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Library of Congress Cataloging‐in‐Publication Data
Name: Peckol, James K., author. | John Wiley & Sons, publisher.
Title: Introduction to fuzzy logic / James K. Peckol.
Description: Hoboken, NJ : Wiley, 2021. | Includes bibliographical references and index.
Identifiers: LCCN 2021011123 (print) | LCCN 2021011124 (ebook) | ISBN 9781119772613 (cloth) | ISBN 9781119772620 (adobe pdf) | ISBN 9781119772637 (epub)
Subjects: LCSH: Fuzzy logic. | Fuzzy sets. | Logic, Symbolic and mathematical.
Classification: LCC QA9.64 .P43 2021 (print) | LCC QA9.64 (ebook) | DDC 511.3/13–dc23
LC record available at https://lccn.loc.gov/2021011123 LC ebook record available at https://lccn.loc.gov/2021011124
Cover Design: Wiley
Cover Image: © Rasi Bhadramani/iStock/Getty Images
Dedication
To my family: Near and Extended, Close and Distant,Present and Departed, So Similar,So Different, So Known, So Surprising …especially to our youngest brother Karl,taken from us out of season during the last voyageof the Edmund Fitzgerald.
Preface
Starting to Think Fuzzy and Beyond
Let's begin with these questions: “Exactly what is fuzzy logic?” “Why is the logic called fuzzy?” “Who might use fuzzy logic?” These are very good questions. People may have heard something about fuzzy logic and other kinds of logic but may not be quite sure what these terms mean or quite understand the applications.
Does fuzzy logic mean that someone's comment in a discussion is very confused? Let's try to answer that question and several of the other more common ones over the course of this text by starting with some simple fuzzy examples.
Our daily language is often routinely fuzzy; yet most of the time we easily understand it. Let's start by looking at some familiar expressions from our everyday exchanges.
Where did you park the car?
I parked up close to the front door of the building.
Please put the box in the trunk of the car.
I can't lift it. It's very heavy.
Are we close to the city yet?
We're roughly about thirty minutes away.
Is that shower warm?
It's very, very hot.
Is he tall?
Yes, he's very, very tall.
Is she smart?
Trust me, she's incredibly smart.
Each of the responses to the questions above is somewhat vague and imprecise yet, for the most part, each provides a reasonable answer that is probably well understood. Each expression in italics is called a fuzzy
linguistic variable rather than a crisp real number or a simple “yes” or “no.” The expressions give a high‐level view of fuzzy logic or fuzzy reasoning. Accompanying such reasoning we also find threshold logic and perceptrons, which model the brain.
In daily life, we find that there are two kinds of imprecision: statistical and nonstatistical. Statistical imprecision is that which arises from such events as the outcome of a coin toss or card game. Nonstatistical imprecision, on the other hand, is that which we find in expressions such as “We're roughly about thirty minutes away.” This latter type of imprecision is what we call fuzzy.
Children learn to understand and to manipulate such instructions at an early age. They quite easily understand phrases such as “Be home by around 5:00.” Perhaps children understand too well. They are adept at turning such a fuzzy expression into one that is also fuzzy. When they arrive home shortly after 6:00, they argue that 6:00 is about 5:00.
As we note, humans are quite facile at understanding fuzzy expressions and linguistic variables. For a computer, however, the opposite is true. With fuzzy logic, threshold logic, and perceptrons, increasingly both computer hardware and software are evolving to more challenging and interesting areas of logic such as neural networks, machine learning, and artificial intelligence.
Despite its amusing and seemingly contradictory name, fuzzy logic is not a logic that is fuzzy. On the contrary, fuzzy logic is a way of capturing the vagueness and imprecision that are so common in everyday human language. This capturing of vagueness and imprecision is also found in threshold logic and has significant application in artificial neurons called perceptrons. Capturing and representing the vagueness and imprecision of everyday language in terms that a computer can understand and work with is one of the objectives of fuzzy logic.
The computers we are all so familiar with operate using classical or crisp logic. Classical logic, around since Aristotle, divides the world into precise, nonoverlapping groups such as: yes–no, up–down, true–false, black–white, etc. Like a light bulb that can only be on or off, a classical logic statement can only be true or false. Those of you who have just said, “Wait a minute, what if the light's on a dimmer?” have just taken the first step to understanding fuzzy logic, threshold logic, and perceptrons. Like the light on a dimmer, a fuzzy logic statement can also be completely true or completely false, but it can also be partially true or partially false.
Fuzzy logic is simply a flexible variation and extension of classical logic. Fuzzy logic can represent statements that are completely true or false, and it can also represent those that are partially true. Classical logic lives in a black‐and‐white world. Fuzzy logic, threshold logic, and perceptrons, like humans, admit shades of gray. This ability to represent degrees of truth makes such tools very powerful for representing vague or imprecise ideas. We can now say, for example, that the tolerance on one capacitor is tighter than that on another or one program runs faster than another and not be concerned about specific values.
Organizing the Book
It is often all too easy to hack together a one‐off crisp logic application that appears to work. Trying to replicate a million or more copies of such a design (with elastic timing constraints, variable path impedance, or flexible data values) very quickly runs into the real‐world gremlins that are waiting for us. A solid, secure, robust, reliable design must always
