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<title>Binary lambda calculus - WOW.com</title>
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<div class="titleCnt"><h1>Binary lambda calculus</h1></div>
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Source: http://en.wikipedia.org/wiki/Binary_lambda_calculus<br />Updated: 2016-12-29T22:49Z
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If notability cannot be established, the article is likely to be <a href="https://en.wikipedia.org/wiki/Merging" class="extiw" title="wikipedia:Merging">merged</a>, <a href="https://en.wikipedia.org/wiki/Redirect" class="extiw" title="wikipedia:Redirect">redirected</a>, or <a href="https://en.wikipedia.org/wiki/Deletion_policy" class="extiw" title="wikipedia:Deletion policy">deleted</a>.<br /><small><span class="plainlinks"><i>Find sources:</i> <a rel="nofollow" class="external text" href="//www.google.com/search?as_eq=wikipedia&q=%22Binary+lambda+calculus%22&num=50">"Binary lambda calculus"</a> â <a rel="nofollow" class="external text" href="//www.google.com/search?q=%22Binary+lambda+calculus%22&tbm=nws">news</a> <b>·</b> <a rel="nofollow" class="external text" href="//www.google.com/search?&q=%22Binary+lambda+calculus%22+site:news.google.com/newspapers&source=newspapers">newspapers</a> <b>·</b> <a rel="nofollow" class="external text" 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title="Help:Maintenance template removal">Learn how and when to remove this template message</a>)</i></small></td></tr></table><p><b>Binary lambda calculus</b> (<b>BLC</b>) is a minimal, <a href="/wiki/Purely_functional_programming_language" title="Purely functional programming language" class="mw-redirect">purely functional programming language</a> invented by John Tromp in 2004,<sup id="cite_ref-RAC_1-0" class="reference"><a href="#cite_note-RAC-1">[1]</a></sup> based on a <a href="/wiki/Binary_encoding" title="Binary encoding" class="mw-redirect">binary encoding</a> of the <a href="/wiki/Untyped_lambda_calculus" title="Untyped lambda calculus" class="mw-redirect">untyped lambda calculus</a> in <a href="/wiki/De_Bruijn_index" title="De Bruijn index">De Bruijn index</a> notation.</p><p></p><div id="toc" class="toc"><div id="toctitle"><h2>Contents</h2></div><ul><li class="toclevel-1 tocsection-1"><a href="#Background"><span class="tocnumber">1</span> <span class="toctext">Background</span></a></li><li class="toclevel-1 tocsection-2"><a href="#Binary_strings_in_BLC"><span class="tocnumber">2</span> <span class="toctext">Binary strings in BLC</span></a></li><li class="toclevel-1 tocsection-3"><a href="#Delimited_versus_undelimited"><span class="tocnumber">3</span> <span class="toctext">Delimited versus undelimited</span></a></li><li class="toclevel-1 tocsection-4"><a href="#Universality"><span class="tocnumber">4</span> <span class="toctext">Universality</span></a></li><li class="toclevel-1 tocsection-5"><a href="#Lambda_encoding"><span class="tocnumber">5</span> <span class="toctext">Lambda encoding</span></a></li><li class="toclevel-1 tocsection-6"><a href="#BLC_Complexity"><span class="tocnumber">6</span> <span class="toctext">BLC Complexity</span></a></li><li class="toclevel-1 tocsection-7"><a href="#Basic_Theorems"><span class="tocnumber">7</span> <span class="toctext">Basic Theorems</span></a></li><li class="toclevel-1 tocsection-8"><a href="#Halting_probability"><span class="tocnumber">8</span> <span class="toctext">Halting probability</span></a></li><li class="toclevel-1 tocsection-9"><a href="#BLC8:_byte_sized_I.2FO"><span class="tocnumber">9</span> <span class="toctext">BLC8: byte sized I/O</span></a></li><li class="toclevel-1 tocsection-10"><a href="#BLC_in_the_IOCCC_2012"><span class="tocnumber">10</span> <span class="toctext">BLC in the IOCCC 2012</span></a></li><li class="toclevel-1 tocsection-11"><a href="#References"><span class="tocnumber">11</span> <span class="toctext">References</span></a></li><li class="toclevel-1 tocsection-12"><a href="#External_links"><span class="tocnumber">12</span> <span class="toctext">External links</span></a></li></ul></div><p></p><h2><span class="mw-headline" id="Background">Background</span></h2><p>BLC is designed to provide a very simple and elegant concrete definition of descriptional complexity (<a href="/wiki/Kolmogorov_complexity" title="Kolmogorov complexity">Kolmogorov complexity</a>), where the complexity of an object is the length of its shortest description.</p><p>This is made precise by identifying a description method with a computable function that transforms bitstrings (descriptions) into objects. Objects are usually also just bitstrings, but can have additional structure as well, e.g., pairs of strings.</p><p>Originally, <a href="/wiki/Turing_machine" title="Turing machine">Turing machines</a>, the most well known formalism for computation, were used for this purpose. But they are somewhat lacking in ease of construction and composability. Another classical computational formalism, the <a href="/wiki/Lambda_calculus" title="Lambda calculus">Lambda calculus</a>, offers distinct advantages in ease of use. BLC is the result of incorporating a notion of binary I/O into lambda calculus, so as to turn it into an effective description method.</p><h2><span class="mw-headline" id="Binary_strings_in_BLC">Binary strings in BLC</span></h2><p>BLC represents bits 0 and 1 are as the standard <a href="/wiki/Lambda_calculus#Logic_and_predicates" title="Lambda calculus">lambda booleans</a> <i>B</i><sub>0</sub> = True and <i>B</i><sub>1</sub> = False:</p><dl><dd>True = <img class="mwe-math-fallback-image-inline tex" alt="\lambda x\, \lambda y.\, x" src="http://upload.wikimedia.org/math/a/c/9/ac986485e66d4afed11d5f6b95e9bfcd.png" /></dd><dd>False = <img class="mwe-math-fallback-image-inline tex" alt="\lambda x\, \lambda y.\, y" src="http://upload.wikimedia.org/math/7/c/1/7c14c1fed4e9ea424e71488b91b953a5.png" /></dd></dl><p>which can be seen to directly implement the <a href="/wiki/If-then-else" title="If-then-else" class="mw-redirect">if-then-else</a> operator.</p><p>The standard pairing function</p><dl><dd><img class="mwe-math-fallback-image-inline tex" alt="\langle,\rangle = \lambda x \,\lambda y \,\lambda z.\, z x y" src="http://upload.wikimedia.org/math/4/d/c/4dc5f3fca0f99cd02b91126ba67d3fe5.png" /></dd></dl><p>applied to two terms <i>M</i> and <i>N</i></p><dl><dd><img class="mwe-math-fallback-image-inline tex" alt="\langle M, N \rangle = \lambda z.\, z M N" src="http://upload.wikimedia.org/math/3/1/e/31e4250f216793e1f7514d9fb484c5c4.png" /></dd></dl><p>can be applied to a boolean to yield the desired component of choice.</p><p>BLC represents a string <i>s</i> = <i>b</i><sub>0</sub><i>b</i><sub>1</sub>â¦<i>b</i><sub>nâ1</sub> by repeated pairing as</p><dl><dd><img class="mwe-math-fallback-image-inline tex" alt="\langle B_{b_0}, \langle B_{b_1} \ldots \langle B_{b_{n-1}}, z\rangle \ldots \rangle \rangle " src="http://upload.wikimedia.org/math/2/2/d/22d3010d61197a5555e16e4f0cbf93b9.png" /> which is denoted as <img class="mwe-math-fallback-image-inline tex" alt="s:z\ " src="http://upload.wikimedia.org/math/4/8/b/48b14906dfc119191a3ef4f1c7759be3.png" />.</dd></dl><p>The <i>z</i> works as a list continuation, that could be a nil list (to end the string) or another string (that would be appended to the original string).</p><h2><span class="mw-headline" id="Delimited_versus_undelimited">Delimited versus undelimited</span></h2><p>Descriptional complexity comes in two distinct flavors, depending on whether the input is considered to be delimited.</p><p>Knowing the end of your input makes it easier to describe objects. For instance, you can just copy the whole input to output. This flavor is called <i>plain</i> or <i>simple</i> complexity.</p><p>But in a sense it is additional information. A file system for instance needs to separately store the length of files. The <a href="/wiki/C_(programming_language)" title="C (programming language)">C language</a> uses the <a href="/wiki/Null_character" title="Null character">null character</a> to denote the end of a string, but this comes at the cost of not having that character available within strings.</p><p>The other flavor is called <i>prefix</i> complexity, named after <a href="/wiki/Prefix_code" title="Prefix code">prefix codes</a>, where the machine needs to figure out, from the input read so far, whether it needs to read more bits. We say that the input is self-delimiting. This works better for communication channels, since one can send multiple descriptions, one after the other, and still tell them apart.</p><p>In the I/O model of BLC, the flavor is dictated by the choice of <i>z</i>. When kept as a free variable, and required to appear as part of the output, then the machine must be working in a self-delimiting manner. If on the other hand z is a lambda term specifically designed to be easy to distinguish from any pairing, then the input becomes delimited. BLC chooses <i>False</i> for this purpose but gives it the more descriptive alternative name of <i>Nil</i>. Dealing with lists that may be Nil is straightforward: since</p><dl><dd><img class="mwe-math-fallback-image-inline tex" alt="\langle x, y \rangle\ M\ N = M\ x\ y\ N" src="http://upload.wikimedia.org/math/c/8/0/c803916f009c25ab3e87c1803674d321.png" />, and</dd><dd><img class="mwe-math-fallback-image-inline tex" alt="Nil\ M\ N = N" src="http://upload.wikimedia.org/math/3/1/c/31c08720893d4e35efc165af3de59124.png" /></dd></dl><p>one can write functions <i>M</i> and <i>N</i> to deal with the two cases, the only caveat being that <i>N</i> will be passed to <i>M</i> as its third argument.</p><h2><span class="mw-headline" id="Universality">Universality</span></h2><p>One can find a description method <i>U</i> such that for any other description method <i>D</i>, there is a constant <i>c</i> (depending only on <i>D</i>) such that no object takes more than <i>c</i> extra bits to describe with method <i>U</i> than with method <i>D</i>. BLC is designed to make these constants relatively small. In fact the constant will be the length of a binary encoding of a <i>D</i>-interpreter written in BLC, and <i>U</i> will be a lambda term that parses this encoding and runs this decoded interpreter on the rest of the input. <i>U</i> won't even have to know whether the description is delimited or not; it works the same either way.</p><p>BLC not only represents bitstrings as lambda calculus terms, but the other way around as well.</p><h2><span class="mw-headline" id="Lambda_encoding">Lambda encoding</span></h2><p>First, lambda terms are written in a particular notation using what is known as <a href="/wiki/De_Bruijn_indices" title="De Bruijn indices" class="mw-redirect">De Bruijn indices</a>. The encoding is then defined recursively as follows</p><dl><dd><img class="mwe-math-fallback-image-inline tex" alt="\widehat{\lambda M}= 00\widehat{M}" src="http://upload.wikimedia.org/math/c/0/4/c04c3b8495df50036c8e1c919653834c.png" /></dd><dd><img class="mwe-math-fallback-image-inline tex" alt="\widehat{M\ N}=01\widehat{M}\widehat{N}" src="http://upload.wikimedia.org/math/7/5/5/75541e294572029270f842cc55b58118.png" /></dd><dd><img class="mwe-math-fallback-image-inline tex" alt="\widehat{i}=1^i0" src="http://upload.wikimedia.org/math/0/b/0/0b0256be2a6bfcdc49ac51688a761805.png" /></dd></dl><p>For instance, the pairing function <img class="mwe-math-fallback-image-inline tex" alt="\lambda x\lambda y\lambda z.xzy" src="http://upload.wikimedia.org/math/8/3/a/83a7998dbe1dde18f7e25589fd7b9fb0.png" /> is written <img class="mwe-math-fallback-image-inline tex" alt="\lambda \lambda \lambda. 1 3 2" src="http://upload.wikimedia.org/math/5/a/9/5a939ad0905d250b2930738168c216ec.png" /> in De Bruijn format, which has encoding <img class="mwe-math-fallback-image-inline tex" alt="00\ 00\ 00\ 01\ 01\ 10\ 1110\ 110" src="http://upload.wikimedia.org/math/b/b/8/bb885c567559f727bb24a04a5592ae52.png" />.</p><p>A <b>closed</b> lambda term is one in which all variables are bound, i.e. without any free variables. In De Bruijn format, this means that an index <i>i</i> can only appear within at least <i>i</i> nested lambdas. The number of closed terms of size <i>n</i> bits is given by sequence <span class="nowrap"><a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="OEIS"><img alt="OEIS" src="http://upload.wikimedia.org/wikipedia/commons/thumb/d/d8/OEISicon_light.svg/11px-OEISicon_light.svg.png" width="11" height="15" srcset="http://upload.wikimedia.org/wikipedia/commons/thumb/d/d8/OEISicon_light.svg/17px-OEISicon_light.svg.png 1.5x, http://upload.wikimedia.org/wikipedia/commons/thumb/d/d8/OEISicon_light.svg/22px-OEISicon_light.svg.png 2x" /></a>â<a href="/wiki/OEIS:A114852" class="new" title="OEIS:A114852 (page does not exist)">A114852</a></span> of the <a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">On-Line Encyclopedia of Integer Sequences</a>.</p><p>The shortest possible closed term is the identity function <img class="mwe-math-fallback-image-inline tex" alt="\widehat{\lambda 1} = 0010" src="http://upload.wikimedia.org/math/3/2/7/327dc3425bb0987548e07f3c4e3e03f4.png" />. In delimited mode, this machine just copies its input to its output.</p><p>The <a href="/wiki/Universal_machine" title="Universal machine" class="mw-redirect">universal machine</a> <i>U</i> in BLC is then, in De Bruijn format (all indices are single digit):</p><dl><dd><img class="mwe-math-fallback-image-inline tex" alt="(\lambda 1 1) (\lambda \lambda \lambda 1 (\lambda \lambda \lambda \lambda 3 (\lambda 5 (3 (\lambda 2 (3 (\lambda \lambda 3 (\lambda 1 2 3))) (4 (\lambda 4 (\lambda 3 1 (2 1))))))" src="http://upload.wikimedia.org/math/4/b/3/4b3a2655a41f74acdb60cfb06a559772.png" /></dd><dd><img class="mwe-math-fallback-image-inline tex" alt="(1 (2 (\lambda 1 2)) (\lambda 4 (\lambda 4 (\lambda 2 (1 4))) 5)))) (3 3) 2) (\lambda 1 ((\lambda 1 1) (\lambda 1 1)))" src="http://upload.wikimedia.org/math/0/b/d/0bd4c10025e33ed7bb9b1a0585e6122e.png" /></dd></dl><p>This is in binary:</p><dl><dd>0101000110100000000101011000000000011110000101111110011110</dd><dd>0001011100111100000011110000101101101110011111000011111000</dd><dd>0101111010011101001011001110000110110000101111100001111100</dd><dd>0011100110111101111100111101110110000110010001101000011010</dd><dd>(only 232 bits (29 bytes) long)</dd></dl><p>A detailed analysis of machine <i>U</i> may be found in.<sup id="cite_ref-RAC_1-1" class="reference"><a href="#cite_note-RAC-1">[1]</a></sup></p><h2><span class="mw-headline" id="BLC_Complexity">BLC Complexity</span></h2><p>In general, complexity of an object can be conditional on several other objects that are provided as additional argument to the universal machine. BLC defines Plain (or simple) complexity <i>KS</i> and prefix complexity <i>KP</i> by</p><dl><dd><img class="mwe-math-fallback-image-inline tex" alt="\begin{align}KS(x|y_1,\ldots,y_{k}) &= \min \{ \ell(p)\ |\ U\ (p:Nil)\ y_1\ \ldots\ y_{k} = \ \,x\ \ \ \ \}\\KP(x|y_1,\ldots,y_{k}) &= \min \{ \ell(p)\ |\ U\ (p:\ z\ \ )\ y_1\ \ldots\ y_{k} = \langle x,z \rangle \}\end{align}" src="http://upload.wikimedia.org/math/5/9/4/594d27bde46bbeda1013e718f2c06ada.png" /></dd></dl><h2><span class="mw-headline" id="Basic_Theorems">Basic Theorems</span></h2><p>The identity program <img class="mwe-math-fallback-image-inline tex" alt="\lambda 1" src="http://upload.wikimedia.org/math/c/d/c/cdc8bd9ab46bcbab3a6cb9e64262910d.png" /> proves that</p><dl><dd><img class="mwe-math-fallback-image-inline tex" alt="KS(x) \leq \ell(x) + 4" src="http://upload.wikimedia.org/math/9/9/4/9940c5fa8f61ff968b53650cd8a4b2d6.png" /></dd></dl><p>The program <img class="mwe-math-fallback-image-inline tex" alt="\lambda \lambda 1 ((\lambda 1 1) (\lambda \lambda \lambda \lambda 2 (4 4) (\lambda \lambda 3 2 (3 2 (2 (5 1 (2 1))))))) (\lambda \lambda 1) (\lambda \lambda \lambda 1 (\lambda 4 (\lambda 4 (\lambda 1 3 2)))) (\lambda \lambda \lambda 1 (3 (\lambda \lambda 1)) 2) (\lambda 1) 2" src="http://upload.wikimedia.org/math/1/4/7/147ce118161941e6de99e628dc8a2925.png" /> proves that</p><dl><dd><img class="mwe-math-fallback-image-inline tex" alt="KP(x|\ell(x)) \leq \ell(x)+188" src="http://upload.wikimedia.org/math/2/8/e/28e9f58bfc0e712b84f1410e8400b0e9.png" /></dd></dl><p>The program</p><dl><dd><img class="mwe-math-fallback-image-inline tex" alt="(\lambda 1 1) (\lambda \lambda \lambda 1 (\lambda 1 (3 (\lambda \lambda 1)) (4 4 (\lambda 1 (\lambda \lambda \lambda 1 (\lambda 4 (\lambda \lambda 5 2 (5 2 (3 1 (2 1)))))) 4 (\lambda 1))))) (\lambda \lambda \lambda 1 (3 ((\lambda 1 1)" src="http://upload.wikimedia.org/math/3/d/4/3d40506d4f90c4d96eab48888b03c9ed.png" /> <img class="mwe-math-fallback-image-inline tex" alt="(\lambda \lambda \lambda \lambda 1 (\lambda 5 5 (\lambda \lambda 3 5 6 (\lambda 1 (\lambda \lambda 6 1 2) 3)) (\lambda \lambda 5 (\lambda 1 4 3))) (3 1)) (\lambda \lambda 1 (\lambda \lambda 2) 2) (\lambda 1)) (\lambda \lambda 1)) 2)" src="http://upload.wikimedia.org/math/b/8/2/b8216bcde39ec68c963ad64d99902492.png" /></dd></dl><p>proves that</p><dl><dd><img class="mwe-math-fallback-image-inline tex" alt="KP(x) \leq \ell(\overline{x})+338" src="http://upload.wikimedia.org/math/8/a/7/8a7e84e30a47618855c3e5662d0b5127.png" /></dd></dl><p>where <img class="mwe-math-fallback-image-inline tex" alt="\overline{x}" src="http://upload.wikimedia.org/math/4/d/8/4d8a563baa616b3bd56826256c46d50e.png" /> is the <a href="/wiki/Levenstein_coding" title="Levenstein coding" class="mw-redirect">Levenstein code</a> for <i>x</i> defined by</p><dl><dd><img class="mwe-math-fallback-image-inline tex" alt="\begin{array}{ll}\overline{0} & = 0 \\\overline{n+1} & = 1\ \overline{\ell(n)}\ n\\\end{array}" src="http://upload.wikimedia.org/math/c/d/2/cd229050f9718731eec112894b44adb4.png" /></dd></dl><p>in which we identify numbers and bitstrings according to lexicographic order. This code has the nice property that for all <i>k</i>,</p><dl><dd><img class="mwe-math-fallback-image-inline tex" alt="\ell(\overline{n}) \leq \ell(n)+\ell(\ell(n))+\cdots+ \ell^{k-1}(n) + O(\ell^k(n))" src="http://upload.wikimedia.org/math/8/c/4/8c4fd3d2ff0442eacea109ef60b5980f.png" /></dd></dl><p>Furthermore, it makes lexicographic order of delimited numbers coincide with numeric order.</p><table class="wikitable" border="1"><tr><th>Number</th><th>String</th><th>Delimited</th></tr><tr><td>0</td><td></td><td>0</td></tr><tr><td>1</td><td>0</td><td>10</td></tr><tr><td>2</td><td>1</td><td>110 0</td></tr><tr><td>3</td><td>00</td><td>110 1</td></tr><tr><td>4</td><td>01</td><td>1110 0 00</td></tr><tr><td>5</td><td>10</td><td>1110 0 01</td></tr><tr><td>6</td><td>11</td><td>1110 0 10</td></tr><tr><td>7</td><td>000</td><td>1110 0 11</td></tr><tr><td>8</td><td>001</td><td>1110 1 000</td></tr><tr><td>9</td><td>010</td><td>1110 1 001</td></tr></table><h2><span class="mw-headline" id="Halting_probability">Halting probability</span></h2><p>The <a href="/wiki/Halting_probability" title="Halting probability" class="mw-redirect">halting probability</a> of the prefix universal machine is defined as the probability it will output any term that has a closed normal form (this includes all translated strings):</p><dl><dd><img class="mwe-math-fallback-image-inline tex" alt="\Omega_{\lambda} = \sum_{\stackrel{U (p:z) = \langle x,z \rangle}{x\in NF}} 2^{-\ell(p)}" src="http://upload.wikimedia.org/math/f/2/8/f288aa1ea044609d14aed130b00a70ec.png" /></dd></dl><p>With some effort, we can determine the first 4 bits of this particular number of wisdom:</p><dl><dd><img class="mwe-math-fallback-image-inline tex" alt="\Omega_{\lambda} = .0001\ldots_2" src="http://upload.wikimedia.org/math/9/3/9/939beca1aff5b4ea0cc956af5e58e45d.png" /></dd></dl><p>where probability <span class="texhtml">.0001<sub>2</sub> = 2<sup>â4</sup></span> is already contributed by programs <span style="font-family: monospace, monospace;">00100</span> and <span style="font-family: monospace, monospace;">00101</span> for terms True and False.</p><h2><span class="mw-headline" id="BLC8:_byte_sized_I.2FO">BLC8: byte sized I/O</span></h2><p>While bit streams are nice in theory, they fare poorly in interfacing with the real world. The language BLC8 is a more practical variation on BLC in which programs operate on a stream of bytes, where each byte is represented as a delimited list of 8 bits in <a href="/wiki/Big-endian" title="Big-endian" class="mw-redirect">big-endian</a> order.</p><h2><span class="mw-headline" id="BLC_in_the_IOCCC_2012">BLC in the IOCCC 2012</span></h2><p>An implementation of both BLC and BLC8 in the <a href="/wiki/C_programming_language" title="C programming language" class="mw-redirect">C programming language</a> won the "Most Functional" award in the 2012 edition of the <a href="/wiki/International_Obfuscated_C_Code_Contest" title="International Obfuscated C Code Contest">International Obfuscated C Code Contest</a>.</p><h2><span class="mw-headline" id="References">References</span></h2><ol class="references"><li id="cite_note-RAC-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-RAC_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-RAC_1-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">John Tromp, Binary Lambda Calculus and Combinatory Logic, in <i>Randomness And Complexity, from Leibniz To Chaitin</i>, ed. Cristian S. Calude, World Scientific Publishing Company, October 2008. (The last reference, to an initial Haskell implementation, is dated 2004) <a rel="nofollow" class="external text" href="http://tromp.github.io/cl/LC.pdf">(pdf version)</a> <a rel="nofollow" class="external text" href="https://web.archive.org/web/20160304083208/http://tromp.github.io/cl/LC.pdf">Archived</a> March 4, 2016, at the <a href="/wiki/Wayback_Machine" title="Wayback Machine">Wayback Machine</a>.</span></li></ol><h2><span class="mw-headline" id="External_links">External links</span></h2><ul><li><a rel="nofollow" class="external text" href="http://tromp.github.io/cl/cl.html">John's Lambda Calculus and Combinatory Logic Playground</a></li><li><a rel="nofollow" class="external text" href="http://www.ioccc.org/2012/tromp/hint.html">A Binary Lambda Calculus interpreter in C for the IOCCC</a></li><li><a rel="nofollow" class="external text" href="http://arxiv.org/abs/1401.0379">Counting Terms in the Binary Lambda Calculus</a></li></ul><!-- NewPP limit reportCPU time usage: 1.551 secondsReal time usage: 5.476 secondsPreprocessor visited node count: 1080/1000000Preprocessor generated node count: 3711/1000000Postâexpand include size: 66298/2097152 bytesTemplate argument size: 15539/2097152 bytesHighest expansion depth: 13/40Expensive parser function count: 4/100Lua time usage: 0.233/10.000 secondsLua memory usage: 1,000 KB/953.67 MB--><!-- Transclusion expansion time report (%,ms,calls,template)100.00% 3245.458 1 - -total 79.19% 2570.105 6 - Template:Ambox 44.75% 1452.306 1 - Template:Multiple_issues 18.80% 610.115 1 - Template:Notability 18.74% 608.144 1 - Template:Article_for_deletion/dated 11.75% 381.212 1 - Template:Find_sources_mainspace 5.97% 193.663 1 - Template:COI 5.10% 165.587 1 - Template:Tone 4.66% 151.093 1 - Template:Original_research 3.35% 108.577 4 - Template:Tlx--><!-- Saved in parser cache with key my_wiki:pcache:idhash:21924619-1!*!0!!en!5!*!math=0 and timestamp 20170111130401 and revision id 757303214 -->
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