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| 22 | <a href="./Home">MLton 20180207</a>\r |
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| 25 | <div id="header">\r |
| 26 | <h1>ValueRestriction</h1>\r |
| 27 | <div id="toc"> |
| 28 | <div id="toctitle">Table of Contents</div> |
| 29 | <noscript><p><b>JavaScript must be enabled in your browser to display the table of contents.</b></p></noscript> |
| 30 | </div>\r |
| 31 | </div>\r |
| 32 | <div id="content">\r |
| 33 | <div id="preamble">\r |
| 34 | <div class="sectionbody">\r |
| 35 | <div class="paragraph"><p>The value restriction is a rule that governs when type inference is\r |
| 36 | allowed to polymorphically generalize a value declaration. In short,\r |
| 37 | the value restriction says that generalization can only occur if the\r |
| 38 | right-hand side of an expression is syntactically a value. For\r |
| 39 | example, in</p></div>\r |
| 40 | <div class="listingblock">\r |
| 41 | <div class="content"><div class="highlight"><pre><span class="k">val</span><span class="w"> </span><span class="n">f</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">x</span><span class="w"></span>\r |
| 42 | <span class="k">val</span><span class="w"> </span><span class="p">_</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="p">(</span><span class="n">f</span><span class="w"> </span><span class="s">"foo"</span><span class="p">;</span><span class="w"> </span><span class="n">f</span><span class="w"> </span><span class="mi">13</span><span class="p">)</span><span class="w"></span>\r |
| 43 | </pre></div></div></div>\r |
| 44 | <div class="paragraph"><p>the expression <span class="monospaced">fn x => x</span> is syntactically a value, so <span class="monospaced">f</span> has\r |
| 45 | polymorphic type <span class="monospaced">'a -> 'a</span> and both calls to <span class="monospaced">f</span> type check. On the\r |
| 46 | other hand, in</p></div>\r |
| 47 | <div class="listingblock">\r |
| 48 | <div class="content"><div class="highlight"><pre><span class="k">val</span><span class="w"> </span><span class="n">f</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="k">let</span><span class="w"> </span><span class="k">in</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">x</span><span class="w"> </span><span class="k">end</span><span class="w"></span>\r |
| 49 | <span class="k">val</span><span class="w"> </span><span class="p">_</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="p">(</span><span class="n">f</span><span class="w"> </span><span class="s">"foo"</span><span class="p">;</span><span class="w"> </span><span class="n">f</span><span class="w"> </span><span class="mi">13</span><span class="p">)</span><span class="w"></span>\r |
| 50 | </pre></div></div></div>\r |
| 51 | <div class="paragraph"><p>the expression <span class="monospaced">let in fn x => end end</span> is not syntactically a value\r |
| 52 | and so <span class="monospaced">f</span> can either have type <span class="monospaced">int -> int</span> or <span class="monospaced">string -> string</span>,\r |
| 53 | but not <span class="monospaced">'a -> 'a</span>. Hence, the program does not type check.</p></div>\r |
| 54 | <div class="paragraph"><p><a href="DefinitionOfStandardML">The Definition of Standard ML</a> spells out\r |
| 55 | precisely which expressions are syntactic values (it refers to such\r |
| 56 | expressions as <em>non-expansive</em>). An expression is a value if it is of\r |
| 57 | one of the following forms.</p></div>\r |
| 58 | <div class="ulist"><ul>\r |
| 59 | <li>\r |
| 60 | <p>\r |
| 61 | a constant (<span class="monospaced">13</span>, <span class="monospaced">"foo"</span>, <span class="monospaced">13.0</span>, …)\r |
| 62 | </p>\r |
| 63 | </li>\r |
| 64 | <li>\r |
| 65 | <p>\r |
| 66 | a variable (<span class="monospaced">x</span>, <span class="monospaced">y</span>, …)\r |
| 67 | </p>\r |
| 68 | </li>\r |
| 69 | <li>\r |
| 70 | <p>\r |
| 71 | a function (<span class="monospaced">fn x => e</span>)\r |
| 72 | </p>\r |
| 73 | </li>\r |
| 74 | <li>\r |
| 75 | <p>\r |
| 76 | the application of a constructor other than <span class="monospaced">ref</span> to a value (<span class="monospaced">Foo v</span>)\r |
| 77 | </p>\r |
| 78 | </li>\r |
| 79 | <li>\r |
| 80 | <p>\r |
| 81 | a type constrained value (<span class="monospaced">v: t</span>)\r |
| 82 | </p>\r |
| 83 | </li>\r |
| 84 | <li>\r |
| 85 | <p>\r |
| 86 | a tuple in which each field is a value <span class="monospaced">(v1, v2, ...)</span>\r |
| 87 | </p>\r |
| 88 | </li>\r |
| 89 | <li>\r |
| 90 | <p>\r |
| 91 | a record in which each field is a value <span class="monospaced">{l1 = v1, l2 = v2, ...}</span>\r |
| 92 | </p>\r |
| 93 | </li>\r |
| 94 | <li>\r |
| 95 | <p>\r |
| 96 | a list in which each element is a value <span class="monospaced">[v1, v2, ...]</span>\r |
| 97 | </p>\r |
| 98 | </li>\r |
| 99 | </ul></div>\r |
| 100 | </div>\r |
| 101 | </div>\r |
| 102 | <div class="sect1">\r |
| 103 | <h2 id="_why_the_value_restriction_exists">Why the value restriction exists</h2>\r |
| 104 | <div class="sectionbody">\r |
| 105 | <div class="paragraph"><p>The value restriction prevents a ref cell (or an array) from holding\r |
| 106 | values of different types, which would allow a value of one type to be\r |
| 107 | cast to another and hence would break type safety. If the restriction\r |
| 108 | were not in place, the following program would type check.</p></div>\r |
| 109 | <div class="listingblock">\r |
| 110 | <div class="content"><div class="highlight"><pre><span class="k">val</span><span class="w"> </span><span class="n">r</span><span class="p">:</span><span class="w"> </span><span class="n">'a</span><span class="w"> </span><span class="n">option</span><span class="w"> </span><span class="n">ref</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">ref</span><span class="w"> </span><span class="n">NONE</span><span class="w"></span>\r |
| 111 | <span class="k">val</span><span class="w"> </span><span class="n">r1</span><span class="p">:</span><span class="w"> </span><span class="n">string</span><span class="w"> </span><span class="n">option</span><span class="w"> </span><span class="n">ref</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">r</span><span class="w"></span>\r |
| 112 | <span class="k">val</span><span class="w"> </span><span class="n">r2</span><span class="p">:</span><span class="w"> </span><span class="n">int</span><span class="w"> </span><span class="n">option</span><span class="w"> </span><span class="n">ref</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">r</span><span class="w"></span>\r |
| 113 | <span class="k">val</span><span class="w"> </span><span class="p">()</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">r1</span><span class="w"> </span><span class="n">:=</span><span class="w"> </span><span class="n">SOME</span><span class="w"> </span><span class="s">"foo"</span><span class="w"></span>\r |
| 114 | <span class="k">val</span><span class="w"> </span><span class="n">v</span><span class="p">:</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">valOf</span><span class="w"> </span><span class="p">(</span><span class="n">!r2</span><span class="p">)</span><span class="w"></span>\r |
| 115 | </pre></div></div></div>\r |
| 116 | <div class="paragraph"><p>The first line violates the value restriction because <span class="monospaced">ref NONE</span> is\r |
| 117 | not a value. All other lines are type correct. By its last line, the\r |
| 118 | program has cast the string <span class="monospaced">"foo"</span> to an integer. This breaks type\r |
| 119 | safety, because now we can add a string to an integer with an\r |
| 120 | expression like <span class="monospaced">v + 13</span>. We could even be more devious, by adding\r |
| 121 | the following two lines, which allow us to threat the string <span class="monospaced">"foo"</span>\r |
| 122 | as a function.</p></div>\r |
| 123 | <div class="listingblock">\r |
| 124 | <div class="content"><div class="highlight"><pre><span class="k">val</span><span class="w"> </span><span class="n">r3</span><span class="p">:</span><span class="w"> </span><span class="p">(</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="w"> </span><span class="n">option</span><span class="w"> </span><span class="n">ref</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">r</span><span class="w"></span>\r |
| 125 | <span class="k">val</span><span class="w"> </span><span class="n">v</span><span class="p">:</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="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">valOf</span><span class="w"> </span><span class="p">(</span><span class="n">!r3</span><span class="p">)</span><span class="w"></span>\r |
| 126 | </pre></div></div></div>\r |
| 127 | <div class="paragraph"><p>Eliminating the explicit <span class="monospaced">ref</span> does nothing to fix the problem. For\r |
| 128 | example, we could replace the declaration of <span class="monospaced">r</span> with the following.</p></div>\r |
| 129 | <div class="listingblock">\r |
| 130 | <div class="content"><div class="highlight"><pre><span class="k">val</span><span class="w"> </span><span class="n">f</span><span class="p">:</span><span class="w"> </span><span class="n">unit</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">option</span><span class="w"> </span><span class="n">ref</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="n">ref</span><span class="w"> </span><span class="n">NONE</span><span class="w"></span>\r |
| 131 | <span class="k">val</span><span class="w"> </span><span class="n">r</span><span class="p">:</span><span class="w"> </span><span class="n">'a</span><span class="w"> </span><span class="n">option</span><span class="w"> </span><span class="n">ref</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">f</span><span class="w"> </span><span class="p">()</span><span class="w"></span>\r |
| 132 | </pre></div></div></div>\r |
| 133 | <div class="paragraph"><p>The declaration of <span class="monospaced">f</span> is well typed, while the declaration of <span class="monospaced">r</span>\r |
| 134 | violates the value restriction because <span class="monospaced">f ()</span> is not a value.</p></div>\r |
| 135 | </div>\r |
| 136 | </div>\r |
| 137 | <div class="sect1">\r |
| 138 | <h2 id="_unnecessarily_rejected_programs">Unnecessarily rejected programs</h2>\r |
| 139 | <div class="sectionbody">\r |
| 140 | <div class="paragraph"><p>Unfortunately, the value restriction rejects some programs that could\r |
| 141 | be accepted.</p></div>\r |
| 142 | <div class="listingblock">\r |
| 143 | <div class="content"><div class="highlight"><pre><span class="k">val</span><span class="w"> </span><span class="n">id</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">'a</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">x</span><span class="w"></span>\r |
| 144 | <span class="k">val</span><span class="w"> </span><span class="n">f</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">'a</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">id</span><span class="w"> </span><span class="n">id</span><span class="w"></span>\r |
| 145 | </pre></div></div></div>\r |
| 146 | <div class="paragraph"><p>The type constraint on <span class="monospaced">f</span> requires <span class="monospaced">f</span> to be polymorphic, which is\r |
| 147 | disallowed because <span class="monospaced">id id</span> is not a value. MLton reports the\r |
| 148 | following type error.</p></div>\r |
| 149 | <div class="listingblock">\r |
| 150 | <div class="content monospaced">\r |
| 151 | <pre>Error: z.sml 2.5-2.5.\r |
| 152 | Type of variable cannot be generalized in expansive declaration: f.\r |
| 153 | type: ['a] -> ['a]\r |
| 154 | in: val 'a f: ('a -> 'a) = id id</pre>\r |
| 155 | </div></div>\r |
| 156 | <div class="paragraph"><p>MLton indicates the inability to make <span class="monospaced">f</span> polymorphic by saying that\r |
| 157 | the type of <span class="monospaced">f</span> cannot be generalized (made polymorphic) its\r |
| 158 | declaration is expansive (not a value). MLton doesn’t explicitly\r |
| 159 | mention the value restriction, but that is the reason. If we leave\r |
| 160 | the type constraint off of <span class="monospaced">f</span></p></div>\r |
| 161 | <div class="listingblock">\r |
| 162 | <div class="content"><div class="highlight"><pre><span class="k">val</span><span class="w"> </span><span class="n">id</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">'a</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">x</span><span class="w"></span>\r |
| 163 | <span class="k">val</span><span class="w"> </span><span class="n">f</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">id</span><span class="w"> </span><span class="n">id</span><span class="w"></span>\r |
| 164 | </pre></div></div></div>\r |
| 165 | <div class="paragraph"><p>then the program succeeds; however, MLton gives us the following\r |
| 166 | warning.</p></div>\r |
| 167 | <div class="listingblock">\r |
| 168 | <div class="content monospaced">\r |
| 169 | <pre>Warning: z.sml 2.5-2.5.\r |
| 170 | Type of variable was not inferred and could not be generalized: f.\r |
| 171 | type: ??? -> ???\r |
| 172 | in: val f = id id</pre>\r |
| 173 | </div></div>\r |
| 174 | <div class="paragraph"><p>This warning indicates that MLton couldn’t polymorphically generalize\r |
| 175 | <span class="monospaced">f</span>, nor was there enough context using <span class="monospaced">f</span> to determine its type.\r |
| 176 | This in itself is not a type error, but it it is a hint that something\r |
| 177 | is wrong with our program. Using <span class="monospaced">f</span> provides enough context to\r |
| 178 | eliminate the warning.</p></div>\r |
| 179 | <div class="listingblock">\r |
| 180 | <div class="content"><div class="highlight"><pre><span class="k">val</span><span class="w"> </span><span class="n">id</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">'a</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">x</span><span class="w"></span>\r |
| 181 | <span class="k">val</span><span class="w"> </span><span class="n">f</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">id</span><span class="w"> </span><span class="n">id</span><span class="w"></span>\r |
| 182 | <span class="k">val</span><span class="w"> </span><span class="p">_</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">f</span><span class="w"> </span><span class="mi">13</span><span class="w"></span>\r |
| 183 | </pre></div></div></div>\r |
| 184 | <div class="paragraph"><p>But attempting to use <span class="monospaced">f</span> as a polymorphic function will fail.</p></div>\r |
| 185 | <div class="listingblock">\r |
| 186 | <div class="content"><div class="highlight"><pre><span class="k">val</span><span class="w"> </span><span class="n">id</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">'a</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">x</span><span class="w"></span>\r |
| 187 | <span class="k">val</span><span class="w"> </span><span class="n">f</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">id</span><span class="w"> </span><span class="n">id</span><span class="w"></span>\r |
| 188 | <span class="k">val</span><span class="w"> </span><span class="p">_</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">f</span><span class="w"> </span><span class="mi">13</span><span class="w"></span>\r |
| 189 | <span class="k">val</span><span class="w"> </span><span class="p">_</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">f</span><span class="w"> </span><span class="s">"foo"</span><span class="w"></span>\r |
| 190 | </pre></div></div></div>\r |
| 191 | <div class="listingblock">\r |
| 192 | <div class="content monospaced">\r |
| 193 | <pre>Error: z.sml 4.9-4.15.\r |
| 194 | Function applied to incorrect argument.\r |
| 195 | expects: [int]\r |
| 196 | but got: [string]\r |
| 197 | in: f "foo"</pre>\r |
| 198 | </div></div>\r |
| 199 | </div>\r |
| 200 | </div>\r |
| 201 | <div class="sect1">\r |
| 202 | <h2 id="_alternatives_to_the_value_restriction">Alternatives to the value restriction</h2>\r |
| 203 | <div class="sectionbody">\r |
| 204 | <div class="paragraph"><p>There would be nothing wrong with treating <span class="monospaced">f</span> as polymorphic in</p></div>\r |
| 205 | <div class="listingblock">\r |
| 206 | <div class="content"><div class="highlight"><pre><span class="k">val</span><span class="w"> </span><span class="n">id</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">'a</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">x</span><span class="w"></span>\r |
| 207 | <span class="k">val</span><span class="w"> </span><span class="n">f</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">id</span><span class="w"> </span><span class="n">id</span><span class="w"></span>\r |
| 208 | </pre></div></div></div>\r |
| 209 | <div class="paragraph"><p>One might think that the value restriction could be relaxed, and that\r |
| 210 | only types involving <span class="monospaced">ref</span> should be disallowed. Unfortunately, the\r |
| 211 | following example shows that even the type <span class="monospaced">'a -> 'a</span> can cause\r |
| 212 | problems. If this program were allowed, then we could cast an integer\r |
| 213 | to a string (or any other type).</p></div>\r |
| 214 | <div class="listingblock">\r |
| 215 | <div class="content"><div class="highlight"><pre><span class="k">val</span><span class="w"> </span><span class="n">f</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">'a</span><span class="w"> </span><span class="p">=</span><span class="w"></span>\r |
| 216 | <span class="w"> </span><span class="k">let</span><span class="w"></span>\r |
| 217 | <span class="w"> </span><span class="k">val</span><span class="w"> </span><span class="n">r</span><span class="p">:</span><span class="w"> </span><span class="n">'a</span><span class="w"> </span><span class="n">option</span><span class="w"> </span><span class="n">ref</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">ref</span><span class="w"> </span><span class="n">NONE</span><span class="w"></span>\r |
| 218 | <span class="w"> </span><span class="k">in</span><span class="w"></span>\r |
| 219 | <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>\r |
| 220 | <span class="w"> </span><span class="k">let</span><span class="w"></span>\r |
| 221 | <span class="w"> </span><span class="k">val</span><span class="w"> </span><span class="n">y</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">!r</span><span class="w"></span>\r |
| 222 | <span class="w"> </span><span class="k">val</span><span class="w"> </span><span class="p">()</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">r</span><span class="w"> </span><span class="n">:=</span><span class="w"> </span><span class="n">SOME</span><span class="w"> </span><span class="n">x</span><span class="w"></span>\r |
| 223 | <span class="w"> </span><span class="k">in</span><span class="w"></span>\r |
| 224 | <span class="w"> </span><span class="k">case</span><span class="w"> </span><span class="n">y</span><span class="w"> </span><span class="k">of</span><span class="w"></span>\r |
| 225 | <span class="w"> </span><span class="n">NONE</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="n">x</span><span class="w"></span>\r |
| 226 | <span class="w"> </span><span class="p">|</span><span class="w"> </span><span class="n">SOME</span><span class="w"> </span><span class="n">y</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="n">y</span><span class="w"></span>\r |
| 227 | <span class="w"> </span><span class="k">end</span><span class="w"></span>\r |
| 228 | <span class="w"> </span><span class="k">end</span><span class="w"></span>\r |
| 229 | <span class="k">val</span><span class="w"> </span><span class="p">_</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">f</span><span class="w"> </span><span class="mi">13</span><span class="w"></span>\r |
| 230 | <span class="k">val</span><span class="w"> </span><span class="p">_</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">f</span><span class="w"> </span><span class="s">"foo"</span><span class="w"></span>\r |
| 231 | </pre></div></div></div>\r |
| 232 | <div class="paragraph"><p>The previous version of Standard ML took a different approach\r |
| 233 | (<a href="References#MilnerEtAl90">MilnerEtAl90</a>, <a href="References#Tofte90">Tofte90</a>, <a href="ImperativeTypeVariable">ImperativeTypeVariable</a>)\r |
| 234 | than the value restriction. It encoded information in the type system\r |
| 235 | about when ref cells would be created, and used this to prevent a ref\r |
| 236 | cell from holding multiple types. Although it allowed more programs\r |
| 237 | to be type checked, this approach had significant drawbacks. First,\r |
| 238 | it was significantly more complex, both for implementers and for\r |
| 239 | programmers. Second, it had an unfortunate interaction with the\r |
| 240 | modularity, because information about ref usage was exposed in module\r |
| 241 | signatures. This either prevented the use of references for\r |
| 242 | implementing a signature, or required information that one would like\r |
| 243 | to keep hidden to propagate across modules.</p></div>\r |
| 244 | <div class="paragraph"><p>In the early nineties, Andrew Wright studied about 250,000 lines of\r |
| 245 | existing SML code and discovered that it did not make significant use\r |
| 246 | of the extended typing ability, and proposed the value restriction as\r |
| 247 | a simpler alternative (<a href="References#Wright95">Wright95</a>). This was adopted in the\r |
| 248 | revised <a href="DefinitionOfStandardML">Definition</a>.</p></div>\r |
| 249 | </div>\r |
| 250 | </div>\r |
| 251 | <div class="sect1">\r |
| 252 | <h2 id="_working_with_the_value_restriction">Working with the value restriction</h2>\r |
| 253 | <div class="sectionbody">\r |
| 254 | <div class="paragraph"><p>One technique that works with the value restriction is\r |
| 255 | <a href="EtaExpansion">EtaExpansion</a>. We can use eta expansion to make our <span class="monospaced">id id</span>\r |
| 256 | example type check follows.</p></div>\r |
| 257 | <div class="listingblock">\r |
| 258 | <div class="content"><div class="highlight"><pre><span class="k">val</span><span class="w"> </span><span class="n">id</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">'a</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">x</span><span class="w"></span>\r |
| 259 | <span class="k">val</span><span class="w"> </span><span class="n">f</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">'a</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">z</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="p">(</span><span class="n">id</span><span class="w"> </span><span class="n">id</span><span class="p">)</span><span class="w"> </span><span class="n">z</span><span class="w"></span>\r |
| 260 | </pre></div></div></div>\r |
| 261 | <div class="paragraph"><p>This solution means that the computation (in this case <span class="monospaced">id id</span>) will\r |
| 262 | be performed each time <span class="monospaced">f</span> is applied, instead of just once when <span class="monospaced">f</span>\r |
| 263 | is declared. In this case, that is not a problem, but it could be if\r |
| 264 | the declaration of <span class="monospaced">f</span> performs substantial computation or creates a\r |
| 265 | shared data structure.</p></div>\r |
| 266 | <div class="paragraph"><p>Another technique that sometimes works is to move a monomorphic\r |
| 267 | computation prior to a (would-be) polymorphic declaration so that the\r |
| 268 | expression is a value. Consider the following program, which fails\r |
| 269 | due to the value restriction.</p></div>\r |
| 270 | <div class="listingblock">\r |
| 271 | <div class="content"><div class="highlight"><pre><span class="k">datatype</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="k">of</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">B</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="n">'a</span><span class="w"></span>\r |
| 272 | <span class="k">val</span><span class="w"> </span><span class="n">x</span><span class="p">:</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="k">if</span><span class="w"> </span><span class="n">true</span><span class="w"> </span><span class="k">then</span><span class="w"> </span><span class="s">"yes"</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="s">"no"</span><span class="p">)</span><span class="w"></span>\r |
| 273 | </pre></div></div></div>\r |
| 274 | <div class="paragraph"><p>It is easy to rewrite this program as</p></div>\r |
| 275 | <div class="listingblock">\r |
| 276 | <div class="content"><div class="highlight"><pre><span class="k">datatype</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="k">of</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">B</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="n">'a</span><span class="w"></span>\r |
| 277 | <span class="k">local</span><span class="w"></span>\r |
| 278 | <span class="w"> </span><span class="k">val</span><span class="w"> </span><span class="n">s</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="k">if</span><span class="w"> </span><span class="n">true</span><span class="w"> </span><span class="k">then</span><span class="w"> </span><span class="s">"yes"</span><span class="w"> </span><span class="k">else</span><span class="w"> </span><span class="s">"no"</span><span class="w"></span>\r |
| 279 | <span class="k">in</span><span class="w"></span>\r |
| 280 | <span class="w"> </span><span class="k">val</span><span class="w"> </span><span class="n">x</span><span class="p">:</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="n">s</span><span class="w"></span>\r |
| 281 | <span class="k">end</span><span class="w"></span>\r |
| 282 | </pre></div></div></div>\r |
| 283 | <div class="paragraph"><p>The following example (taken from <a href="References#Wright95">Wright95</a>) creates a ref\r |
| 284 | cell to count the number of times a function is called.</p></div>\r |
| 285 | <div class="listingblock">\r |
| 286 | <div class="content"><div class="highlight"><pre><span class="k">val</span><span class="w"> </span><span class="n">count</span><span class="p">:</span><span class="w"> </span><span class="p">(</span><span class="n">'a</span><span class="w"> </span><span class="p">-></span><span class="w"> </span><span class="n">'a</span><span class="p">)</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="p">-></span><span class="w"> </span><span class="n">'a</span><span class="p">)</span><span class="w"> </span><span class="n">*</span><span class="w"> </span><span class="p">(</span><span class="n">unit</span><span class="w"> </span><span class="p">-></span><span class="w"> </span><span class="n">int</span><span class="p">)</span><span class="w"> </span><span class="p">=</span><span class="w"></span>\r |
| 287 | <span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="n">f</span><span class="w"> </span><span class="p">=></span><span class="w"></span>\r |
| 288 | <span class="w"> </span><span class="k">let</span><span class="w"></span>\r |
| 289 | <span class="w"> </span><span class="k">val</span><span class="w"> </span><span class="n">r</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">ref</span><span class="w"> </span><span class="mi">0</span><span class="w"></span>\r |
| 290 | <span class="w"> </span><span class="k">in</span><span class="w"></span>\r |
| 291 | <span class="w"> </span><span class="p">(</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="p">(</span><span class="n">r</span><span class="w"> </span><span class="n">:=</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="n">+</span><span class="w"> </span><span class="n">!r</span><span class="p">;</span><span class="w"> </span><span class="n">f</span><span class="w"> </span><span class="n">x</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="n">!r</span><span class="p">)</span><span class="w"></span>\r |
| 292 | <span class="w"> </span><span class="k">end</span><span class="w"></span>\r |
| 293 | <span class="k">val</span><span class="w"> </span><span class="n">id</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">'a</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">x</span><span class="w"></span>\r |
| 294 | <span class="k">val</span><span class="w"> </span><span class="p">(</span><span class="n">countId</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">'a</span><span class="p">,</span><span class="w"> </span><span class="n">numCalls</span><span class="p">)</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="n">count</span><span class="w"> </span><span class="n">id</span><span class="w"></span>\r |
| 295 | </pre></div></div></div>\r |
| 296 | <div class="paragraph"><p>The example does not type check, due to the value restriction.\r |
| 297 | However, it is easy to rewrite the program, staging the ref cell\r |
| 298 | creation before the polymorphic code.</p></div>\r |
| 299 | <div class="listingblock">\r |
| 300 | <div class="content"><div class="highlight"><pre><span class="k">datatype</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">T</span><span class="w"> </span><span class="k">of</span><span class="w"> </span><span class="n">int</span><span class="w"> </span><span class="n">ref</span><span class="w"></span>\r |
| 301 | <span class="k">val</span><span class="w"> </span><span class="n">count1</span><span class="p">:</span><span class="w"> </span><span class="n">unit</span><span class="w"> </span><span class="p">-></span><span class="w"> </span><span class="n">t</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="n">T</span><span class="w"> </span><span class="p">(</span><span class="n">ref</span><span class="w"> </span><span class="mi">0</span><span class="p">)</span><span class="w"></span>\r |
| 302 | <span class="k">val</span><span class="w"> </span><span class="n">count2</span><span class="p">:</span><span class="w"> </span><span class="n">t</span><span class="w"> </span><span class="n">*</span><span class="w"> </span><span class="p">(</span><span class="n">'a</span><span class="w"> </span><span class="p">-></span><span class="w"> </span><span class="n">'a</span><span class="p">)</span><span class="w"> </span><span class="p">-></span><span class="w"> </span><span class="p">(</span><span class="n">unit</span><span class="w"> </span><span class="p">-></span><span class="w"> </span><span class="n">int</span><span class="p">)</span><span class="w"> </span><span class="n">*</span><span class="w"> </span><span class="p">(</span><span class="n">'a</span><span class="w"> </span><span class="p">-></span><span class="w"> </span><span class="n">'a</span><span class="p">)</span><span class="w"> </span><span class="p">=</span><span class="w"></span>\r |
| 303 | <span class="w"> </span><span class="k">fn</span><span class="w"> </span><span class="p">(</span><span class="n">T</span><span class="w"> </span><span class="n">r</span><span class="p">,</span><span class="w"> </span><span class="n">f</span><span class="p">)</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="w"> </span><span class="p">=></span><span class="w"> </span><span class="n">!r</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="p">(</span><span class="n">r</span><span class="w"> </span><span class="n">:=</span><span class="w"> </span><span class="mi">1</span><span class="w"> </span><span class="n">+</span><span class="w"> </span><span class="n">!r</span><span class="p">;</span><span class="w"> </span><span class="n">f</span><span class="w"> </span><span class="n">x</span><span class="p">))</span><span class="w"></span>\r |
| 304 | <span class="k">val</span><span class="w"> </span><span class="n">id</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">'a</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">x</span><span class="w"></span>\r |
| 305 | <span class="k">val</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">count1</span><span class="w"> </span><span class="p">()</span><span class="w"></span>\r |
| 306 | <span class="k">val</span><span class="w"> </span><span class="n">countId</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">'a</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">z</span><span class="w"> </span><span class="p">=></span><span class="w"> </span><span class="p">#</span><span class="mi">2</span><span class="w"> </span><span class="p">(</span><span class="n">count2</span><span class="w"> </span><span class="p">(</span><span class="n">t</span><span class="p">,</span><span class="w"> </span><span class="n">id</span><span class="p">))</span><span class="w"> </span><span class="n">z</span><span class="w"></span>\r |
| 307 | <span class="k">val</span><span class="w"> </span><span class="n">numCalls</span><span class="w"> </span><span class="p">=</span><span class="w"> </span><span class="p">#</span><span class="mi">1</span><span class="w"> </span><span class="p">(</span><span class="n">count2</span><span class="w"> </span><span class="p">(</span><span class="n">t</span><span class="p">,</span><span class="w"> </span><span class="n">id</span><span class="p">))</span><span class="w"></span>\r |
| 308 | </pre></div></div></div>\r |
| 309 | <div class="paragraph"><p>Of course, one can hide the constructor <span class="monospaced">T</span> inside a <span class="monospaced">local</span> or behind\r |
| 310 | a signature.</p></div>\r |
| 311 | </div>\r |
| 312 | </div>\r |
| 313 | <div class="sect1">\r |
| 314 | <h2 id="_also_see">Also see</h2>\r |
| 315 | <div class="sectionbody">\r |
| 316 | <div class="ulist"><ul>\r |
| 317 | <li>\r |
| 318 | <p>\r |
| 319 | <a href="ImperativeTypeVariable">ImperativeTypeVariable</a>\r |
| 320 | </p>\r |
| 321 | </li>\r |
| 322 | </ul></div>\r |
| 323 | </div>\r |
| 324 | </div>\r |
| 325 | </div>\r |
| 326 | <div id="footnotes"><hr></div>\r |
| 327 | <div id="footer">\r |
| 328 | <div id="footer-text">\r |
| 329 | </div>\r |
| 330 | <div id="footer-badges">\r |
| 331 | </div>\r |
| 332 | </div>\r |
| 333 | </body>\r |
| 334 | </html>\r |