Combs method
Encyclopedia
The Combs method is a method of writing fuzzy logic
rules described by William E. Combs in 1997. It is designed to prevent combinatorial explosion
in fuzzy logic rules.
The Combs method takes advantage of the logical equality
.
, whereas the Combs method would need only 
rules. For example, if we have five sets and five variables to consider to produce one output, covering all the cases would require 3125 rules in a traditional system, while the Combs method would require only 25 rules, taming the combinatorial explosion
that occurs when more inputs or more sets are added to the system.
This article will focus on the Combs method itself. To learn more about the way rules are traditionally formed, see fuzzy logic
and fuzzy associative matrix
.
The table translates to:
[ IF Fear IS Unafraid THEN Friendship IS Enemies OR
IF Fear IS ModerateFear THEN Friendship IS Neutral OR
IF Fear IS Afraid THEN Friendship IS GoodFriends]
OR
[ IF Trust IS Distrusting THEN Friendship IS Enemies OR
IF Trust IS ModerateTrust THEN Friendship IS Neutral OR
IF Trust IS Trusting THEN Friendship IS GoodFriends]
OR
[ IF Love IS Unloving THEN Friendship IS Enemies OR
IF Love IS ModerateLove THEN Friendship IS Neutral OR
IF Love IS Loving THEN Friendship IS GoodFriends]
In this case, because the table follows a straightforward pattern in the output, it could be rewritten as:
Each column of the table maps to the output provided in the last row. To obtain the output of the system, we just average the outputs of each rule for that output. For example, to calculate how much the computer is Enemies with the player, we take the average of how much the computer is Unafraid, Distrusting, and Unloving of the player. When all three averages are obtained, the result can then be defuzzified
by any of the traditional means.
Fuzzy logic
Fuzzy logic is a form of many-valued logic; it deals with reasoning that is approximate rather than fixed and exact. In contrast with traditional logic theory, where binary sets have two-valued logic: true or false, fuzzy logic variables may have a truth value that ranges in degree between 0 and 1...
rules described by William E. Combs in 1997. It is designed to prevent combinatorial explosion
Combinatorial explosion
In administration and computing, a combinatorial explosion is the rapidly accelerating increase in lines of communication as organizations are added in a process...
in fuzzy logic rules.
The Combs method takes advantage of the logical equality
.Equality proof
The simplest proof of given equality involves usage of truth tables:  |  |  |  |  |  |  |
|---|---|---|---|---|---|---|
| T | T | T | T | T | T | T |
| T | T | N | F | F | F | F |
| T | F | T | T | T | T | T |
| T | F | F | T | F | T | T |
| F | T | T | T | T | T | T |
| F | T | F | T | T | F | T |
| F | F | T | T | T | T | T |
| F | F | F | T | T | T | T |
Combinatorial explosion
Suppose we have a fuzzy system that considers N variables at a time, each of which can fit into at least one of S sets. The number of rules necessary to cover all the cases in a traditional fuzzy system is, whereas the Combs method would need only rules. For example, if we have five sets and five variables to consider to produce one output, covering all the cases would require 3125 rules in a traditional system, while the Combs method would require only 25 rules, taming the combinatorial explosionCombinatorial explosion
In administration and computing, a combinatorial explosion is the rapidly accelerating increase in lines of communication as organizations are added in a process...
that occurs when more inputs or more sets are added to the system.
This article will focus on the Combs method itself. To learn more about the way rules are traditionally formed, see fuzzy logic
Fuzzy logic
Fuzzy logic is a form of many-valued logic; it deals with reasoning that is approximate rather than fixed and exact. In contrast with traditional logic theory, where binary sets have two-valued logic: true or false, fuzzy logic variables may have a truth value that ranges in degree between 0 and 1...
and fuzzy associative matrix
Fuzzy associative matrix
A fuzzy associative matrix expresses fuzzy logic rules in tabular form. These rules usually take two variables as input, mapping cleanly to a two-dimensional matrix, although theoretically a matrix of any number of dimensions is possible....
.
Example
Suppose we were designing an artificial personality system that determined how friendly the personality is supposed to be towards a person in a strategic video game. The personality would consider its own fear, trust, and love in the other person. A set of rules in the Combs system might look like this:| Fear | Unafraid THEN Enemies | Moderate Fear THEN Neutral | Afraid THEN Good Friends |
|---|---|---|---|
| Trust | Distrusting THEN Enemies | Moderate Trust THEN Neutral | Trusting THEN Good Friends |
| Love | Unloving THEN Enemies | Moderate Love THEN Neutral | Loving THEN Good Friends |
The table translates to:
IF Fear IS ModerateFear THEN Friendship IS Neutral OR
IF Fear IS Afraid THEN Friendship IS GoodFriends
OR
IF Trust IS ModerateTrust THEN Friendship IS Neutral OR
IF Trust IS Trusting THEN Friendship IS GoodFriends
OR
IF Love IS ModerateLove THEN Friendship IS Neutral OR
IF Love IS Loving THEN Friendship IS GoodFriends
In this case, because the table follows a straightforward pattern in the output, it could be rewritten as:
| Fear | Unafraid | Moderate Fear | Afraid |
|---|---|---|---|
| Trust | Distrusting | Moderate Trust | Trusting |
| Love | Unloving | Moderate Love | Loving |
| Friendship | Enemies | Neutral | Good Friends |
Each column of the table maps to the output provided in the last row. To obtain the output of the system, we just average the outputs of each rule for that output. For example, to calculate how much the computer is Enemies with the player, we take the average of how much the computer is Unafraid, Distrusting, and Unloving of the player. When all three averages are obtained, the result can then be defuzzified
Defuzzification
Defuzzification is the process of producing a quantifiable result in fuzzy logic, given fuzzy sets and corresponding membership degrees. It is typically needed in fuzzy control systems. These will have a number of rules that transform a number of variables into a fuzzy result, that is, the result...
by any of the traditional means.