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<?xml version="1.0" encoding="utf-8"?>
<feed xmlns="http://www.w3.org/2005/Atom">
<title><![CDATA[修身齐家]]></title>
<link href="http://twqingsong.github.com/atom.xml" rel="self"/>
<link href="http://twqingsong.github.com/"/>
<updated>2013-03-17T23:49:38+08:00</updated>
<id>http://twqingsong.github.com/</id>
<author>
<name><![CDATA[yqsqq]]></name>
</author>
<generator uri="http://octopress.org/">Octopress</generator>
<entry>
<title type="html"><![CDATA[Lambda演算-编程语言算术操作实现]]></title>
<link href="http://twqingsong.github.com/blog/2013/03/15/lambdayan-suan-bian-cheng-yu-yan-suan-zhu-cao-zuo-shi-xian/"/>
<updated>2013-03-15T10:46:00+08:00</updated>
<id>http://twqingsong.github.com/blog/2013/03/15/lambdayan-suan-bian-cheng-yu-yan-suan-zhu-cao-zuo-shi-xian</id>
<content type="html"><![CDATA[<h3><a target="_blank" href="http://zh.wikipedia.org/wiki/%CE%9B%E6%BC%94%E7%AE%97 ">Lambda演算</a></h3>
<p>λ演算，λ演算是一套用于研究函数定义、函数应用和递归的形式系统。它由 Alonzo Church 和 Stephen Cole Kleene 在 20 世纪三十年代引入，Church 运用 lambda 演算在 1936 年给出 判定性问题 (Entscheidungsproblem) 的一个否定的答案。这种演算可以用来清晰地定义什么是一个可计算函数。关于两个 lambda 演算表达式是否等价的命题无法通过一个通用的算法来解决，这是不可判定性能够证明的头一个问题，甚至还在停机问题之先。Lambda 演算对函数式编程有巨大的影响，特别是Lisp 语言。</p>
<h3>定义常量</h3>
<p>我们这里使用ruby的lambda函数，为了编写方便，我们给lambda定义一个别名 L；同时，为了能方便地看到输出结果，我们定义了printL函数</p>
<figure class='code'><figcaption><span></span></figcaption><div class="highlight"><table><tr><td class="gutter"><pre class="line-numbers"><span class='line-number'>1</span>
<span class='line-number'>2</span>
<span class='line-number'>3</span>
<span class='line-number'>4</span>
<span class='line-number'>5</span>
</pre></td><td class='code'><pre><code class='ruby'><span class='line'><span class="k">alias</span> <span class="n">L</span> <span class="nb">lambda</span>
</span><span class='line'>
</span><span class='line'><span class="k">def</span> <span class="nf">printL</span> <span class="n">n</span>
</span><span class='line'> <span class="nb">puts</span> <span class="n">n</span><span class="o">[</span><span class="n">L</span><span class="p">{</span><span class="o">|</span><span class="n">x</span><span class="o">|</span> <span class="n">x</span> <span class="o">+</span> <span class="mi">1</span><span class="p">}</span><span class="o">][</span><span class="mi">0</span><span class="o">]</span>
</span><span class='line'><span class="k">end</span>
</span></code></pre></td></tr></table></div></figure>
<p>我们先来看 Church数的定义。<br/>
0 ≡ λf.λx.x,<br/>
1 ≡ λf.λx.f x<br/>
2 ≡ λf.λx.f (f x)<br/>
3 ≡ λf.λx.f (f (f x))<br/>
..<br/>
n ≡ λf.λx.fn x<br/>
..<br/>
由此可见Church整数是一个高阶函数，以单一参数函数f为参数，返回另一个单一参数的函数。
据此，我们定义几个需要用到的整数，例如0-5：</p>
<figure class='code'><figcaption><span></span></figcaption><div class="highlight"><table><tr><td class="gutter"><pre class="line-numbers"><span class='line-number'>1</span>
<span class='line-number'>2</span>
<span class='line-number'>3</span>
<span class='line-number'>4</span>
<span class='line-number'>5</span>
<span class='line-number'>6</span>
</pre></td><td class='code'><pre><code class='ruby'><span class='line'><span class="vg">$_0</span> <span class="o">=</span> <span class="n">L</span><span class="p">{</span><span class="o">|</span><span class="n">f</span><span class="o">|</span> <span class="n">L</span><span class="p">{</span><span class="o">|</span><span class="n">x</span><span class="o">|</span> <span class="n">x</span><span class="p">}}</span>
</span><span class='line'><span class="vg">$_1</span> <span class="o">=</span> <span class="n">L</span><span class="p">{</span><span class="o">|</span><span class="n">f</span><span class="o">|</span> <span class="n">L</span><span class="p">{</span><span class="o">|</span><span class="n">x</span><span class="o">|</span> <span class="n">f</span><span class="o">[</span><span class="n">x</span><span class="o">]</span><span class="p">}}</span>
</span><span class='line'><span class="vg">$_2</span> <span class="o">=</span> <span class="n">L</span><span class="p">{</span><span class="o">|</span><span class="n">f</span><span class="o">|</span> <span class="n">L</span><span class="p">{</span><span class="o">|</span><span class="n">x</span><span class="o">|</span> <span class="n">f</span><span class="o">[</span><span class="n">f</span><span class="o">[</span><span class="n">x</span><span class="o">]]</span><span class="p">}}</span>
</span><span class='line'><span class="vg">$_3</span> <span class="o">=</span> <span class="n">L</span><span class="p">{</span><span class="o">|</span><span class="n">f</span><span class="o">|</span> <span class="n">L</span><span class="p">{</span><span class="o">|</span><span class="n">x</span><span class="o">|</span> <span class="n">f</span><span class="o">[</span><span class="n">f</span><span class="o">[</span><span class="n">f</span><span class="o">[</span><span class="n">x</span><span class="o">]]]</span><span class="p">}}</span>
</span><span class='line'><span class="vg">$_4</span> <span class="o">=</span> <span class="n">L</span><span class="p">{</span><span class="o">|</span><span class="n">f</span><span class="o">|</span> <span class="n">L</span><span class="p">{</span><span class="o">|</span><span class="n">x</span><span class="o">|</span> <span class="n">f</span><span class="o">[</span><span class="n">f</span><span class="o">[</span><span class="n">f</span><span class="o">[</span><span class="n">f</span><span class="o">[</span><span class="n">x</span><span class="o">]]]]</span><span class="p">}}</span>
</span><span class='line'><span class="vg">$_5</span> <span class="o">=</span> <span class="n">L</span><span class="p">{</span><span class="o">|</span><span class="n">f</span><span class="o">|</span> <span class="n">L</span><span class="p">{</span><span class="o">|</span><span class="n">x</span><span class="o">|</span> <span class="n">f</span><span class="o">[</span><span class="n">f</span><span class="o">[</span><span class="n">f</span><span class="o">[</span><span class="n">f</span><span class="o">[</span><span class="n">f</span><span class="o">[</span><span class="n">x</span><span class="o">]]]]]</span><span class="p">}}</span>
</span></code></pre></td></tr></table></div></figure>
<p>Church布尔值:<br/>
TRUE = λx.λy.x<br/>
FALSE = λx.λy.y</p>
<p>条件分支:<br/>
IFTHENELSE = λp.λa.λb. p a b</p>
<figure class='code'><figcaption><span></span></figcaption><div class="highlight"><table><tr><td class="gutter"><pre class="line-numbers"><span class='line-number'>1</span>
<span class='line-number'>2</span>
<span class='line-number'>3</span>
</pre></td><td class='code'><pre><code class='ruby'><span class='line'><span class="vg">$_TRUE</span> <span class="o">=</span> <span class="n">L</span><span class="p">{</span><span class="o">|</span><span class="n">x</span><span class="o">|</span> <span class="n">L</span><span class="p">{</span><span class="o">|</span><span class="n">y</span><span class="o">|</span> <span class="n">x</span><span class="p">}}</span>
</span><span class='line'><span class="vg">$_FALSE</span> <span class="o">=</span> <span class="n">L</span><span class="p">{</span><span class="o">|</span><span class="n">x</span><span class="o">|</span> <span class="n">L</span><span class="p">{</span><span class="o">|</span><span class="n">y</span><span class="o">|</span> <span class="n">y</span><span class="p">}}</span>
</span><span class='line'><span class="vg">$_IF</span> <span class="o">=</span> <span class="n">L</span><span class="p">{</span><span class="o">|</span><span class="nb">p</span><span class="o">|</span> <span class="n">L</span><span class="p">{</span><span class="o">|</span><span class="n">a</span><span class="o">|</span> <span class="n">L</span><span class="p">{</span><span class="o">|</span><span class="n">b</span><span class="o">|</span> <span class="nb">p</span><span class="o">[</span><span class="n">a</span><span class="o">][</span><span class="n">b</span><span class="o">]</span><span class="p">}}}</span>
</span></code></pre></td></tr></table></div></figure>
<p>有序对（2元组）数据类型可以用TRUE、FALSE来定义。<br/>
CONS := λm.λn.λs (s m n)<br/>
CAR := λp.p TRUE<br/>
CDR := λp.p FALSE</p>
<figure class='code'><figcaption><span></span></figcaption><div class="highlight"><table><tr><td class="gutter"><pre class="line-numbers"><span class='line-number'>1</span>
<span class='line-number'>2</span>
<span class='line-number'>3</span>
</pre></td><td class='code'><pre><code class='ruby'><span class='line'><span class="vg">$_CONS</span> <span class="o">=</span> <span class="n">L</span><span class="p">{</span><span class="o">|</span><span class="n">m</span><span class="o">|</span> <span class="n">L</span><span class="p">{</span><span class="o">|</span><span class="n">n</span><span class="o">|</span> <span class="n">L</span><span class="p">{</span><span class="o">|</span><span class="n">s</span><span class="o">|</span> <span class="n">s</span><span class="o">[</span><span class="n">m</span><span class="o">][</span><span class="n">n</span><span class="o">]</span><span class="p">}}}</span>
</span><span class='line'><span class="vg">$_CAR</span> <span class="o">=</span> <span class="n">L</span><span class="p">{</span><span class="o">|</span><span class="nb">p</span><span class="o">|</span> <span class="nb">p</span><span class="o">[</span><span class="vg">$_TRUE</span><span class="o">]</span><span class="p">}</span>
</span><span class='line'><span class="vg">$_CDR</span> <span class="o">=</span> <span class="n">L</span><span class="p">{</span><span class="o">|</span><span class="nb">p</span><span class="o">|</span> <span class="nb">p</span><span class="o">[</span><span class="vg">$_FALSE</span><span class="o">]</span><span class="p">}</span>
</span></code></pre></td></tr></table></div></figure>
<h3>算术运算</h3>
<p>我们先来看看自增的定义：<br/>
INC = λn.λf.λx.f(n f x)<br/>
我们可以定义自增的函数，接收一个参数n, 返回n+1。有了自增，我们就可以实现加法了<br/>
ADD = λn.λm.m INC n<br/>
自增和加法的代码如下:</p>
<figure class='code'><figcaption><span></span></figcaption><div class="highlight"><table><tr><td class="gutter"><pre class="line-numbers"><span class='line-number'>1</span>
<span class='line-number'>2</span>
<span class='line-number'>3</span>
<span class='line-number'>4</span>
<span class='line-number'>5</span>
<span class='line-number'>6</span>
</pre></td><td class='code'><pre><code class='ruby'><span class='line'><span class="vg">$_INC</span> <span class="o">=</span> <span class="n">L</span><span class="p">{</span><span class="o">|</span><span class="n">n</span><span class="o">|</span> <span class="n">L</span><span class="p">{</span><span class="o">|</span><span class="n">f</span><span class="o">|</span> <span class="n">L</span><span class="p">{</span><span class="o">|</span><span class="n">x</span><span class="o">|</span> <span class="n">f</span><span class="o">[</span><span class="n">n</span><span class="o">[</span><span class="n">f</span><span class="o">][</span><span class="n">x</span><span class="o">]]</span><span class="p">}}}</span>
</span><span class='line'><span class="vg">$_ADD</span> <span class="o">=</span> <span class="n">L</span><span class="p">{</span><span class="o">|</span><span class="n">n</span><span class="o">|</span> <span class="n">L</span><span class="p">{</span><span class="o">|</span><span class="n">m</span><span class="o">|</span> <span class="n">m</span><span class="o">[</span><span class="vg">$_INC</span><span class="o">][</span><span class="n">n</span><span class="o">]</span><span class="p">}}</span>
</span><span class='line'>
</span><span class='line'><span class="c1">#验证结果 4 + 5</span>
</span><span class='line'><span class="n">printL</span> <span class="vg">$_ADD</span><span class="o">[</span><span class="vg">$_4</span><span class="o">][</span><span class="vg">$_5</span><span class="o">]</span>
</span><span class='line'><span class="c1">#输出 9</span>
</span></code></pre></td></tr></table></div></figure>
<p>乘法的定义：<br/>
MULT = λm.λn.m (ADD n) 0<br/>
乘法的代码如下:</p>
<figure class='code'><figcaption><span></span></figcaption><div class="highlight"><table><tr><td class="gutter"><pre class="line-numbers"><span class='line-number'>1</span>
<span class='line-number'>2</span>
<span class='line-number'>3</span>
<span class='line-number'>4</span>
<span class='line-number'>5</span>
</pre></td><td class='code'><pre><code class='ruby'><span class='line'><span class="vg">$_MULT</span> <span class="o">=</span> <span class="n">L</span><span class="p">{</span><span class="o">|</span><span class="n">m</span><span class="o">|</span> <span class="n">L</span><span class="p">{</span><span class="o">|</span><span class="n">n</span><span class="o">|</span> <span class="n">m</span><span class="o">[</span><span class="vg">$_ADD</span><span class="o">[</span><span class="n">n</span><span class="o">]][</span><span class="vg">$_0</span><span class="o">]</span><span class="p">}}</span>
</span><span class='line'>
</span><span class='line'><span class="c1">#验证结果 4*5</span>
</span><span class='line'><span class="n">printL</span> <span class="vg">$_MULT</span><span class="o">[</span><span class="vg">$_4</span><span class="o">][</span><span class="vg">$_5</span><span class="o">]</span>
</span><span class='line'><span class="c1">#输出 20</span>
</span></code></pre></td></tr></table></div></figure>
<p>阶乘的定义:<br/>
POW = λb.λe.e b<br/>
阶乘的代码如下:</p>
<figure class='code'><figcaption><span></span></figcaption><div class="highlight"><table><tr><td class="gutter"><pre class="line-numbers"><span class='line-number'>1</span>
<span class='line-number'>2</span>
<span class='line-number'>3</span>
<span class='line-number'>4</span>
</pre></td><td class='code'><pre><code class='ruby'><span class='line'><span class="vg">$_POW</span> <span class="o">=</span> <span class="n">L</span><span class="p">{</span><span class="o">|</span><span class="n">b</span><span class="o">|</span> <span class="n">L</span><span class="p">{</span><span class="o">|</span><span class="n">e</span><span class="o">|</span> <span class="n">e</span><span class="o">[</span><span class="n">b</span><span class="o">]</span><span class="p">}}</span>
</span><span class='line'><span class="c1"># 验证结果 3的四次幂</span>
</span><span class='line'><span class="n">printL</span> <span class="vg">$_POW</span><span class="o">[</span><span class="vg">$_3</span><span class="o">][</span><span class="vg">$_4</span><span class="o">]</span>
</span><span class='line'><span class="c1"># 输出 81 </span>
</span></code></pre></td></tr></table></div></figure>
<p></p>
<p>减法代码：</p>
<figure class='code'><figcaption><span></span></figcaption><div class="highlight"><table><tr><td class="gutter"><pre class="line-numbers"><span class='line-number'>1</span>
<span class='line-number'>2</span>
<span class='line-number'>3</span>
<span class='line-number'>4</span>
<span class='line-number'>5</span>
<span class='line-number'>6</span>
<span class='line-number'>7</span>
<span class='line-number'>8</span>
<span class='line-number'>9</span>
<span class='line-number'>10</span>
<span class='line-number'>11</span>
<span class='line-number'>12</span>
</pre></td><td class='code'><pre><code class='ruby'><span class='line'><span class="vg">$_WRAP</span> <span class="o">=</span> <span class="n">L</span><span class="p">{</span><span class="o">|</span><span class="n">f</span><span class="o">|</span> <span class="n">L</span><span class="p">{</span><span class="o">|</span><span class="nb">p</span><span class="o">|</span> <span class="vg">$_CONS</span> <span class="o">[</span><span class="vg">$_FALSE</span><span class="o">]</span> <span class="o">[</span><span class="vg">$_IF</span><span class="o">[</span><span class="vg">$_CAR</span><span class="o">[</span><span class="nb">p</span><span class="o">]][</span><span class="vg">$_CDR</span><span class="o">[</span><span class="nb">p</span><span class="o">]][</span><span class="n">f</span><span class="o">[</span><span class="vg">$_SND</span><span class="o">[</span><span class="nb">p</span><span class="o">]]]]</span><span class="p">}}</span>
</span><span class='line'><span class="vg">$_SUB1</span> <span class="o">=</span> <span class="n">L</span><span class="p">{</span><span class="o">|</span><span class="n">n</span><span class="o">|</span> <span class="n">L</span><span class="p">{</span><span class="o">|</span><span class="n">f</span><span class="o">|</span> <span class="n">L</span><span class="p">{</span><span class="o">|</span><span class="n">x</span><span class="o">|</span> <span class="vg">$_CDR</span><span class="o">[</span><span class="n">n</span><span class="o">[</span><span class="vg">$_WRAP</span><span class="o">[</span><span class="n">f</span><span class="o">]][</span><span class="vg">$_CONS</span><span class="o">[</span><span class="vg">$_TRUE</span><span class="o">][</span><span class="n">x</span><span class="o">]]]</span><span class="p">}}}</span>
</span><span class='line'><span class="vg">$_SUB</span> <span class="o">=</span> <span class="n">L</span><span class="p">{</span><span class="o">|</span><span class="n">n</span><span class="o">|</span> <span class="n">L</span><span class="p">{</span><span class="o">|</span><span class="n">m</span><span class="o">|</span> <span class="n">m</span><span class="o">[</span><span class="vg">$_SUB1</span><span class="o">][</span><span class="n">n</span><span class="o">]</span><span class="p">}}</span>
</span><span class='line'>
</span><span class='line'>
</span><span class='line'><span class="c1">#验证 3自减1</span>
</span><span class='line'><span class="n">printL</span> <span class="vg">$_SUB1</span><span class="o">[</span><span class="vg">$_3</span><span class="o">]</span>
</span><span class='line'><span class="c1">#输出 2</span>
</span><span class='line'>
</span><span class='line'><span class="c1">#验证 5 减去 2</span>
</span><span class='line'><span class="n">printL</span> <span class="vg">$_SUB</span><span class="o">[</span><span class="vg">$_5</span><span class="o">][</span><span class="vg">$_2</span><span class="o">]</span>
</span><span class='line'><span class="c1">#输出 3</span>
</span></code></pre></td></tr></table></div></figure>
]]></content>
</entry>
<entry>
<title type="html"><![CDATA[About me]]></title>
<link href="http://twqingsong.github.com/blog/2013/03/03/about-me/"/>
<updated>2013-03-03T22:37:00+08:00</updated>
<id>http://twqingsong.github.com/blog/2013/03/03/about-me</id>
<content type="html"><![CDATA[<p>Hello everyone,</p>
<p>This is my new Github blog address, welcome to visit!</p>
<p>For contact me, send email to [email protected].</p>
<p><img class="left" src="http://twqingsong.github.com/images/2013/02/avatar.png" width="300" height="300"></p>
<p>Thanks!</p>
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