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class="mw-indicator"><div class="mw-parser-output"><span typeof="mw:File"><a href="/wiki/Help:Development_stages" title="25% developed"><img alt="25% developed" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/25_percent.svg/24px-25_percent.svg.png" decoding="async" width="24" height="24" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/ce/25_percent.svg/36px-25_percent.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/ce/25_percent.svg/48px-25_percent.svg.png 2x" data-file-width="9" data-file-height="9" /></a></span></div></div> </div> <div id="siteSub" class="noprint">From Wikibooks, open books for an open world</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><p><b>Finite Model Theory</b> (FMT) is a subarea of Model Theory (MT). MT is the branch of mathematical logic which deals with the relation between a formal language (syntax) and its interpretations (semantics). FMT is a restriction of MT to finite structures, such as finite graphs or strings. Since many central theorems of MT do not hold when restricted to finite structures, FMT is quite different from MT in methods and application areas. FMT has become an "unusually effective" instrument in computer science, for example in database theory, model checking or for gaining new perspectives on computational complexity. </p><p>The three main areas of FMT are presented here: Expressive Power of Languages, Descriptive Complexity, and Random Structures. But first the results fundamental for all areas are introduced on the level of first order languages. </p> <table class="plainlinks noprint messagebox notice" style="width:250px; float:right; clear:right; margin:0px; margin-left:10px;"> <tbody><tr style="vertical-align:middle;"><td style="padding:0.1em; text-align:center; vertical-align:middle; width:45px; border:none;"> <span typeof="mw:File"><span><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/80/Wikipedia-logo-v2.svg/40px-Wikipedia-logo-v2.svg.png" decoding="async" width="40" height="37" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/80/Wikipedia-logo-v2.svg/60px-Wikipedia-logo-v2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/80/Wikipedia-logo-v2.svg/80px-Wikipedia-logo-v2.svg.png 2x" data-file-width="103" data-file-height="94" /></span></span></td> <td style="color:black; text-align:left; vertical-align:middle; padding:0.5em; padding-left:0em; border:none;"> <p><a href="https://en.wikipedia.org/wiki/" class="extiw" title="w:">Wikipedia</a> has related information at <a class="external text" href="https://en.wikipedia.org/wiki/Finite_model_theory"><i><b>Finite model theory</b></i></a> </p> </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Basics">Basics</h2></div> <p><a href="/wiki/Finite_Model_Theory/Motivation" title="Finite Model Theory/Motivation">Why?</a> (Motivation) </p><p><a href="/wiki/Finite_Model_Theory/Character" title="Finite Model Theory/Character">What is it?</a> (Definition and Background) </p><p><a href="/wiki/Finite_Model_Theory/Model_Theory" title="Finite Model Theory/Model Theory">Why is it special?</a> (... compared to 'common' Model Theory) </p><p><a href="/wiki/Finite_Model_Theory/Logics_and_Structures" title="Finite Model Theory/Logics and Structures">What is it about?</a> (Typical Logics and Structures studied) </p><p><a href="/wiki/Finite_Model_Theory/Preliminaries" title="Finite Model Theory/Preliminaries">What is required?</a> (Preliminaries) </p><p><a href="/wiki/Finite_Model_Theory/Basics" title="Finite Model Theory/Basics">What to start with?</a> (Basic Concepts) </p> <div class="mw-heading mw-heading2"><h2 id="Ehrenfeucht-Fraisse-Games_for_FO">Ehrenfeucht-Fraisse-Games for FO</h2></div> <p><a href="/wiki/Finite_Model_Theory/FO_Discrimination" title="Finite Model Theory/FO Discrimination">The Problem</a> (Expressibility of Properties) </p><p><a href="/wiki/Finite_Model_Theory/FO_BFP" title="Finite Model Theory/FO BFP">The Idea I</a> (Fraisse's Theorem) </p><p><a href="/wiki/Finite_Model_Theory/FO_EFG" title="Finite Model Theory/FO EFG">The Idea II</a> (Ehrenfeucht's Games) </p><p><a href="/wiki/Finite_Model_Theory/FO_EFM" title="Finite Model Theory/FO EFM">The Tool</a> (Ehrenfeucht-Fraisse-Method) </p><p><a href="/wiki/Finite_Model_Theory/FO_Localities" title="Finite Model Theory/FO Localities">Some Utilities</a> (Localities) </p><p><a href="/w/index.php?title=Finite_Model_Theory/FO_Structures&action=edit&redlink=1" class="new" title="Finite Model Theory/FO Structures (does not exist)">Some Solutions</a> (for some Structures) </p><p>Complexity? </p> <div class="mw-heading mw-heading2"><h2 id="Expressive_Power_of_Languages">Expressive Power of Languages</h2></div> <p>Typical problem: Given a finite graph, can the property of being an acyclic be expressed in a first order language? </p> <div class="mw-heading mw-heading2"><h2 id="Descriptive_Complexity">Descriptive Complexity</h2></div> <div class="mw-heading mw-heading2"><h2 id="Random_Structures">Random Structures</h2></div> <p><br /> </p><p><br /> </p> <div class="boilerplate metadata; plainlinks" id="stub"> <div class="PrettyTextBox" style="vertical-align:bottom; margin:0px 25px; padding:5px 20px;"> <span typeof="mw:File"><a href="/wiki/File:E-to-the-i-pi.svg" class="mw-file-description" title="Mathematics stub"><img alt="Mathematics stub" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/35/E-to-the-i-pi.svg/40px-E-to-the-i-pi.svg.png" decoding="async" width="40" height="35" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/35/E-to-the-i-pi.svg/60px-E-to-the-i-pi.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/35/E-to-the-i-pi.svg/80px-E-to-the-i-pi.svg.png 2x" data-file-width="512" data-file-height="453" /></a></span> <i>This <a href="/wiki/Subject:Mathematics" title="Subject:Mathematics">mathematics</a> module is a <a href="/wiki/Help:Stub" class="mw-redirect" title="Help:Stub">stub</a>. 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