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<title>Automated Mathematician</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Automated Mathematician</span></span>

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<p>The <b>Automated Mathematician</b> (AM) is one of the earliest successful <a href="Discovery_system_(AI_research)" class="mw-redirect" title="Discovery system (AI research)">discovery systems</a>.<sup id="cite_ref-rh_1-0" class="reference"><a href="#cite_note-rh-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> It was created by <a href="Douglas_Lenat" title="Douglas Lenat">Douglas Lenat</a> in <a href="Lisp_programming_language" class="mw-redirect" title="Lisp programming language">Lisp</a>,<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> and in 1977 led to Lenat being awarded the <a href="IJCAI_Computers_and_Thought_Award" title="IJCAI Computers and Thought Award">IJCAI Computers and Thought Award</a>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>

</p><p>AM worked by generating and modifying short Lisp programs which were then interpreted as defining various mathematical concepts;<sup id="cite_ref-koza_4-0" class="reference"><a href="#cite_note-koza-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> for example, a program that tested equality between the length of two lists was considered to represent the concept of numerical equality, while a program that produced a list whose length was the product of the lengths of two other lists was interpreted as representing the concept of multiplication. The system had elaborate heuristics for choosing which programs to extend and modify, based on the experiences of working mathematicians in solving mathematical problems.

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

<p>Lenat claimed that the system was composed of hundreds of data structures called "concepts", together with hundreds of "heuristic rules" and a simple flow of control: "AM repeatedly selects the top task from the agenda and tries to carry it out. This is the whole control structure!" Yet the heuristic rules were not always represented as separate data structures; some had to be intertwined with the control flow logic. Some rules had preconditions that depended on the history, or otherwise could not be represented in the framework of the explicit rules.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup>

</p><p>What's more, the published versions of the rules often involve vague terms that are not defined further, such as "If two expressions are structurally similar, ..." (Rule 218) or "... replace the value obtained by some other (very similar) value..." (Rule 129).<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup>

</p><p>Another source of information is the user, via Rule 2: "If the user has recently referred to X, then boost the priority of any tasks involving X." Thus, it appears quite possible that much of the real discovery work is buried in unexplained procedures.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup>

</p><p>Lenat claimed that the system had rediscovered both <a href="Goldbach's_conjecture" title="Goldbach's conjecture">Goldbach's conjecture</a> and the <a href="Fundamental_theorem_of_arithmetic" title="Fundamental theorem of arithmetic">fundamental theorem of arithmetic</a>. Later critics accused Lenat of over-interpreting the output of AM. In his paper <i>Why AM and Eurisko appear to work</i>, Lenat conceded that any system that generated enough short Lisp programs would generate ones that could be interpreted by an external observer as representing equally sophisticated mathematical concepts. However, he argued that this property was in itself interesting—and that a promising direction for further research would be to look for other languages in which short random strings were likely to be useful.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup>

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

<p>This intuition was the basis of AM's successor <a href="Eurisko" title="Eurisko">Eurisko</a>, which attempted to generalize the search for mathematical concepts to the search for useful <a href="Heuristic" title="Heuristic">heuristics</a>.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup>

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<ul><li><a href="Computer-assisted_proof" title="Computer-assisted proof">Computer-assisted proof</a></li>

<li><a href="Automated_theorem_proving" title="Automated theorem proving">Automated theorem proving</a></li>

<li><a href="Symbolic_mathematics" class="mw-redirect" title="Symbolic mathematics">Symbolic mathematics</a></li>

<li><a href="Experimental_mathematics" title="Experimental mathematics">Experimental mathematics</a></li>

<li><a href="HR_(software)" title="HR (software)">HR (software)</a> and <a href="Graffiti_(program)" title="Graffiti (program)">Graffiti (program)</a>, related math discovery systems</li></ul>

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

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</style><cite id="CITEREFRitchieHanna1984" class="citation journal cs1">Ritchie, G.D.; Hanna, F.K. (August 1984). "am: A case study in AI methodology". <i>Artificial Intelligence</i>. <b>23</b> (3): <span class="nowrap">249–</span>268. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0004-3702%2884%2990015-8">10.1016/0004-3702(84)90015-8</a>.</cite></span>

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<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><cite id="CITEREFLenat1976" class="citation thesis cs1">Lenat, Douglas Bruce (1976). <a rel="nofollow" class="external text" href="https://dl.acm.org/doi/book/10.5555/907763"><i>Am: An artificial intelligence approach to discovery in mathematics as heuristic search</i></a> (Thesis).</cite></span>

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<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><cite id="CITEREFLenat1977" class="citation cs2">Lenat, Douglas B. (1977), "The Ubiquity of Discovery (Computers and Thought Lecture)", <a rel="nofollow" class="external text" href="https://web.archive.org/web/20140806035424/http://ijcai.org/Past%20Proceedings/IJCAI-77-VOL2/PDF/099.pdf"><i>IJCAI</i></a> <span class="cs1-format">(PDF)</span>, pp. <span class="nowrap">1093–</span>1103, archived from <a rel="nofollow" class="external text" href="http://www.ijcai.org/Past%20Proceedings/IJCAI-77-VOL2/PDF/099.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2014-08-06</cite>.</span>

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<li id="cite_note-koza-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-koza_4-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFKoza1992" class="citation cs2">Koza, John R. (1992), "9.3 AM and Euroski", <a rel="nofollow" class="external text" href="https://books.google.com/books?id=Bhtxo60BV0EC&pg=PA232"><i>Genetic Programming: On the Programming of Computers by Means of Natural Selection</i></a>, MIT Press, pp. <span class="nowrap">232–</span>236, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780262111706</bdi></cite>.</span>

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<li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><a href="#CITEREFRitchieHanna1984">Ritchie & Hanna (1984)</a>, pp. 256–257.</span>

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<li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><a href="#CITEREFRitchieHanna1984">Ritchie & Hanna (1984)</a>, p. 258.</span>

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<li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text"><a href="#CITEREFRitchieHanna1984">Ritchie & Hanna (1984)</a>, p. 252.</span>

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<li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><cite id="CITEREFLenatBrown1984" class="citation journal cs1">Lenat, Douglas B.; Brown, John Seely (August 1984). "Why am and eurisko appear to work". <i>Artificial Intelligence</i>. <b>23</b> (3): <span class="nowrap">269–</span>294. <a href="CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.565.8830">10.1.1.565.8830</a></span>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0004-3702%2884%2990016-X">10.1016/0004-3702(84)90016-X</a>.</cite></span>

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<li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><cite id="CITEREFHenderson2007" class="citation cs2">Henderson, Harry (2007), "The Automated Mathematician", <a rel="nofollow" class="external text" href="https://books.google.com/books?id=vKmIiICDIwgC&pg=PA93"><i>Artificial Intelligence: Mirrors for the Mind</i></a>, Milestones in Discovery and Invention, Infobase Publishing, pp. <span class="nowrap">93–</span>94, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9781604130591</bdi></cite>.</span>

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<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li>Edmund Furse; <a rel="nofollow" class="external text" href="https://web.archive.org/web/20060528011654/http://www.comp.glam.ac.uk/pages/staff/efurse/Abstracts/Why-did-AM-halt.html">Why did AM run out of steam?</a></li>

<li>Ken Haase's Ph.D. Thesis; <a rel="nofollow" class="external text" href="https://web.archive.org/web/20030505231346/http://web.media.mit.edu/~haase/thesis/thesis.html">Invention and Exploration in Discovery</a>, a rational reconstruction of Doug Lenat's seminal AM program and an analysis of the relationship between invention and exploration in discovery.</li>

<li>open source Prolog claimed re-implementation of Lenat's AM available at <a rel="nofollow" class="external free" href="https://github.com/akkartik/am-utexas">https://github.com/akkartik/am-utexas</a></li>

<li><a rel="nofollow" class="external text" href="https://github.com/white-flame/am">Source code of Douglas Lenat's AM from SAIL archives circa 1977, hosted on GitHub</a> (from <a rel="nofollow" class="external autonumber" href="https://white-flame.com/am-eurisko.html">[1]</a>)</li></ul></div><div><div class="zim-footer">

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