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<title>Combs method</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Combs method</span></span>

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</style><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="Coombs'_method" title="Coombs' method">Coombs' method</a>.</div>

<p>The <b>Combs method</b> is a rule base reduction method of writing <a href="Fuzzy_logic" title="Fuzzy logic">fuzzy logic</a> rules described by William E. Combs in 1997. It is designed to prevent <a href="Combinatorial_explosion" title="Combinatorial explosion">combinatorial explosion</a> in fuzzy logic rules.<sup id="cite_ref-Ross2005_1-0" class="reference"><a href="#cite_note-Ross2005-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

</p><p>The Combs method takes advantage of the <a href="Propositional_logic" title="Propositional logic">logical</a> equality <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ((p\land q)\Rightarrow r)\iff ((p\Rightarrow r)\lor (q\Rightarrow r))}">

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<annotation encoding="application/x-tex">{\displaystyle ((p\land q)\Rightarrow r)\iff ((p\Rightarrow r)\lor (q\Rightarrow r))}</annotation>

</semantics>

</math></span><img src="./69469f62d62e501ab238ee0fbea467db1fde58fe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:39.575ex; height:2.843ex;" alt="{\displaystyle ((p\land q)\Rightarrow r)\iff ((p\Rightarrow r)\lor (q\Rightarrow r))}" loading="lazy"></span>.

</p>

<div class="mw-heading mw-heading2"><h2 id="Equality_proof">Equality proof</h2></div>

<p>The simplest proof of given equality involves usage of truth tables:

</p>

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<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

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</math></span><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"></span></th>

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<annotation encoding="application/x-tex">{\displaystyle q}</annotation>

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</math></span><img src="./06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy"></span></th>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r}</annotation>

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</math></span><img src="./0d1ecb613aa2984f0576f70f86650b7c2a132538.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:1.676ex;" alt="{\displaystyle r}" loading="lazy"></span></th>

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<annotation encoding="application/x-tex">{\displaystyle p\land q\Rightarrow r}</annotation>

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</math></span><img src="./50a9088e42ae635aae603697e801ee55c3915406.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:9.574ex; height:2.343ex;" alt="{\displaystyle p\land q\Rightarrow r}" loading="lazy"></span></th>

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<annotation encoding="application/x-tex">{\displaystyle p\Rightarrow r}</annotation>

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</math></span><img src="./eca2661c42392e36642f0e5d670409bfa340dc8b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.922ex; height:2.176ex;" alt="{\displaystyle p\Rightarrow r}" loading="lazy"></span></th>

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<annotation encoding="application/x-tex">{\displaystyle q\Rightarrow r}</annotation>

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</math></span><img src="./43dbd9b0f1e40896f69f4a0cdc52b99a82f934ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.732ex; height:2.176ex;" alt="{\displaystyle q\Rightarrow r}" loading="lazy"></span></th>

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<annotation encoding="application/x-tex">{\displaystyle (p\Rightarrow r)\lor (q\Rightarrow r)}</annotation>

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</math></span><img src="./17a6abae7dacab9c97e58ee5e9aa3240eb902f51.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.766ex; height:2.843ex;" alt="{\displaystyle (p\Rightarrow r)\lor (q\Rightarrow r)}" loading="lazy"></span>

</th></tr>

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<td>T

</td></tr>

<tr>

<td>F

</td></tr>

<tr>

<td>T

</td></tr>

<tr>

<td>T

</td></tr>

<tr>

<td>T

</td></tr>

<tr>

<td>T

</td></tr>

<tr>

<td>T

</td></tr>

<tr>

<td>T

</td></tr></tbody></table>

<div class="mw-heading mw-heading2"><h2 id="Combinatorial_explosion">Combinatorial explosion</h2></div>

<p>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 <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{N}}">

<msup>

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<annotation encoding="application/x-tex">{\displaystyle S^{N}}</annotation>

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</math></span><img src="./a1e9b324b2b9bb76e58fd0dc2cf9db713a8d647f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.213ex; height:2.676ex;" alt="{\displaystyle S^{N}}" loading="lazy"></span>, whereas the Combs method would need only <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\times N}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S\times N}</annotation>

</semantics>

</math></span><img src="./5c79de6159ca042567e30ab3107e986265f5f6df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.403ex; height:2.176ex;" alt="{\displaystyle S\times N}" loading="lazy"></span> 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 <a href="Combinatorial_explosion" title="Combinatorial explosion">combinatorial explosion</a> that occurs when more inputs or more sets are added to the system.

</p><p>This article will focus on the Combs method itself. To learn more about the way rules are traditionally formed, see <a href="Fuzzy_logic" title="Fuzzy logic">fuzzy logic</a> and <a href="Fuzzy_associative_matrix" title="Fuzzy associative matrix">fuzzy associative matrix</a>.

</p>

<div class="mw-heading mw-heading2"><h2 id="Example">Example</h2></div>

<p>Suppose we were designing an <a href="Artificial_personality" class="mw-redirect" title="Artificial personality">artificial personality</a> 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:

</p>

<th>Fear

</th>

<td>Unafraid THEN Enemies</td>

<td>Moderate Fear THEN Neutral</td>

<td>Afraid THEN Good Friends

</td></tr>

<tr>

<th>Trust

</th>

<td>Distrusting THEN Enemies</td>

<td>Moderate Trust THEN Neutral</td>

<td>Trusting THEN Good Friends

</td></tr>

<tr>

<th>Love

</th>

<td>Unloving THEN Enemies</td>

<td>Moderate Love THEN Neutral</td>

<td>Loving THEN Good Friends

</td></tr></tbody></table>

<p>The table translates to:

</p>

<pre>[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 ]

[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]

[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]

</pre>

<p>In this case, because the table follows a straightforward pattern in the output, it could be rewritten as:

</p>

<th>Fear

</th>

<td>Unafraid</td>

<td>Moderate Fear</td>

<td>Afraid

</td></tr>

<tr>

<th>Trust

</th>

<td>Distrusting</td>

<td>Moderate Trust</td>

<td>Trusting

</td></tr>

<tr>

<th>Love

</th>

<td>Unloving</td>

<td>Moderate Love</td>

<td>Loving

</td></tr>

<tr>

<th>Friendship

</th>

<td><i>Enemies</i></td>

<td><i>Neutral</i></td>

<td><i>Good Friends</i>

</td></tr></tbody></table>

<p>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 <a href="Defuzzification" title="Defuzzification">defuzzified</a> by any of the traditional means.

</p>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFTimothy_J._Ross2005" class="citation book cs1">Timothy J. Ross (8 April 2005). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=3zcgIKPl8L0C&pg=PA282"><i>Fuzzy Logic with Engineering Applications</i></a>. John Wiley & Sons. pp. 282–. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-470-86076-2</bdi>.</cite></span>

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<ul><li><a rel="nofollow" class="external text" href="http://gaia.ecs.csus.edu/~hellerm/EEE222/Atricles/Combs_Fuzzy_Logic/Combs_Rapid_Inference.htm">The Combs Method for Rapid Inference</a> (the original paper by William E. Combs)</li>

<li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20021212184408/http://gaia.ecs.csus.edu/~hellerm/EEE222/Atricles/Combs_Fuzzy_Logic/Combs_Rapid_Inference.htm">The Combs Method for Rapid Inference</a> (Archive of the original paper by William E. Combs)</li></ul>

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</style><div id="Logic605" style="font-size:114%;margin:0 4em"><a href="Logic" title="Logic">Logic</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div>

<ul><li><a href="History_of_logic" title="History of logic">History</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Major fields</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Logic_in_computer_science" title="Logic in computer science">Computer science</a></li>

<li><a href="Formal_semantics_(natural_language)" title="Formal semantics (natural language)">Formal semantics (natural language)</a></li>

<li><a href="Inference" title="Inference">Inference</a></li>

<li><a href="Philosophy_of_logic" title="Philosophy of logic">Philosophy of logic</a></li>

<li><a href="Formal_proof" title="Formal proof">Proof</a></li>

<li><a href="Semantics_of_logic" title="Semantics of logic">Semantics of logic</a></li>

<li><a href="Syntax_(logic)" title="Syntax (logic)">Syntax</a></li></ul>

</div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Logics</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Classical_logic" title="Classical logic">Classical</a></li>

<li><a href="Informal_logic" title="Informal logic">Informal</a>

<ul><li><a href="Critical_thinking" title="Critical thinking">Critical thinking</a></li>

<li><a href="Reason" title="Reason">Reason</a></li></ul></li>

<li><a href="Mathematical_logic" title="Mathematical logic">Mathematical</a></li>

<li><a href="Non-classical_logic" title="Non-classical logic">Non-classical</a></li>

<li><a href="Philosophical_logic" title="Philosophical logic">Philosophical</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theories</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Argumentation_theory" title="Argumentation theory">Argumentation</a></li>

<li><a href="Metalogic" title="Metalogic">Metalogic</a></li>

<li><a href="Metamathematics" title="Metamathematics">Metamathematics</a></li>

</div></td></tr></tbody></table><div>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Foundations</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Abductive_reasoning" title="Abductive reasoning">Abduction</a></li>

<li><a href="Analytic%E2%80%93synthetic_distinction" title="Analytic–synthetic distinction">Analytic and synthetic propositions</a></li>

<li><a href="Antecedent_(logic)" title="Antecedent (logic)">Antecedent</a></li>

<li><a href="Consequent" title="Consequent">Consequent</a></li>

<li><a href="Contradiction" title="Contradiction">Contradiction</a>

<ul><li><a href="Paradox" title="Paradox">Paradox</a></li>

<li><a href="Antinomy" title="Antinomy">Antinomy</a></li></ul></li>

<li><a href="Deductive_reasoning" title="Deductive reasoning">Deduction</a></li>

<li><a href="Deductive_closure" title="Deductive closure">Deductive closure</a></li>

<li><a href="Definition" title="Definition">Definition</a></li>

<li><a href="Description" title="Description">Description</a></li>

<li><a href="Dichotomy" title="Dichotomy">Dichotomy</a></li>

<li><a href="Logical_consequence" title="Logical consequence">Entailment</a>

<ul><li><a href="Linguistic_entailment" title="Linguistic entailment">Linguistic</a></li></ul></li>

<li><a href="Inductive_reasoning" title="Inductive reasoning">Induction</a></li>

<li><a href="Logical_truth" title="Logical truth">Logical truth</a></li>

<li><a href="Necessity_and_sufficiency" title="Necessity and sufficiency">Necessity and sufficiency</a></li>

<li><a href="Premise" title="Premise">Premise</a></li>

<li><a href="Probability" title="Probability">Probability</a></li>

<li><a href="Proposition" title="Proposition">Proposition</a></li>

<li><a href="Reference" title="Reference">Reference</a></li>

<li><a href="Statement_(logic)" class="mw-redirect" title="Statement (logic)">Statement</a></li>

<li><a href="Substitution_(logic)" title="Substitution (logic)">Substitution</a></li>

<li><a href="Truth" title="Truth">Truth</a></li>

<li><a href="Validity_(logic)" title="Validity (logic)">Validity</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Lists</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Index_of_logic_articles" class="mw-redirect" title="Index of logic articles">Topics</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="List_of_mathematical_logic_topics" title="List of mathematical logic topics">Mathematical logic</a></li>

<li><a href="List_of_Boolean_algebra_topics" title="List of Boolean algebra topics">Boolean algebra</a></li>

<li><a href="List_of_set_theory_topics" title="List of set theory topics">Set theory</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Other</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="List_of_logicians" title="List of logicians">Logicians</a></li>

<li><a href="List_of_rules_of_inference" title="List of rules of inference">Rules of inference</a></li>

<li><a href="List_of_paradoxes" title="List of paradoxes">Paradoxes</a></li>

<li><a href="List_of_fallacies" title="List of fallacies">Fallacies</a></li>

<li><a href="List_of_logic_symbols" title="List of logic symbols">Logic symbols</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div>

<ul><li><span class="noviewer" typeof="mw:File"><span title="Category"></span></span> Category</li>

<li><span class="noviewer" typeof="mw:File"><span title="Outline"></span></span> <a href="Outline_of_logic" title="Outline of logic">Outline</a></li>

<li><span class="noviewer" typeof="mw:File"></span> <a href="Portal%3APhilosophy" title="Portal:Philosophy">Portal</a></li>

<li><span class="noviewer" typeof="mw:File"><span title="WikiProject"></span></span> WikiProject</li>

<li><a class="external text external" href="https://en.wikipedia.org/w/index.php?title=Special:Recentchangeslinked&target=Template:Logic&hidebots=0">changes</a></li></ul>

</div></td></tr></tbody></table></div></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2024-07-20" href="https://en.wikipedia.org/wiki/?title=Combs_method&oldid=1235584482">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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