Combs Method: Fundamentals and Applications

By Fouad Sabry

What Is Combs Method

In 1997, William E. Combs described a method for writing fuzzy logic rules that he called the Combs method. This method is a rule base reduction method. Its purpose is to thwart the combinatorial explosion that might occur with fuzzy logic rules.

How You Will Benefit

(I) Insights, and validations about the following topics:

Chapter 1: Combs method

Chapter 2: Finite-state machine

Chapter 3: Modus ponens

Chapter 4: Relational model

Chapter 5: Fuzzy control system

Chapter 6: Fuzzy logic

Chapter 7: Negation

Chapter 8: Sequent calculus

Chapter 9: List of rules of inference

Chapter 10: Double negation

(II) Answering the public top questions about combs method.

(III) Real world examples for the usage of combs method in many fields.

(IV) 17 appendices to explain, briefly, 266 emerging technologies in each industry to have 360-degree full understanding of combs method' technologies.

Who This Book Is For

Professionals, undergraduate and graduate students, enthusiasts, hobbyists, and those who want to go beyond basic knowledge or information for any kind of combs method.

• Language: English
• Publisher: One Billion Knowledgeable
• Release date: Jun 25, 2023

Book preview

Chapter 1: Combs method

William E. Combs described the Combs approach in 1997 as a means of condensing the number of rules needed to implement fuzzy logic. The goal of this system is to limit the number of possible permutations of fuzzy logic rules.

The Combs method takes advantage of the logical equality

{\displaystyle ((p\land q)\Rightarrow r)\iff ((p\Rightarrow r)\lor (q\Rightarrow r))}

.

Truth tables are the simplest tool for demonstrating a theorem of equality:

Let's pretend N variables are being considered simultaneously by a fuzzy system, each of which is contained in some subset of S.

The number of rules necessary to cover all the cases in a traditional fuzzy system is {\displaystyle S^{N}} , whereas the Combs method would need only {\displaystyle S\times N} rules.

For example, To get a single result from five inputs, five sets, and five variables,, In a standard system, covering all possible scenarios would necessitate 3125 rules, whereas only 25 rules are needed for the Combs approach, slowing down the system's combinatorial explosion as new inputs or sets are introduced.

The Combs technique will be the main topic of discussion here. See fuzzy logic and fuzzy associative matrix for more on the conventional method of rule formation.

Let's pretend for a moment that we were developing a friendly-ness meter for an artificial personality in a real-time strategy game. One's own feelings of apprehension, confidence, and affection toward the other would be taken into account. This is an example of how the Combs system's rules might look::

This table can be interpreted as:

[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]

Since the output of the table is rather predictable, it can be rewritten as:

The last row of the table contains the corresponding output for each column. The system's output is obtained by arithmetically averaging the results of each rule producing that outcome. To determine the computer's Enemy rating toward the player, for instance, we would average the computer's Fearless, Distrustful, and Hateful ratings for the player. Once the three averages are calculated, the result can be defuzzed in the usual way.

{End Chapter 1}

Chapter 2: Finite-state machine

A mathematical model of computation, often known as a finite-state machine (FSM), finite-state automaton (FSA, plural: automata), or just a state machine. A state of this hypothetical machine is strictly one of a small set of possible possibilities. It is possible for the FSM to transition from one state to another in response to certain inputs. Every non-deterministic finite-state machine is equivalent to a deterministic one that can be built.

Many technologies in modern civilization behave like state machines because they follow a set of instructions that are executed in response to a series of events. Examples include combination locks, which can only be opened by entering a specific sequence of numbers, and vending machines, which only dispense items when the correct combination of coins is deposited; elevators, whose sequence of stops is determined by riders' requests for specific floors; traffic lights, whose sequence changes when cars are waiting at the light; and so on.

When compared to other models of computation, such the Turing computer, the finite-state machine is inadequate. Because of this difference in processing capability, there are computations that a Turing machine can perform but an FSM cannot. This is due to the fact that the number of states in an FSM equals the upper bound on its storage capacity. In terms of processing capacity, a finite-state machine is equivalent to a restricted Turing machine in which the head is limited to read operations and always moves from left to right. The area of automata theory encompasses the study of FSMs.

Turnstiles are a good example of the kinds of simple mechanisms that can be depicted using state machines. A turnstile is a gate with three spinning arms at waist height, one across the opening. These gates are commonly found in subways and at amusement parks to regulate who can enter certain areas. The arms are initially locked in place, preventing customers from passing through the entrance. A single client can pass through the turnstile at a time if they deposit a coin or token into the appropriate slot. The arms will remain locked until another coin is placed after the client passes through.

The turnstile, if viewed as a state machine, can be in either the Locked or Unlocked state. You can change its status by doing one of two things: dropping a coin into the slot (coin) or pushing the arm (push). Once the arm has entered the locked condition, further attempts to move it by pushing on it will have no impact. Providing the machine with a coin as an input causes it to transition from the Locked state to the Unlocked state. If a machine is already in the unlocked condition and further coins are inserted, those coins will not be used to unlock the machine. A customer's push through the arms triggers a return to the Locked state.

A state-transition table can be used to depict the turnstile state machine, detailing the various states, the transitions between them (determined by the inputs), and the outputs achieved by the machine:

A state diagram, or directed graph, can also be used to illustrate the turnstile state machine (above). A node for each country in the map (circle). The changes from one condition to another are represented by edges (arrows). The input that makes the indicated change is described next to each arrow. Coin input in the Unlocked state, for example, is represented by a circular arrow that loops back around to the Unlocked state. Initial condition is shown by an arrow from the black dot into the Locked node.

A system is said to be in state when it is in the requisite waiting for transition state. A transition is a sequence of operations to be carried out when a certain condition is met or an event occurs. If you have your audio system set to the radio state and press the next button, you'll switch to the next station. When in the CD state, the next stimulus will cause the system to advance to the next song. Varied states have different responses to the same stimuli.

Actions can be linked to states in some models of finite-state machines:

a procedure carried out when entering the state; action taken as part of leaving a state.

There are several variations of state-transition tables in use. Below is the most popular illustration: the next state (represented by the combination of the current state (represented by B) and input (represented by Y)) (e.g. C). Information on the full activity cannot be added to the table except through footnotes, which provide only a partial picture. Using state tables, we may define an FSM that includes all the details of an activity (see also virtual finite-state machine).

State machines can be described using a notation in the Unified Modeling Language. UML state machines improve upon and preserve the main benefits of conventional finite-state machines. UML state machines expand the idea of actions by introducing novel ideas such as orthogonal areas and stacked states. Mealy and Moore machine properties can be found in UML state machines. Like Mealy machines, they allow for state-dependent actions to be taken in response to events, and like Moore machines, they also allow for state-dependent entry and exit actions.

Particulars & Detailed Description The ITU-standard language makes use of pictorial symbols