Fuzzy Logic Systems

The fuzzy logic module provides implementations of fuzzy sets, membership functions, and fuzzy inference systems for reasoning under uncertainty. Fuzzy logic extends classical Boolean logic to handle partial truth values between 0 and 1.

Overview

Fuzzy logic is particularly useful for:

Control systems with uncertain or imprecise inputs
Decision-making under vagueness
Rule-based expert systems
Combining neural networks with symbolic reasoning

Fuzzy Sets

Fuzzy sets define membership functions that map input values to degrees of membership in [0, 1].

Triangular Fuzzy Set

from neurenix.fuzzy import TriangularSet
import neurenix as nx

# Define a triangular fuzzy set
fuzzy_set = TriangularSet(a=0.0, b=5.0, c=10.0)

# Evaluate membership
x = nx.linspace(0, 10, 100)
membership = fuzzy_set(x)

Parameters:

a (float): Left foot of triangle
b (float): Peak of triangle
c (float): Right foot of triangle

Trapezoidal Fuzzy Set

from neurenix.fuzzy import TrapezoidalSet

fuzzy_set = TrapezoidalSet(a=0.0, b=3.0, c=7.0, d=10.0)
membership = fuzzy_set(x)

Gaussian Fuzzy Set

from neurenix.fuzzy import GaussianSet

fuzzy_set = GaussianSet(mean=5.0, std=2.0)
membership = fuzzy_set(x)

Bell-Shaped Fuzzy Set

from neurenix.fuzzy import BellSet

fuzzy_set = BellSet(a=2.0, b=4.0, c=5.0)
membership = fuzzy_set(x)

Sigmoid Fuzzy Set

from neurenix.fuzzy import SigmoidSet

fuzzy_set = SigmoidSet(a=1.0, c=5.0)
membership = fuzzy_set(x)

Linguistic Variables

Linguistic variables represent fuzzy concepts with natural language terms.

from neurenix.fuzzy import LinguisticVariable, TriangularSet
import neurenix as nx

# Define temperature variable
temperature = LinguisticVariable(
    name="temperature",
    universe=nx.linspace(0, 100, 1000)
)

# Add fuzzy sets for linguistic terms
temperature.add_set("cold", TriangularSet(0, 0, 30))
temperature.add_set("warm", TriangularSet(20, 40, 60))
temperature.add_set("hot", TriangularSet(50, 100, 100))

# Fuzzify a crisp value
fuzzy_values = temperature.fuzzify(25.0)
print(fuzzy_values)  # {'cold': 0.17, 'warm': 0.25, 'hot': 0.0}

Fuzzy Rules

Define IF-THEN rules for fuzzy inference.

from neurenix.fuzzy import FuzzyRule, FuzzyRuleSet

# Create rules
rule1 = FuzzyRule(
    antecedents=[("temperature", "cold")],
    consequent=("heater", "high")
)

rule2 = FuzzyRule(
    antecedents=[("temperature", "warm")],
    consequent=("heater", "medium")
)

rule3 = FuzzyRule(
    antecedents=[("temperature", "hot")],
    consequent=("heater", "off")
)

# Create rule set
rule_set = FuzzyRuleSet()
rule_set.add_rule(rule1)
rule_set.add_rule(rule2)
rule_set.add_rule(rule3)

Fuzzy Inference Systems

Mamdani Inference System

The most common fuzzy inference method, using fuzzy sets for both inputs and outputs.

from neurenix.fuzzy import MamdaniSystem, LinguisticVariable, TriangularSet
import neurenix as nx

# Create Mamdani system
fis = MamdaniSystem(
    name="Temperature Control",
    defuzzification_method="centroid"
)

# Define input variable
temperature = LinguisticVariable(
    name="temperature",
    universe=nx.linspace(0, 100, 1000)
)
temperature.add_set("cold", TriangularSet(0, 0, 30))
temperature.add_set("warm", TriangularSet(20, 50, 80))
temperature.add_set("hot", TriangularSet(70, 100, 100))

fis.add_input_variable(temperature)

# Define output variable
heater = LinguisticVariable(
    name="heater",
    universe=nx.linspace(0, 100, 1000)
)
heater.add_set("off", TriangularSet(0, 0, 20))
heater.add_set("medium", TriangularSet(30, 50, 70))
heater.add_set("high", TriangularSet(80, 100, 100))

fis.add_output_variable(heater)

# Add rules
from neurenix.fuzzy import FuzzyRule

fis.add_rule(FuzzyRule(
    antecedents=[("temperature", "cold")],
    consequent=("heater", "high")
))

fis.add_rule(FuzzyRule(
    antecedents=[("temperature", "warm")],
    consequent=("heater", "medium")
))

fis.add_rule(FuzzyRule(
    antecedents=[("temperature", "hot")],
    consequent=("heater", "off")
))

# Evaluate
inputs = {"temperature": 25.0}
outputs = fis.evaluate(inputs)
print(f"Heater setting: {outputs['heater']:.2f}%")

Sugeno Inference System

Uses crisp output functions instead of fuzzy sets.

from neurenix.fuzzy import SugenoSystem

# Create Sugeno system
fis = SugenoSystem(name="Tip Calculator")

# Add input variables
fis.add_input_variable(service)  # Defined as above
fis.add_input_variable(food)     # Defined as above

# Add output variable
tip = LinguisticVariable(name="tip", universe=nx.linspace(0, 30, 1000))
fis.add_output_variable(tip)

# Define output functions (crisp values or linear functions)
def low_tip(inputs):
    return 5.0

def medium_tip(inputs):
    return 15.0

def high_tip(inputs):
    return 0.2 * inputs["service"] + 0.1 * inputs["food"]

fis.add_output_function("tip", "low", low_tip)
fis.add_output_function("tip", "medium", medium_tip)
fis.add_output_function("tip", "high", high_tip)

# Add rules and evaluate
fis.add_rule(FuzzyRule(
    antecedents=[("service", "poor"), ("food", "poor")],
    consequent=("tip", "low")
))

outputs = fis.evaluate({"service": 3, "food": 8})

Tsukamoto Inference System

Uses monotonic membership functions.

from neurenix.fuzzy import TsukamotoSystem

fis = TsukamotoSystem(name="Speed Control")

# Define variables with monotonic membership functions
# Add rules and evaluate
outputs = fis.evaluate(inputs)

Defuzzification Methods

Convert fuzzy outputs back to crisp values.

from neurenix.fuzzy import (
    centroid,
    bisector,
    mean_of_maximum,
    smallest_of_maximum,
    largest_of_maximum
)
import neurenix as nx

# Sample fuzzy output
universe = nx.linspace(0, 100, 1000)
membership = fuzzy_set(universe)

# Centroid (center of area)
value = centroid(universe, membership)

# Bisector (divides area in half)
value = bisector(universe, membership)

# Mean of maximum (average of max membership)
value = mean_of_maximum(universe, membership)

Example: Fuzzy Controller

import neurenix as nx
from neurenix.fuzzy import (
    MamdaniSystem,
    LinguisticVariable,
    TriangularSet,
    FuzzyRule
)

# Create fuzzy controller for inverted pendulum
fis = MamdaniSystem(name="Pendulum Controller")

# Define angle input
angle = LinguisticVariable("angle", nx.linspace(-90, 90, 1000))
angle.add_set("left", TriangularSet(-90, -90, 0))
angle.add_set("center", TriangularSet(-30, 0, 30))
angle.add_set("right", TriangularSet(0, 90, 90))
fis.add_input_variable(angle)

# Define angular velocity input
velocity = LinguisticVariable("velocity", nx.linspace(-100, 100, 1000))
velocity.add_set("negative", TriangularSet(-100, -100, 0))
velocity.add_set("zero", TriangularSet(-20, 0, 20))
velocity.add_set("positive", TriangularSet(0, 100, 100))
fis.add_input_variable(velocity)

# Define force output
force = LinguisticVariable("force", nx.linspace(-10, 10, 1000))
force.add_set("left_push", TriangularSet(-10, -10, 0))
force.add_set("no_push", TriangularSet(-2, 0, 2))
force.add_set("right_push", TriangularSet(0, 10, 10))
fis.add_output_variable(force)

# Define control rules
rules = [
    FuzzyRule([("angle", "left"), ("velocity", "negative")], ("force", "right_push")),
    FuzzyRule([("angle", "left"), ("velocity", "zero")], ("force", "right_push")),
    FuzzyRule([("angle", "center"), ("velocity", "zero")], ("force", "no_push")),
    FuzzyRule([("angle", "right"), ("velocity", "positive")], ("force", "left_push")),
]

for rule in rules:
    fis.add_rule(rule)

# Control loop
for t in range(1000):
    current_angle = get_angle()
    current_velocity = get_velocity()
    
    output = fis.evaluate({
        "angle": current_angle,
        "velocity": current_velocity
    })
    
    apply_force(output["force"])

Example: Decision Making

from neurenix.fuzzy import MamdaniSystem, LinguisticVariable, TrapezoidalSet

# Investment decision system
fis = MamdaniSystem(name="Investment Advisor")

# Risk tolerance
risk = LinguisticVariable("risk", nx.linspace(0, 10, 1000))
risk.add_set("low", TrapezoidalSet(0, 0, 2, 4))
risk.add_set("medium", TrapezoidalSet(3, 5, 5, 7))
risk.add_set("high", TrapezoidalSet(6, 8, 10, 10))
fis.add_input_variable(risk)

# Market condition
market = LinguisticVariable("market", nx.linspace(0, 10, 1000))
market.add_set("bear", TrapezoidalSet(0, 0, 3, 5))
market.add_set("neutral", TrapezoidalSet(4, 5, 5, 6))
market.add_set("bull", TrapezoidalSet(5, 7, 10, 10))
fis.add_input_variable(market)

# Investment allocation
allocation = LinguisticVariable("allocation", nx.linspace(0, 100, 1000))
allocation.add_set("conservative", TrapezoidalSet(0, 0, 20, 40))
allocation.add_set("balanced", TrapezoidalSet(30, 50, 50, 70))
allocation.add_set("aggressive", TrapezoidalSet(60, 80, 100, 100))
fis.add_output_variable(allocation)

# Rules
fis.add_rule(FuzzyRule([("risk", "low"), ("market", "bear")], ("allocation", "conservative")))
fis.add_rule(FuzzyRule([("risk", "high"), ("market", "bull")], ("allocation", "aggressive")))
# ... more rules

# Get recommendation
result = fis.evaluate({"risk": 7.0, "market": 8.0})
print(f"Recommended allocation: {result['allocation']:.1f}% stocks")

Integration with Neural Networks

import neurenix as nx
from neurenix.fuzzy import MamdaniSystem
from neurenix.nn import Module

class NeuroFuzzySystem(Module):
    def __init__(self, fis: MamdaniSystem):
        super().__init__()
        self.fis = fis
        self.neural_preprocessor = nx.nn.Sequential(
            nx.nn.Linear(10, 32),
            nx.nn.ReLU(),
            nx.nn.Linear(32, 2)  # Outputs for fuzzy system
        )
    
    def forward(self, x):
        # Neural preprocessing
        features = self.neural_preprocessor(x)
        
        # Fuzzy inference
        outputs = []
        for sample in features:
            result = self.fis.evaluate({
                "input1": sample[0].item(),
                "input2": sample[1].item()
            })
            outputs.append(result["output"])
        
        return nx.tensor(outputs)

Best Practices

Membership Functions: Choose appropriate shapes (triangular for simplicity, Gaussian for smoothness)
Universe of Discourse: Define sufficient resolution for accurate calculations
Rule Coverage: Ensure rules cover all important input combinations
Defuzzification: Use centroid for balanced results, mean-of-maximum for responsive control
Validation: Test with known inputs to verify expected behavior

​Fuzzy Logic Systems

​Overview

​Fuzzy Sets

​Triangular Fuzzy Set

​Trapezoidal Fuzzy Set

​Gaussian Fuzzy Set

​Bell-Shaped Fuzzy Set

​Sigmoid Fuzzy Set

​Linguistic Variables

​Fuzzy Rules

​Fuzzy Inference Systems

​Mamdani Inference System

​Sugeno Inference System

​Tsukamoto Inference System

​Defuzzification Methods

​Example: Fuzzy Controller

​Example: Decision Making

​Integration with Neural Networks

​Best Practices

​See Also