Fuzzy Systems Design

Why do we teach fuzzy systems?

Because the holy grail of technology is autonomous systems.

The five capabilities for autonomous systems are:

1. Sensing
2. Reasoning & Decision-Making
3. Learning
4. Memory & Recall
5. Response / Actuation

Fuzzy methods provide an intuitive, accessible and highly effective method for building automated reasoning system.  These include expert systems, classifiers & pattern recognisers, recommendation systems, control systems of all kinds, business systems, and linguistic interfaces.  This was the 'AI' and machine learning of the 1980s and 1990s and was so successful that articles are no longer written about it - the technology is considered mainstream.

However, it is still not taught until graduate school, and typically only within specialised electrical engineering courses.

Can young learners engage with fuzzy logic?

Absolutely.  Children as young as 12 years old can understand and appreciate fuzzy logic and fuzzy design and get very excited at being able to codify the way they think in a way that enables a computer to make decisions that follow these guidelines.

The basic idea

Sets & membership functions.

Membership functions indicate degree (or grade) of membership in a so-called 'fuzzy set'.

The membership functions extend the usual (crisp) notion of set theory to fuzzy set theory, in which all the rest is identical: union ('or') becomes join (max), intersection ('and') becomes meet (min), complement is 1-, and all the usual properties of the operations hold including De Morgan laws.

Membership functions can be formulated using intuition, statistics, surveys/polling, experimentation (modeling), theoretical closed-form expressions, or built up from logical/set operations or fuzzy inference rules.

Relations

A fuzzy relation indicates strength of relation between all elements of a Cartesian product.  It can be formed using min ('AND') rule between the Cartesian product of two fuzzy sets, or using similarity methods from data science that provide the strength of similarity between any two elements of the Cartesian product.

Fuzzy relations can be composed using max-min or max-prod composition rules.