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22 | <a href="./Home">MLton 20180207</a>\r | |
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25 | <div id="header">\r | |
26 | <h1>VariableArityPolymorphism</h1>\r | |
27 | </div>\r | |
28 | <div id="content">\r | |
29 | <div id="preamble">\r | |
30 | <div class="sectionbody">\r | |
31 | <div class="paragraph"><p><a href="StandardML">Standard ML</a> programmers often face the problem of how to\r | |
32 | provide a variable-arity polymorphic function. For example, suppose\r | |
33 | one is defining a combinator library, e.g. for parsing or pickling.\r | |
34 | The signature for such a library might look something like the\r | |
35 | following.</p></div>\r | |
36 | <div class="listingblock">\r | |
37 | <div class="content"><div class="highlight"><pre><span class="k">signature</span><span class="w"> </span><span class="n">COMBINATOR</span><span class="w"> </span><span class="p">=</span><span class="w"></span>\r | |
38 | <span class="w"> </span><span class="k">sig</span><span class="w"></span>\r | |
39 | <span class="w"> </span><span class="k">type</span><span class="w"> </span><span class="n">'a</span><span class="w"> </span><span class="n">t</span><span class="w"></span>\r | |
40 | \r | |
41 | <span class="w"> </span><span class="k">val</span><span class="w"> </span><span class="n">int</span><span class="p">:</span><span class="w"> </span><span class="n">int</span><span class="w"> </span><span class="n">t</span><span class="w"></span>\r | |
42 | <span class="w"> </span><span class="k">val</span><span class="w"> </span><span class="n">real</span><span class="p">:</span><span class="w"> </span><span class="n">real</span><span class="w"> </span><span class="n">t</span><span class="w"></span>\r | |
43 | <span class="w"> </span><span class="k">val</span><span class="w"> </span><span class="n">string</span><span class="p">:</span><span class="w"> </span><span class="n">string</span><span class="w"> </span><span class="n">t</span><span class="w"></span>\r | |
44 | <span class="w"> </span><span class="k">val</span><span class="w"> </span><span class="n">unit</span><span class="p">:</span><span class="w"> </span><span class="n">unit</span><span class="w"> </span><span class="n">t</span><span class="w"></span>\r | |
45 | <span class="w"> </span><span class="k">val</span><span class="w"> </span><span class="n">tuple2</span><span class="p">:</span><span class="w"> </span><span class="n">'a1</span><span class="w"> </span><span class="n">t</span><span class="w"> </span><span class="n">*</span><span class="w"> </span><span class="n">'a2</span><span class="w"> </span><span class="n">t</span><span class="w"> </span><span class="p">-></span><span class="w"> </span><span class="p">(</span><span class="n">'a1</span><span class="w"> </span><span class="n">*</span><span class="w"> </span><span class="n">'a2</span><span class="p">)</span><span class="w"> </span><span class="n">t</span><span class="w"></span>\r | |
46 | <span class="w"> </span><span class="k">val</span><span class="w"> </span><span class="n">tuple3</span><span class="p">:</span><span class="w"> </span><span class="n">'a1</span><span class="w"> </span><span class="n">t</span><span class="w"> </span><span class="n">*</span><span class="w"> </span><span class="n">'a2</span><span class="w"> </span><span class="n">t</span><span class="w"> </span><span class="n">*</span><span class="w"> </span><span class="n">'a3</span><span class="w"> </span><span class="n">t</span><span class="w"> </span><span class="p">-></span><span class="w"> </span><span class="p">(</span><span class="n">'a1</span><span class="w"> </span><span class="n">*</span><span class="w"> </span><span class="n">'a2</span><span class="w"> </span><span class="n">*</span><span class="w"> </span><span class="n">'a3</span><span class="p">)</span><span class="w"> </span><span class="n">t</span><span class="w"></span>\r | |
47 | <span class="w"> </span><span class="k">val</span><span class="w"> </span><span class="n">tuple4</span><span class="p">:</span><span class="w"> </span><span class="n">'a1</span><span class="w"> </span><span class="n">t</span><span class="w"> </span><span class="n">*</span><span class="w"> </span><span class="n">'a2</span><span class="w"> </span><span class="n">t</span><span class="w"> </span><span class="n">*</span><span class="w"> </span><span class="n">'a3</span><span class="w"> </span><span class="n">t</span><span class="w"> </span><span class="n">*</span><span class="w"> </span><span class="n">'a4</span><span class="w"> </span><span class="n">t</span><span class="w"></span>\r | |
48 | <span class="w"> </span><span class="p">-></span><span class="w"> </span><span class="p">(</span><span class="n">'a1</span><span class="w"> </span><span class="n">*</span><span class="w"> </span><span class="n">'a2</span><span class="w"> </span><span class="n">*</span><span class="w"> </span><span class="n">'a3</span><span class="w"> </span><span class="n">*</span><span class="w"> </span><span class="n">'a4</span><span class="p">)</span><span class="w"> </span><span class="n">t</span><span class="w"></span>\r | |
49 | <span class="w"> </span><span class="p">...</span><span class="w"></span>\r | |
50 | <span class="w"> </span><span class="k">end</span><span class="w"></span>\r | |
51 | </pre></div></div></div>\r | |
52 | <div class="paragraph"><p>The question is how to define a variable-arity tuple combinator.\r | |
53 | Traditionally, the only way to take a variable number of arguments in\r | |
54 | SML is to put the arguments in a list (or vector) and pass that. So,\r | |
55 | one might define a tuple combinator with the following signature.</p></div>\r | |
56 | <div class="listingblock">\r | |
57 | <div class="content"><div class="highlight"><pre><span class="k">val</span><span class="w"> </span><span class="n">tupleN</span><span class="p">:</span><span class="w"> </span><span class="n">'a</span><span class="w"> </span><span class="n">list</span><span class="w"> </span><span class="p">-></span><span class="w"> </span><span class="n">'a</span><span class="w"> </span><span class="n">list</span><span class="w"> </span><span class="n">t</span><span class="w"></span>\r | |
58 | </pre></div></div></div>\r | |
59 | <div class="paragraph"><p>The problem with this approach is that as soon as one places values in\r | |
60 | a list, they must all have the same type. So, programmers often take\r | |
61 | an alternative approach, and define a family of <span class="monospaced">tuple<N></span> functions,\r | |
62 | as we see in the <span class="monospaced">COMBINATOR</span> signature above.</p></div>\r | |
63 | <div class="paragraph"><p>The family-of-functions approach is ugly for many reasons. First, it\r | |
64 | clutters the signature with a number of functions when there should\r | |
65 | really only be one. Second, it is <em>closed</em>, in that there are a fixed\r | |
66 | number of tuple combinators in the interface, and should a client need\r | |
67 | a combinator for a large tuple, he is out of luck. Third, this\r | |
68 | approach often requires a lot of duplicate code in the implementation\r | |
69 | of the combinators.</p></div>\r | |
70 | <div class="paragraph"><p>Fortunately, using <a href="Fold01N">Fold01N</a> and <a href="ProductType">products</a>, one can\r | |
71 | provide an interface and implementation that solves all these\r | |
72 | problems. Here is a simple pickling module that converts values to\r | |
73 | strings.</p></div>\r | |
74 | <div class="listingblock">\r | |
75 | <div class="content"><div class="highlight"><pre><span class="k">structure</span><span class="w"> </span><span class="n">Pickler</span><span class="w"> </span><span class="p">=</span><span class="w"></span>\r | |
76 | <span class="w"> </span><span class="k">struct</span><span class="w"></span>\r | |
77 | <span class="w"> </span><span class="k">type</span><span class="w"> </span><span class="n">'a</span><span class="w"> </span><span class="n">t</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">'a</span><span class="w"> </span><span class="p">-></span><span class="w"> </span><span class="n">string</span><span class="w"></span>\r | |
78 | \r | |
79 | <span class="w"> </span><span class="k">val</span><span class="w"> </span><span class="n">unit</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="p">()</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="s">""</span><span class="w"></span>\r | |
80 | \r | |
81 | <span class="w"> </span><span class="k">val</span><span class="w"> </span><span class="n">int</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">Int</span><span class="p">.</span><span class="n">toString</span><span class="w"></span>\r | |
82 | \r | |
83 | <span class="w"> </span><span class="k">val</span><span class="w"> </span><span class="n">real</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">Real</span><span class="p">.</span><span class="n">toString</span><span class="w"></span>\r | |
84 | \r | |
85 | <span class="w"> </span><span class="k">val</span><span class="w"> </span><span class="n">string</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">id</span><span class="w"></span>\r | |
86 | \r | |
87 | <span class="w"> </span><span class="k">type</span><span class="w"> </span><span class="n">'a</span><span class="w"> </span><span class="n">accum</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">'a</span><span class="w"> </span><span class="n">*</span><span class="w"> </span><span class="n">string</span><span class="w"> </span><span class="n">list</span><span class="w"> </span><span class="p">-></span><span class="w"> </span><span class="n">string</span><span class="w"> </span><span class="n">list</span><span class="w"></span>\r | |
88 | \r | |
89 | <span class="w"> </span><span class="k">val</span><span class="w"> </span><span class="n">tuple</span><span class="w"> </span><span class="p">=</span><span class="w"></span>\r | |
90 | <span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="n">z</span><span class="w"> </span><span class="p">=></span><span class="w"></span>\r | |
91 | <span class="w"> </span><span class="n">Fold01N</span><span class="p">.</span><span class="n">fold</span><span class="w"></span>\r | |
92 | <span class="w"> </span><span class="p">{</span><span class="n">finish</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="n">ps</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="n">x</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="n">concat</span><span class="w"> </span><span class="p">(</span><span class="n">rev</span><span class="w"> </span><span class="p">(</span><span class="n">ps</span><span class="w"> </span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="w"> </span><span class="p">[]))),</span><span class="w"></span>\r | |
93 | <span class="w"> </span><span class="n">start</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="n">p</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="w"> </span><span class="n">l</span><span class="p">)</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="n">p</span><span class="w"> </span><span class="n">x</span><span class="w"> </span><span class="n">::</span><span class="w"> </span><span class="n">l</span><span class="p">,</span><span class="w"></span>\r | |
94 | <span class="w"> </span><span class="n">zero</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">unit</span><span class="p">}</span><span class="w"></span>\r | |
95 | <span class="w"> </span><span class="n">z</span><span class="w"></span>\r | |
96 | \r | |
97 | <span class="w"> </span><span class="k">val</span><span class="w"> </span><span class="n">`</span><span class="w"> </span><span class="p">=</span><span class="w"></span>\r | |
98 | <span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="n">z</span><span class="w"> </span><span class="p">=></span><span class="w"></span>\r | |
99 | <span class="w"> </span><span class="n">Fold01N</span><span class="p">.</span><span class="n">step1</span><span class="w"></span>\r | |
100 | <span class="w"> </span><span class="p">{</span><span class="n">combine</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="p">(</span><span class="k">fn</span><span class="w"> </span><span class="p">(</span><span class="n">p</span><span class="p">,</span><span class="w"> </span><span class="n">p'</span><span class="p">)</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="p">(</span><span class="n">x</span><span class="w"> </span><span class="n">&</span><span class="w"> </span><span class="n">x'</span><span class="p">,</span><span class="w"> </span><span class="n">l</span><span class="p">)</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="n">p'</span><span class="w"> </span><span class="n">x'</span><span class="w"> </span><span class="n">::</span><span class="w"> </span><span class="s">","</span><span class="w"> </span><span class="n">::</span><span class="w"> </span><span class="n">p</span><span class="w"> </span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="w"> </span><span class="n">l</span><span class="p">))}</span><span class="w"></span>\r | |
101 | <span class="w"> </span><span class="n">z</span><span class="w"></span>\r | |
102 | <span class="w"> </span><span class="k">end</span><span class="w"></span>\r | |
103 | </pre></div></div></div>\r | |
104 | <div class="paragraph"><p>If one has <span class="monospaced">n</span> picklers of types</p></div>\r | |
105 | <div class="listingblock">\r | |
106 | <div class="content"><div class="highlight"><pre><span class="k">val</span><span class="w"> </span><span class="n">p1</span><span class="p">:</span><span class="w"> </span><span class="n">a1</span><span class="w"> </span><span class="n">Pickler</span><span class="p">.</span><span class="n">t</span><span class="w"></span>\r | |
107 | <span class="k">val</span><span class="w"> </span><span class="n">p2</span><span class="p">:</span><span class="w"> </span><span class="n">a2</span><span class="w"> </span><span class="n">Pickler</span><span class="p">.</span><span class="n">t</span><span class="w"></span>\r | |
108 | <span class="p">...</span><span class="w"></span>\r | |
109 | <span class="k">val</span><span class="w"> </span><span class="n">pn</span><span class="p">:</span><span class="w"> </span><span class="n">an</span><span class="w"> </span><span class="n">Pickler</span><span class="p">.</span><span class="n">t</span><span class="w"></span>\r | |
110 | </pre></div></div></div>\r | |
111 | <div class="paragraph"><p>then one can construct a pickler for n-ary products as follows.</p></div>\r | |
112 | <div class="listingblock">\r | |
113 | <div class="content"><div class="highlight"><pre><span class="n">tuple</span><span class="w"> </span><span class="n">`p1</span><span class="w"> </span><span class="n">`p2</span><span class="w"> </span><span class="p">...</span><span class="w"> </span><span class="n">`pn</span><span class="w"> </span><span class="n">$</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="p">(</span><span class="n">a1</span><span class="w"> </span><span class="n">&</span><span class="w"> </span><span class="n">a2</span><span class="w"> </span><span class="n">&</span><span class="w"> </span><span class="p">...</span><span class="w"> </span><span class="n">&</span><span class="w"> </span><span class="n">an</span><span class="p">)</span><span class="w"> </span><span class="n">Pickler</span><span class="p">.</span><span class="n">t</span><span class="w"></span>\r | |
114 | </pre></div></div></div>\r | |
115 | <div class="paragraph"><p>For example, with <span class="monospaced">Pickler</span> in scope, one can prove the following\r | |
116 | equations.</p></div>\r | |
117 | <div class="listingblock">\r | |
118 | <div class="content"><div class="highlight"><pre><span class="s">""</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">tuple</span><span class="w"> </span><span class="n">$</span><span class="w"> </span><span class="p">()</span><span class="w"></span>\r | |
119 | <span class="s">"1"</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">tuple</span><span class="w"> </span><span class="n">`int</span><span class="w"> </span><span class="n">$</span><span class="w"> </span><span class="mi">1</span><span class="w"></span>\r | |
120 | <span class="s">"1,2.0"</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">tuple</span><span class="w"> </span><span class="n">`int</span><span class="w"> </span><span class="n">`real</span><span class="w"> </span><span class="n">$</span><span class="w"> </span><span class="p">(</span><span class="mi">1</span><span class="w"> </span><span class="n">&</span><span class="w"> </span><span class="mf">2.0</span><span class="p">)</span><span class="w"></span>\r | |
121 | <span class="s">"1,2.0,three"</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">tuple</span><span class="w"> </span><span class="n">`int</span><span class="w"> </span><span class="n">`real</span><span class="w"> </span><span class="n">`string</span><span class="w"> </span><span class="n">$</span><span class="w"> </span><span class="p">(</span><span class="mi">1</span><span class="w"> </span><span class="n">&</span><span class="w"> </span><span class="mf">2.0</span><span class="w"> </span><span class="n">&</span><span class="w"> </span><span class="s">"three"</span><span class="p">)</span><span class="w"></span>\r | |
122 | </pre></div></div></div>\r | |
123 | <div class="paragraph"><p>Here is the signature for <span class="monospaced">Pickler</span>. It shows why the <span class="monospaced">accum</span> type is\r | |
124 | useful.</p></div>\r | |
125 | <div class="listingblock">\r | |
126 | <div class="content"><div class="highlight"><pre><span class="k">signature</span><span class="w"> </span><span class="n">PICKLER</span><span class="w"> </span><span class="p">=</span><span class="w"></span>\r | |
127 | <span class="w"> </span><span class="k">sig</span><span class="w"></span>\r | |
128 | <span class="w"> </span><span class="k">type</span><span class="w"> </span><span class="n">'a</span><span class="w"> </span><span class="n">t</span><span class="w"></span>\r | |
129 | \r | |
130 | <span class="w"> </span><span class="k">val</span><span class="w"> </span><span class="n">int</span><span class="p">:</span><span class="w"> </span><span class="n">int</span><span class="w"> </span><span class="n">t</span><span class="w"></span>\r | |
131 | <span class="w"> </span><span class="k">val</span><span class="w"> </span><span class="n">real</span><span class="p">:</span><span class="w"> </span><span class="n">real</span><span class="w"> </span><span class="n">t</span><span class="w"></span>\r | |
132 | <span class="w"> </span><span class="k">val</span><span class="w"> </span><span class="n">string</span><span class="p">:</span><span class="w"> </span><span class="n">string</span><span class="w"> </span><span class="n">t</span><span class="w"></span>\r | |
133 | <span class="w"> </span><span class="k">val</span><span class="w"> </span><span class="n">unit</span><span class="p">:</span><span class="w"> </span><span class="n">unit</span><span class="w"> </span><span class="n">t</span><span class="w"></span>\r | |
134 | \r | |
135 | <span class="w"> </span><span class="k">type</span><span class="w"> </span><span class="n">'a</span><span class="w"> </span><span class="n">accum</span><span class="w"></span>\r | |
136 | <span class="w"> </span><span class="k">val</span><span class="w"> </span><span class="n">`</span><span class="w"> </span><span class="p">:</span><span class="w"> </span><span class="p">(</span><span class="n">'a</span><span class="w"> </span><span class="n">accum</span><span class="p">,</span><span class="w"> </span><span class="n">'b</span><span class="w"> </span><span class="n">t</span><span class="p">,</span><span class="w"> </span><span class="p">(</span><span class="n">'a</span><span class="p">,</span><span class="w"> </span><span class="n">'b</span><span class="p">)</span><span class="w"> </span><span class="n">prod</span><span class="w"> </span><span class="n">accum</span><span class="p">,</span><span class="w"></span>\r | |
137 | <span class="w"> </span><span class="n">'z1</span><span class="p">,</span><span class="w"> </span><span class="n">'z2</span><span class="p">,</span><span class="w"> </span><span class="n">'z3</span><span class="p">,</span><span class="w"> </span><span class="n">'z4</span><span class="p">,</span><span class="w"> </span><span class="n">'z5</span><span class="p">,</span><span class="w"> </span><span class="n">'z6</span><span class="p">,</span><span class="w"> </span><span class="n">'z7</span><span class="p">)</span><span class="w"> </span><span class="n">Fold01N</span><span class="p">.</span><span class="n">step1</span><span class="w"></span>\r | |
138 | <span class="w"> </span><span class="k">val</span><span class="w"> </span><span class="n">tuple</span><span class="p">:</span><span class="w"> </span><span class="p">(</span><span class="n">'a</span><span class="w"> </span><span class="n">t</span><span class="p">,</span><span class="w"> </span><span class="n">'a</span><span class="w"> </span><span class="n">accum</span><span class="p">,</span><span class="w"> </span><span class="n">'b</span><span class="w"> </span><span class="n">accum</span><span class="p">,</span><span class="w"> </span><span class="n">'b</span><span class="w"> </span><span class="n">t</span><span class="p">,</span><span class="w"> </span><span class="n">unit</span><span class="w"> </span><span class="n">t</span><span class="p">,</span><span class="w"></span>\r | |
139 | <span class="w"> </span><span class="n">'z1</span><span class="p">,</span><span class="w"> </span><span class="n">'z2</span><span class="p">,</span><span class="w"> </span><span class="n">'z3</span><span class="p">,</span><span class="w"> </span><span class="n">'z4</span><span class="p">,</span><span class="w"> </span><span class="n">'z5</span><span class="p">)</span><span class="w"> </span><span class="n">Fold01N</span><span class="p">.</span><span class="n">t</span><span class="w"></span>\r | |
140 | <span class="w"> </span><span class="k">end</span><span class="w"></span>\r | |
141 | \r | |
142 | <span class="k">structure</span><span class="w"> </span><span class="n">Pickler</span><span class="p">:</span><span class="w"> </span><span class="n">PICKLER</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">Pickler</span><span class="w"></span>\r | |
143 | </pre></div></div></div>\r | |
144 | </div>\r | |
145 | </div>\r | |
146 | </div>\r | |
147 | <div id="footnotes"><hr></div>\r | |
148 | <div id="footer">\r | |
149 | <div id="footer-text">\r | |
150 | </div>\r | |
151 | <div id="footer-badges">\r | |
152 | </div>\r | |
153 | </div>\r | |
154 | </body>\r | |
155 | </html>\r |