): 25, 26, 26, 27, 27, 27, 28, 28, 29, 30 °C. A simple membership value for "hot" at temperature $t$ is the fraction of students whose threshold is at most $t$ .

At 26 °C: 3 thresholds are at most 26 (25, 26, 26), so $\mu = \frac{3}{10} = 0.3$ .
At 27 °C: 6 thresholds, so $\mu = 0.6$ .
At 28 °C: 8, so $\mu = 0.8$ ; at 30 °C, all 10, so $\mu = 1$ ; at 24 °C, none, so $\mu = 0$ .

This short Python program, run with python3, does the same count for every temperature:

thresholds = [25, 26, 26, 27, 27, 27, 28, 28, 29, 30]

def mu_hot(t):
    return sum(1 for h in thresholds if h <= t) / len(thresholds)

for t in range(24, 31):
    print(t, mu_hot(t))

The values rise from 0 to 1 without a jump: a fuzzy set built from how people actually use the word.

Warning

Fuzzy does not mean careless. Fuzzy logic is exact mathematics about vague words. The membership values are precise numbers, the rules are written down, and the result is the same every time for the same inputs. What is "fuzzy" is the boundary of the word, not the calculation.

Tip

Two questions tell them apart. "How well does it fit the word?" is fuzzy membership. "How likely is it to happen?" is probability. If the number would turn into a plain 0 or 1 once you look, it was a probability.

✦

Hands-on: measure a vague word

Choose a vague word with a measurable quantity: a tall student (height in cm), a long journey to school (minutes), a heavy school bag (kg).
Ask at least ten classmates for the value at which the word starts to apply for them. Record their answers.
Type them into the Python program above, in place of thresholds, and change the range to suit your quantity. Run it.
Draw the membership curve on graph paper: the quantity across, $μ \mu$ up.
Draw the crisp version that uses the median answer as a cut-off, on the same axes.

Check: your curve rises from 0 to 1, and you can read off the membership at any value and explain why the crisp line gives a strange answer just below the cut-off.

Practice

Define a fuzzy set and a degree of membership.
Give two reasons why a crisp cut-off is a poor model of the word "hot".
Membership in "hot" rises from 0 at 24 °C to 1 at 30 °C. Find $μ \mu$ at 25 °C, 28.5 °C and 33 °C.
Explain the difference between "the road is 0.8 congested" and "there is a probability of 0.8 that the road is congested".
Who introduced fuzzy sets, and in what year? Name one early application of fuzzy control.
Is fuzzy logic a form of machine learning? Explain.
A teacher says: "Fuzzy logic gives vague answers, so it is useless for engineering." Evaluate the statement.

Show answersHide answers

A set in which each element has a degree of membership, a number from 0 to 1 that says how fully it belongs. The degree of membership $\mu(x)$ is that number.
It jumps from not hot to hot within a tiny change (27.9 to 28.0 °C), unlike the way people use the word; and a controller using it switches abruptly and can flicker on and off near the cut-off.
$\mu(25) = \frac{1}{6} \approx 0.17$ ; $\mu(28.5) = \frac{4.5}{6} = 0.75$ ; $\mu(33) = 1$ because 33 is beyond 30.
The first says the present state fits the word "congested" to degree 0.8: the road is fairly busy. The second says that it is uncertain whether the road is congested, and that it is congested in 80% of such cases; once you look, it either is or is not.
Lotfi A. Zadeh, 1965. Examples: Mamdani and Assilian's steam engine controller (1975); the Sendai subway's train control (1987); washing machines or rice cookers.
No. Fuzzy logic applies rules written by experts in words; it does not learn from data by itself. It can be combined with learning (neuro-fuzzy systems), but on its own it is knowledge-based AI.
Wrong. Fuzzy logic computes exact numbers from written rules, and gives the same output for the same input. It models vague words precisely, and its smooth outputs are why it has been used for decades in train braking, appliances and cameras.

Sources and credits

Fuzzy logic, Wikipedia contributors, Wikipedia, licensed under CC BY-SA 4.0. Adapted by Ganzaa.
Fuzzy set, Wikipedia contributors, Wikipedia, licensed under CC BY-SA 4.0. Adapted by Ganzaa.
Lotfi A. Zadeh, Wikipedia contributors, Wikipedia, licensed under CC BY-SA 4.0. Adapted by Ganzaa.
L. A. Zadeh, "Fuzzy Sets", Information and Control, volume 8, 1965 (cited, not copied).
Video: Fuzzy Logic, Computerphile, embedded from YouTube.
The scenarios, graph, survey data (illustrative), program, worked examples, hands-on task and practice items were written and tested by Ganzaa.

This lesson adapts CC BY-SA material and is shared under CC BY-SA 4.0.

1When does a day become hot?

2Crisp and fuzzy, side by side

3Fuzzy is not probability

4Where fuzzy logic sits in AI

5Hands-on: measure a vague word

When does a day become hot?

Crisp and fuzzy, side by side

Fuzzy is not probability

Where fuzzy logic sits in AI

Hands-on: measure a vague word