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New article: Inductive lattices
| author | Dylan Holmes <ocsenave@gmail.com> |
|---|---|
| date | Tue, 01 Nov 2011 01:55:26 -0500 |
| parents | b4de894a1e2e |
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1 <?xml version="1.0" encoding="utf-8"?>2 <!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN"3 "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd">4 <html xmlns="http://www.w3.org/1999/xhtml"5 lang="en" xml:lang="en">6 <head>7 <title>LAMBDA: The Ultimate Imperative</title>8 <meta http-equiv="Content-Type" content="text/html;charset=utf-8"/>9 <meta name="generator" content="Org-mode"/>10 <meta name="generated" content="2011-07-15 02:49:04 EDT"/>11 <meta name="author" content="Guy Lewis Steele Jr. and Gerald Jay Sussman"/>12 <meta name="description" content=""/>13 <meta name="keywords" content=""/>14 <style type="text/css">15 <!--/*--><![CDATA[/*><!--*/16 html { font-family: Times, serif; font-size: 12pt; }17 .title { text-align: center; }18 .todo { color: red; }19 .done { color: green; }20 .tag { background-color: #add8e6; font-weight:normal }21 .target { }22 .timestamp { color: #bebebe; }23 .timestamp-kwd { color: #5f9ea0; }24 .right {margin-left:auto; margin-right:0px; text-align:right;}25 .left {margin-left:0px; margin-right:auto; text-align:left;}26 .center {margin-left:auto; margin-right:auto; text-align:center;}27 p.verse { margin-left: 3% }28 pre {29 border: 1pt solid #AEBDCC;30 background-color: #F3F5F7;31 padding: 5pt;32 font-family: courier, monospace;33 font-size: 90%;34 overflow:auto;35 }36 table { border-collapse: collapse; }37 td, th { vertical-align: top; }38 th.right { text-align:center; }39 th.left { text-align:center; }40 th.center { text-align:center; }41 td.right { text-align:right; }42 td.left { text-align:left; }43 td.center { text-align:center; }44 dt { font-weight: bold; }45 div.figure { padding: 0.5em; }46 div.figure p { text-align: center; }47 textarea { overflow-x: auto; }48 .linenr { font-size:smaller }49 .code-highlighted {background-color:#ffff00;}50 .org-info-js_info-navigation { border-style:none; }51 #org-info-js_console-label { font-size:10px; font-weight:bold;52 white-space:nowrap; }53 .org-info-js_search-highlight {background-color:#ffff00; color:#000000;54 font-weight:bold; }55 /*]]>*/-->56 </style>57 <script type="text/javascript">58 <!--/*--><![CDATA[/*><!--*/59 function CodeHighlightOn(elem, id)60 {61 var target = document.getElementById(id);62 if(null != target) {63 elem.cacheClassElem = elem.className;64 elem.cacheClassTarget = target.className;65 target.className = "code-highlighted";66 elem.className = "code-highlighted";67 }68 }69 function CodeHighlightOff(elem, id)70 {71 var target = document.getElementById(id);72 if(elem.cacheClassElem)73 elem.className = elem.cacheClassElem;74 if(elem.cacheClassTarget)75 target.className = elem.cacheClassTarget;76 }77 /*]]>*///-->78 </script>79 <script type="text/javascript" src="http://orgmode.org/mathjax/MathJax.js">80 <!--/*--><![CDATA[/*><!--*/81 MathJax.Hub.Config({82 // Only one of the two following lines, depending on user settings83 // First allows browser-native MathML display, second forces HTML/CSS84 // config: ["MMLorHTML.js"], jax: ["input/TeX"],85 jax: ["input/TeX", "output/HTML-CSS"],86 extensions: ["tex2jax.js","TeX/AMSmath.js","TeX/AMSsymbols.js",87 "TeX/noUndefined.js"],88 tex2jax: {89 inlineMath: [ ["\\(","\\)"] ],90 displayMath: [ ['$$','$$'], ["\\[","\\]"], ["\\begin{displaymath}","\\end{displaymath}"] ],91 skipTags: ["script","noscript","style","textarea","pre","code"],92 ignoreClass: "tex2jax_ignore",93 processEscapes: false,94 processEnvironments: true,95 preview: "TeX"96 },97 showProcessingMessages: true,98 displayAlign: "center",99 displayIndent: "2em",101 "HTML-CSS": {102 scale: 100,103 availableFonts: ["STIX","TeX"],104 preferredFont: "TeX",105 webFont: "TeX",106 imageFont: "TeX",107 showMathMenu: true,108 },109 MMLorHTML: {110 prefer: {111 MSIE: "MML",112 Firefox: "MML",113 Opera: "HTML",114 other: "HTML"115 }116 }117 });118 /*]]>*///-->119 </script>120 </head>121 <body>123 <div id="content">127 <div id="table-of-contents">128 <h2>Table of Contents</h2>129 <div id="text-table-of-contents">130 <ul>131 <li><a href="#sec-1">1 Abstract </a></li>132 <li><a href="#sec-2">2 Introduction </a>133 <ul>134 <li><a href="#sec-2-1">2.1 Simple Loops </a>135 <ul>136 <li><a href="#sec-2-1-1">2.1.1 Simple Recursion </a></li>137 <li><a href="#sec-2-1-2">2.1.2 Iteration </a></li>138 </ul></li>139 </ul>140 </li>141 <li><a href="#sec-3">3 Imperative Programming </a>142 <ul>143 <li><a href="#sec-3-1">3.1 Compound Statements </a></li>144 <li><a href="#sec-3-2">3.2 2.2. The G0 TO Statement </a></li>145 <li><a href="#sec-3-3">3.3 2.3. Simple Assignment </a></li>146 <li><a href="#sec-3-4">3.4 Compound Expressions ' </a></li>147 </ul>148 </li>149 <li><a href="#sec-4">4 Continuations </a>150 <ul>151 <li><a href="#sec-4-1">4.1 Continuation-Passing Recursion </a></li>152 <li><a href="#sec-4-2">4.2 Escape Expressions </a></li>153 </ul>154 </li>155 </ul>156 </div>157 </div>159 <div id="outline-container-1" class="outline-2">160 <h2 id="sec-1"><span class="section-number-2">1</span> Abstract </h2>161 <div class="outline-text-2" id="text-1">164 <p>165 We demonstrate how to model the following common programming constructs in166 terms of an applicative order language similar to LISP:167 </p><ul>168 <li>Simple Recursion169 </li>170 <li>Iteration171 </li>172 <li>Compound Statements and Expressions173 </li>174 <li>GO TO and Assignment175 </li>176 <li>Continuation—Passing177 </li>178 <li>Escape Expressions179 </li>180 <li>Fluid Variables181 </li>182 <li>Call by Name, Call by Need, and Call by Reference183 </li>184 </ul>186 <p>The models require only (possibly self-referent) lambda application,187 conditionals, and (rarely) assignment. No complex data structures such as188 stacks are used. The models are transparent, involving only local syntactic189 transformations.190 Some of these models, such as those for GO TO and assignment, are already well191 known, and appear in the work of Landin, Reynolds, and others. The models for192 escape expressions, fluid variables, and call by need with side effects are193 new. This paper is partly tutorial in intent, gathering all the models194 together for purposes of context.195 This report describes research done at the Artificial Intelligence Laboratory196 of the Massachusetts Institute of Teehnology. Support for the laboratory's197 artificial intelligence research is provided in part by the Advanced Research198 Projects Agency of the Department of Defense under Office of Naval Research199 contract N000l4-75-C-0643.200 </p>201 </div>203 </div>205 <div id="outline-container-2" class="outline-2">206 <h2 id="sec-2"><span class="section-number-2">2</span> Introduction </h2>207 <div class="outline-text-2" id="text-2">210 <p>211 We catalogue a number of common programming constructs. For each212 construct we examine "typical" usage in well-known programming languages, and213 then capture the essence of the semantics of the construct in terms of a214 common meta-language.215 The lambda calculus {Note Alonzowins} is often used as such a meta-216 language. Lambda calculus offers clean semantics, but it is clumsy because it217 was designed to be a minimal language rather than a convenient one. All218 lambda calculus "functions" must take exactly one "argument"; the only "data219 type" is lambda expressions; and the only "primitive operation‘ is variable‘220 substitution. While its utter simplicity makes lambda calculus ideal for221 logicians, it is too primitive for use by programmers. The meta-language we222 use is a programming language called SCHEME {Note Schemepaper) which is based223 on lambda calculus.224 SCHEME is a dialect of LISP. [McCarthy 62] It is an expression-225 oriented, applicative order, interpreter-based language which allows one to226 manipulate programs as data. It differs from most current dialects of LISP in227 that it closes all lambda expressions in the environment of their definition228 or declaration, rather than in the execution environment. {Note Closures}229 This preserves the substitution semantics of lambda calculus, and has the230 consequence that all variables are lexically scoped, as in ALGOL. [Naur 63]231 Another difference is that SCHEME is implemented in such a way that tail-232 recursions execute without net growth of the interpreter stack. {Note233 Schemenote} We have chosen to use LISP syntax rather than, say, ALGOL syntax234 because we want to treat programs as data for the purpose of describing235 transformations on the code. LISP supplies names for the parts of an236 executable expression and standard operators for constructing expressions and237 extracting their components. The use of LISP syntax makes the structure of238 such expressions manifest. We use ALGOL as an expository language, because it239 is familiar to many people, but ALGOL is not sufficiently powerful to express240 the necessary concepts; in particular, it does not allow functions to return241 functions as values. we are thus forced to use a dialect of LISP in many242 cases.243 We will consider various complex programming language constructs and244 show how to model them in terms of only a few simple ones. As far as possible245 we will use only three control constructs from SCHEME: LAMBDA expressions, as246 in LISP, which are Just functions with lexically scoped free variables;247 LABELS, which allows declaration of mutually recursive procedures (Note248 Labelsdef}; and IF, a primitive conditional expression. For more complex249 modelling we will introduce an assignment primitive (ASET).i We will freely250 assume the existence of other comon primitives, such as arithmetic functions.251 The constructs we will examine are divided into four broad classes.252 The first is sfinph?Lo0pl; this contains simple recursions and iterations, and253 an introduction to the notion of continuations. The second is hmponflivo254 Connrucls; this includes compound statements, G0 T0, and simple variable255 assignments. The third is continuations, which encompasses the distinction between statements and expressions, escape operators (such as Landin's J-256 operator [Landin 65] and Reynold's escape expression [Reynolds 72]). and fluid257 (dynamically bound) variables. The fourth is Parameter Passing Mechanisms, such258 as ALGOL call—by-name and FORTRAN call-by-location.259 Some of the models presented here are already well-known, particularly260 those for G0 T0 and assignment. [McCarthy 60] [Landin 65] [Reynolds 72]261 Those for escape operators, fluid variables, and call-by-need with side262 effects are new.263 </p>264 </div>266 <div id="outline-container-2-1" class="outline-3">267 <h3 id="sec-2-1"><span class="section-number-3">2.1</span> Simple Loops </h3>268 <div class="outline-text-3" id="text-2-1">270 <p>By <i>simple loops</i> we mean constructs which enable programs to execute the271 same piece of code repeatedly in a controlled manner. Variables may be made272 to take on different values during each repetition, and the number of273 repetitions may depend on data given to the program.274 </p>275 </div>277 <div id="outline-container-2-1-1" class="outline-4">278 <h4 id="sec-2-1-1"><span class="section-number-4">2.1.1</span> Simple Recursion </h4>279 <div class="outline-text-4" id="text-2-1-1">281 <p>One of the easiest ways to produce a looping control structure is to282 use a recursive function, one which calls itself to perform a subcomputation.283 For example, the familiar factorial function may be written recursively in284 ALGOL: '285 </p>288 <pre class="src src-algol">integer procedure fact(n) value n: integer n:289 fact := if n=0 then 1 else n*fact(n-1);291 </pre>296 <p>297 The invocation <code>fact(n)</code> computes the product of the integers from 1 to n using298 the identity n!=n(n-1)! (n>0). If \(n\) is zero, 1 is returned; otherwise <code>fact</code>:299 calls itself recursively to compute \((n-1)!\) , then multiplies the result by \(n\)300 and returns it.301 </p>302 <p>303 This same function may be written in Clojure as follows:304 </p>307 <pre class="src src-clojure">(<span style="color: #f0dfaf; font-weight: bold;">ns</span> categorical.imperative)308 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">fact</span>[n]309 (<span style="color: #f0dfaf; font-weight: bold;">if</span> (<span style="color: #8cd0d3;">=</span> n 0) 1 (<span style="color: #8cd0d3;">*</span> n (fact (<span style="color: #8cd0d3;">dec</span> n))))310 )312 </pre>318 <p>319 Clojure does not require an assignment to the ``variable'' fan: to return a value320 as ALGOL does. The IF primitive is the ALGOL if-then-else rendered in LISP321 syntax. Note that the arithmetic primitives are prefix operators in SCHEME.322 </p>323 </div>325 </div>327 <div id="outline-container-2-1-2" class="outline-4">328 <h4 id="sec-2-1-2"><span class="section-number-4">2.1.2</span> Iteration </h4>329 <div class="outline-text-4" id="text-2-1-2">331 <p>There are many other ways to compute factorial. One important way is332 through the use of <i>iteration</i>.333 A comon iterative construct is the DO loop. The most general form we334 have seen in any programming language is the MacLISP DO [Moon 74]. It permits335 the simultaneous initialization of any number of control variables and the336 simultaneous stepping of these variables by arbitrary functions at each337 iteration step. The loop is terminated by an arbitrary predicate, and an338 arbitrary value may be returned. The DO loop may have a body, a series of339 expressions executed for effect on each iteration. A version of the MacLISP340 DO construct has been adopted in SCHEME.341 </p>342 <p>343 The general form of a SCHEME DO is:344 </p>347 <pre class="src src-nothing">(DO ((<var1> (init1> <stepl>)348 ((var2> <init2> <step2>)349 (<varn> <intn> (stepn>))350 (<pred> (value>)351 (optional body>)353 </pre>357 <p>358 The semantics of this are that the variables are bound and initialized to the359 values of the <initi> expressions, which must all be evaluated in the360 environment outside the D0; then the predicate <pred> is evaluated in the new361 environment, and if TRUE, the (value) is evaluated and returned. Otherwise362 the (optional body) is evaluated, then each of the steppers <stepi> is363 evaluated in the current environment, all the variables made to have the364 results as their values, the predicate evaluated again, and so on.365 Using D0 loops in both ALGOL and SCHEME, we may express FACT by means366 of iteration.367 </p>370 <pre class="src src-algol">integer procedure fact(n); value n; integer n:371 begin372 integer m, ans;373 ans := 1;374 for m := n step -l until 0 do ans := m*ans;375 fact := ans;376 end;378 </pre>384 <pre class="src src-clojure">(<span style="color: #f0dfaf; font-weight: bold;">in-ns</span> 'categorical.imperative)385 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">fact-do</span>[n]386 )388 </pre>393 <p>394 Note that the SCHEME D0 loop in FACT has no body – the stepping functions do395 all the work. The ALGOL DO loop has an assignment in its body; because an396 ALGOL DO loop can step only one variable, we need the assignment to step the397 the variable "manually'.398 In reality the SCHEME DO construct is not a primitive; it is a macro399 which expands into a function which performs the iteration by tail—recursion.400 Consider the following definition of FACT in SCHEME. Although it appears to401 be recursive, since it "calls itself", it is entirely equivalent to the DO402 loop given above, for it is the code that the DO macro expands into! It403 captures the essence of our intuitive notion of iteration, because execution404 of this program will not produce internal structures (e.g. stacks or variable405 bindings) which increase in size with the number of iteration steps.406 </p>410 <pre class="src src-clojure">(<span style="color: #f0dfaf; font-weight: bold;">in-ns</span> 'categorical.imperative)411 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">fact-do-expand</span>[n]412 (<span style="color: #f0dfaf; font-weight: bold;">let</span> [fact1413 (<span style="color: #8cd0d3;">fn</span>[m ans]414 (<span style="color: #f0dfaf; font-weight: bold;">if</span> (zero? m) ans415 (<span style="color: #f0dfaf; font-weight: bold;">recur</span> (<span style="color: #8cd0d3;">dec</span> m) (<span style="color: #8cd0d3;">*</span> m ans))))]416 (fact1 n 1)))418 </pre>423 <p>424 From this we can infer a general way to express iterations in SCHEME in425 a manner isomorphic to the HacLISP D0. The expansion of the general D0 loop426 </p>429 <pre class="src src-nothing">(DO ((<varl> <init1> <step1>)430 (<var2> (init2) <step2>)431 ...432 (<varn> <initn> <stepn>))433 (<pred> <va1ue>)434 <body>)436 </pre>440 <p>441 is this:442 </p>445 <pre class="src src-nothing">(let [doloop446 (fn [dummy <var1> <var2> ... <varn>)447 (if <pred> <value>448 (recur <body> <step1> <step2> ... <stepn>)))]450 (doloop nil <init1> <init2> ... <initn>))452 </pre>456 <p>457 The identifiers <code>doloop</code> and <code>dummy</code> are chosen so as not to conflict with any458 other identifiers in the program.459 Note that, unlike most implementations of D0, there are no side effects460 in the steppings of the iteration variables. D0 loops are usually modelled461 using assignment statements. For example:462 </p>465 <pre class="src src-nothing">for r :1 0 step b until c do <statement>;467 </pre>472 <p>473 can be modelled as follows: [Naur 63]474 </p>477 <pre class="src src-nothing">begin478 x := a;479 L: if (x-c)*sign(n) > 0 then go to Endloop;480 <statement>;481 x := x+b;482 go to L:483 Endloop:484 end;486 </pre>490 <p>491 Later we will see how such assignment statements can in general be492 modelled without using side effects.493 </p>494 </div>495 </div>496 </div>498 </div>500 <div id="outline-container-3" class="outline-2">501 <h2 id="sec-3"><span class="section-number-2">3</span> Imperative Programming </h2>502 <div class="outline-text-2" id="text-3">504 <p>Lambda calculus (and related languages, such as ``pure LISP'') is often505 used for modelling the applicative constructs of programming languages.506 However, they are generally thought of as inappropriate for modelling507 imperative constructs. In this section we show how imperative constructs may508 be modelled by applicative SCHEME constructs.509 </p>510 </div>512 <div id="outline-container-3-1" class="outline-3">513 <h3 id="sec-3-1"><span class="section-number-3">3.1</span> Compound Statements </h3>514 <div class="outline-text-3" id="text-3-1">516 <p>The simplest kind of imperative construct is the statement sequencer,517 for example the compound statement in ALGOL:518 </p>521 <pre class="src src-algol">begin522 S1;523 S2;524 end526 </pre>531 <p>532 This construct has two interesting properties:533 </p><ul>534 <li>(1) It performs statement S1 before S2, and so may be used for535 sequencing.536 </li>537 <li>(2) S1 is useful only for its side effects. (In ALGOL, S2 must also538 be a statement, and so is also useful only for side effects, but539 other languages have compound expressions containing a statement540 followed by an expression.)541 </li>542 </ul>545 <p>546 The ALGOL compound statement may actually contain any number of statements,547 but such statements can be expressed as a series of nested two-statement548 compounds. That is:549 </p>552 <pre class="src src-algol">begin553 S1;554 S2;555 ...556 Sn-1;557 Sn;558 end560 </pre>564 <p>565 is equivalent to:566 </p>570 <pre class="src src-algol">begin571 S1;572 begin573 S2;574 begin575 ...577 begin578 Sn-1;579 Sn;580 end;581 end;582 end:583 end585 </pre>590 <p>591 It is not immediately apparent that this sequencing can be expressed in a592 purely applicative language. We can, however, take advantage of the implicit593 sequencing of applicative order evaluation. Thus, for example, we may write a594 two-statement sequence as follows:595 </p>599 <pre class="src src-clojure">((<span style="color: #8cd0d3;">fn</span>[dummy] S2) S1)601 </pre>606 <p>607 where <code>dummy</code> is an identifier not used in S2. From this it is608 manifest that the value of S1 is ignored, and so is useful only for609 side effects. (Note that we did not claim that S1 is expressed in a610 purely applicative language, but only that the sequencing can be so611 expressed.) From now on we will use the form <code>(BLOCK S1 S2)</code> as an612 abbreviation for this expression, and <code>(BLOCK S1 S2...Sn-1 Sn)</code> as an613 abbreviation for614 </p>618 <pre class="src src-algol">(BLOCK S1 (BLOCK S2 (BLOCK ... (BLOCK Sn-1 Sn)...)))620 </pre>625 </div>627 </div>629 <div id="outline-container-3-2" class="outline-3">630 <h3 id="sec-3-2"><span class="section-number-3">3.2</span> 2.2. The G0 TO Statement </h3>631 <div class="outline-text-3" id="text-3-2">634 <p>635 A more general imperative structure is the compound statement with636 labels and G0 T0s. Consider the following code fragment due to637 Jacopini, taken from Knuth: [Knuth 74]638 </p>641 <pre class="src src-algol">begin642 L1: if B1 then go to L2; S1;643 if B2 then go to L2; S2;644 go to L1;645 L2: S3;647 </pre>652 <p>653 It is perhaps surprising that this piece of code can be <i>syntactically</i>654 transformed into a purely applicative style. For example, in SCHEME we655 could write:656 </p>660 <pre class="src src-clojure">(<span style="color: #f0dfaf; font-weight: bold;">in-ns</span> 'categorical.imperative)661 (<span style="color: #f0dfaf; font-weight: bold;">let</span>662 [L1 (<span style="color: #8cd0d3;">fn</span>[]663 (<span style="color: #f0dfaf; font-weight: bold;">if</span> B1 (L2) (<span style="color: #f0dfaf; font-weight: bold;">do</span> S1664 (<span style="color: #f0dfaf; font-weight: bold;">if</span> B2 (L2) (<span style="color: #f0dfaf; font-weight: bold;">do</span> S2 (L1))))))665 L2 (<span style="color: #8cd0d3;">fn</span>[] S3)]666 (L1))668 </pre>673 <p>674 As with the D0 loop, this transformation depends critically on SCHEME's675 treatment of tail-recursion and on lexical scoping of variables. The labels676 are names of functions of no arguments. In order to ‘go to" the labeled code,677 we merely call the function named by that label.678 </p>679 </div>681 </div>683 <div id="outline-container-3-3" class="outline-3">684 <h3 id="sec-3-3"><span class="section-number-3">3.3</span> 2.3. Simple Assignment </h3>685 <div class="outline-text-3" id="text-3-3">687 <p>Of course, all this sequencing of statements is useless unless the688 statements have side effects. An important side effect is assignment. For689 example, one often uses assignment to place intermediate results in a named690 location (i.e. a variable) so that they may be used more than once later691 without recomputing them:692 </p>696 <pre class="src src-algol">begin697 real a2, sqrtdisc;698 a2 := 2*a;699 sqrtdisc := sqrt(b^2 - 4*a*c);700 root1 := (- b + sqrtdisc) / a2;701 root2 := (- b - sqrtdisc) / a2;702 print(root1);703 print(root2);704 print(root1 + root2);705 end707 </pre>712 <p>713 It is well known that such naming of intermediate results may be accomplished714 by calling a function and binding the formal parameter variables to the715 results:716 </p>720 <pre class="src src-clojure">(<span style="color: #8cd0d3;">fn</span> [a2 sqrtdisc]721 ((<span style="color: #8cd0d3;">fn</span>[root1 root2]722 (<span style="color: #f0dfaf; font-weight: bold;">do</span> (<span style="color: #8cd0d3;">println</span> root1)723 (<span style="color: #8cd0d3;">println</span> root2)724 (<span style="color: #8cd0d3;">println</span> (<span style="color: #8cd0d3;">+</span> root1 root2))))725 (<span style="color: #8cd0d3;">/</span> (<span style="color: #8cd0d3;">+</span> (<span style="color: #8cd0d3;">-</span> b) sqrtdisc) a2)726 (<span style="color: #8cd0d3;">/</span> (<span style="color: #8cd0d3;">-</span> (<span style="color: #8cd0d3;">-</span> b) sqrtdisc) a2))728 (<span style="color: #8cd0d3;">*</span> 2 a)729 (sqrt (<span style="color: #8cd0d3;">-</span> (<span style="color: #8cd0d3;">*</span> b b) (<span style="color: #8cd0d3;">*</span> 4 a c))))731 </pre>735 <p>736 This technique can be extended to handle all simple variable assignments which737 appear as statements in blocks, even if arbitrary G0 T0 statements also appear738 in such blocks. {Note Mccarthywins}739 </p>740 <p>741 For example, here is a program which uses G0 TO statements in the form742 presented before; it determines the parity of a non-negative integer by743 counting it down until it reaches zero.744 </p>748 <pre class="src src-algol">begin749 L1: if a = 0 then begin parity := 0; go to L2; end;750 a := a - 1;751 if a = 0 then begin parity := 1; go to L2; end;752 a := a - 1;753 go to L1;754 L2: print(parity);756 </pre>761 <p>762 This can be expressed in SCHEME:763 </p>767 <pre class="src src-clojure">(<span style="color: #f0dfaf; font-weight: bold;">let</span>768 [L1 (<span style="color: #8cd0d3;">fn</span> [a parity]769 (<span style="color: #f0dfaf; font-weight: bold;">if</span> (zero? a) (L2 a 0)770 (L3 (<span style="color: #8cd0d3;">dec</span> a) parity)))771 L3 (<span style="color: #8cd0d3;">fn</span> [a parity]772 (<span style="color: #f0dfaf; font-weight: bold;">if</span> (zero? a) (L2 a 1)773 (L1 (<span style="color: #8cd0d3;">dec</span> a) parity)))774 L2 (<span style="color: #8cd0d3;">fn</span> [a parity]775 (<span style="color: #8cd0d3;">println</span> parity))]776 (L1 a parity))778 </pre>783 <p>784 The trick is to pass the set of variables which may be altered as arguments to785 the label functions. {Note Flowgraph} It may be necessary to introduce new786 labels (such as L3) so that an assignment may be transformed into the binding787 for a function call. At worst, one may need as many labels as there are788 statements (not counting the eliminated assignment and GO TO statements).789 </p>790 </div>792 </div>794 <div id="outline-container-3-4" class="outline-3">795 <h3 id="sec-3-4"><span class="section-number-3">3.4</span> Compound Expressions ' </h3>796 <div class="outline-text-3" id="text-3-4">798 <p>At this point we are almost in a position to model the most general799 form of compound statement. In LISP, this is called the 'PROG feature". In800 addition to statement sequencing and G0 T0 statements, one can return a <i>value</i>801 from a PROG by using the RETURN statement.802 </p>803 <p>804 Let us first consider the simplest compound statement, which in SCHEME805 we call BLOCK. Recall that806 <code>(BLOCK s1 sz)</code> is defined to be <code>((lambda (dummy) s2) s1)</code>807 </p>808 <p>809 Notice that this not only performs Sl before S2, but also returns the value of810 S2. Furthermore, we defined <code>(BLOCK S1 S2 ... Sn)</code> so that it returns the value811 of <code>Sn</code>. Thus BLOCK may be used as a compound expression, and models a LISP812 PROGN, which is a PROG with no G0 T0 statements and an implicit RETURN of the813 last ``statement'' (really an expression).814 </p>815 <p>816 Host LISP compilers compile D0 expressions by macro-expansion. We have817 already seen one way to do this in SCHEME using only variable binding. A more818 common technique is to expand the D0 into a PROG, using variable assignments819 instead of bindings. Thus the iterative factorial program:820 </p>824 <pre class="src src-clojure">(oarxnz FACT .825 (LAMaoA (n) .826 (D0 ((M N (<span style="color: #8cd0d3;">-</span> H 1))827 (Ans 1 (<span style="color: #8cd0d3;">*</span> M Ans)))828 ((<span style="color: #8cd0d3;">-</span> H 0) A<span style="color: #cc9393;">"$))))</span>830 </pre>835 <p>836 would expand into:837 </p>841 <pre class="src src-clojure">(DEFINE FACT842 . (LAMeoA (M)843 (PRO6 (M Ans)844 (sszro M n845 Ans 1) ~846 LP (tr (<span style="color: #8cd0d3;">-</span> M 0) (RETURN Ans))847 (ssero M (<span style="color: #8cd0d3;">-</span> n 1)848 Ans (' M Ans))849 (60 LP))))851 </pre>856 <p>857 where SSETQ is a simultaneous multiple assignment operator. (SSETQ is not a858 SCHEME (or LISP) primitive. It can be defined in terms of a single assignment859 operator, but we are more interested here in RETURN than in simultaneous860 assignment. The SSETQ's will all be removed anyway and modelled by lambda861 binding.) We can apply the same technique we used before to eliminate G0 T0862 statements and assignments from compound statements:863 </p>867 <pre class="src src-clojure">(DEFINE FACT868 (LAHBOA (I)869 (LABELS ((L1 (LAManA (M Ans)870 (LP n 1)))871 (LP (LAMaoA (M Ans)872 (IF (<span style="color: #8cd0d3;">-</span> M o) (nztunn Ans)873 (£2 H An$))))874 (L2 (LAMaoA (M An )875 (LP (<span style="color: #8cd0d3;">-</span> <span style="color: #cc9393;">" 1) (' H flN$)))))</span>876 <span style="color: #e37170; background-color: #332323;">(</span><span style="color: #cc9393;">L1 NIL NlL))))</span>878 </pre>881 <p>882 clojure883 </p>884 <p>885 We still haven't done anything about RETURN. Let's see…886 </p><ul>887 <li>==> the value of (FACT 0) is the value of (Ll NIL NIL)888 </li>889 <li>==> which is the value of (LP 0 1)890 </li>891 <li>==> which is the value of (IF (= 0 0) (RETURN 1) (L2 0 1))892 </li>893 <li>==> which is the value of (RETURN 1)894 </li>895 </ul>898 <p>899 Notice that if RETURN were the <i>identity function</i> (LAMBDA (X) X), we would get the right answer. This is in fact a900 general truth: if we Just replace a call to RETURN with its argument, then901 our old transformation on compound statements extends to general compound902 expressions, i.e. PROG.903 </p>904 </div>905 </div>907 </div>909 <div id="outline-container-4" class="outline-2">910 <h2 id="sec-4"><span class="section-number-2">4</span> Continuations </h2>911 <div class="outline-text-2" id="text-4">913 <p>Up to now we have thought of SCHEME's LAMBDA expressions as functions,914 and of a combination such as <code>(G (F X Y))</code> as meaning ``apply the function F to915 the values of X and Y, and return a value so that the function G can be916 applied and return a value …'' But notice that we have seldom used LAMBDA917 expressions as functions. Rather, we have used them as control structures and918 environment modifiers. For example, consider the expression:919 <code>(BLOCK (PRINT 3) (PRINT 4))</code>920 </p>921 <p>922 This is defined to be an abbreviation for:923 <code>((LAMBDA (DUMMY) (PRINT 4)) (PRINT 3))</code>924 </p>925 <p>926 We do not care about the value of this BLOCK expression; it follows that we927 do not care about the value of the <code>(LAMBDA (DUMMY) ...)</code>. We are not using928 LAMBDA as a function at all.929 </p>930 <p>931 It is possible to write useful programs in terms of LAHBDA expressions932 in which we never care about the value of <i>any</i> lambda expression. We have933 already demonstrated how one could represent any "FORTRAN" program in these934 terms: all one needs is PROG (with G0 and SETQ), and PRINT to get the answers935 out. The ultimate generalization of this imperative programing style is936 <i>continuation-passing</i>. (Note Churchwins} .937 </p>939 </div>941 <div id="outline-container-4-1" class="outline-3">942 <h3 id="sec-4-1"><span class="section-number-3">4.1</span> Continuation-Passing Recursion </h3>943 <div class="outline-text-3" id="text-4-1">945 <p>Consider the following alternative definition of FACT. It has an extra946 argument, the continuation, which is a function to call with the answer, when947 we have it, rather than return a value948 </p>952 <pre class="src src-algol.">procedure Inc!(n, c); value n, c;953 integer n: procedure c(integer value);954 if n-=0 then c(l) else955 begin956 procedure l(!mp(d) value a: integer a;957 c(mm);958 Iacl(n-1, romp):959 end;961 </pre>967 <pre class="src src-clojure">(<span style="color: #f0dfaf; font-weight: bold;">in-ns</span> 'categorical.imperative)968 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">fact-cps</span>[n k]969 (<span style="color: #f0dfaf; font-weight: bold;">if</span> (zero? n) (k 1)970 (<span style="color: #f0dfaf; font-weight: bold;">recur</span> (<span style="color: #8cd0d3;">dec</span> n) (<span style="color: #8cd0d3;">fn</span>[a] (k (<span style="color: #8cd0d3;">*</span> n a))))))972 </pre>975 <p>976 clojure977 It is fairly clumsy to use this version of‘ FACT in ALGOL; it is necessary to978 do something like this:979 </p>983 <pre class="src src-algol">begin984 integer ann985 procedure :emp(x); value 2:; integer x;986 ans :1 x;987 ]'act(3. temp);988 comment Now the variable <span style="color: #cc9393;">"am"</span> has 6;989 end;991 </pre>996 <p>997 Procedure <code>fact</code> does not return a value, nor does <code>temp</code>; we must use a side998 effect to get the answer out.999 </p>1000 <p>1001 <code>FACT</code> is somewhat easier to use in SCHEME. We can call it like an1002 ordinary function in SCHEME if we supply an identity function as the second1003 argument. For example, <code>(FACT 3 (LAMBDA (X) X))</code> returns 6. Alternatively, we1004 could write <code>(FACT 3 (LAHBDA (X) (PRINT X)))</code>; we do not care about the value1005 of this, but about what gets printed. A third way to use the value is to1006 write1007 <code>(FACT 3 (LAMBDA (x) (SQRT x)))</code>1008 instead of1009 <code>(SQRT (FACT 3 (LAMBDA (x) x)))</code>1010 </p>1011 <p>1012 In either of these cases we care about the value of the continuation given to1013 FACT. Thus we care about the value of FACT if and only if we care about the1014 value of its continuation!1015 </p>1016 <p>1017 We can redefine other functions to take continuations in the same way.1018 For example, suppose we had arithmetic primitives which took continuations; to1019 prevent confusion, call the version of "+" which takes a continuation '++",1020 etc. Instead of writing1021 <code>(- (+ B Z)(* 4 A C))</code>1022 we can write1023 </p>1026 <pre class="src src-clojure">(<span style="color: #f0dfaf; font-weight: bold;">in-ns</span> 'categorical.imperative)1027 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">enable-continuation</span> <span style="color: #8fb28f;">"Makes f take a continuation as an additional argument."</span>[f]1028 (<span style="color: #8cd0d3;">fn</span>[& args] ((<span style="color: #8cd0d3;">fn</span>[k] (k (<span style="color: #8cd0d3;">apply</span> f (<span style="color: #8cd0d3;">reverse</span> (<span style="color: #8cd0d3;">rest</span> (<span style="color: #8cd0d3;">reverse</span> args)))))) (<span style="color: #8cd0d3;">last</span> args)) ))1029 (<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">+&</span> (enable-continuation +))1030 (<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">*&</span> (enable-continuation *))1031 (<span style="color: #f0dfaf; font-weight: bold;">def</span> <span style="color: #f0dfaf;">-&</span> (enable-continuation -))1034 (<span style="color: #f0dfaf; font-weight: bold;">defn</span> <span style="color: #f0dfaf;">quadratic</span>[a b c k]1035 (*& b b1036 (<span style="color: #8cd0d3;">fn</span>[x] (*& 4 a c1037 (<span style="color: #8cd0d3;">fn</span>[y]1038 (-& x y k))))))1040 </pre>1045 <p>1046 where <code>k</code> is the continuation for the entire expression.1047 </p>1048 <p>1049 This is an obscure way to write an algebraic expression, and we would1050 not advise writing code this way in general, but continuation-passing brings1051 out certain important features of the computation:1052 </p><ul>1053 <li><sup><a class="footref" name="fnr.1" href="#fn.1">1</a></sup> The operations to be performed appear in the order in which they are1054 </li>1055 </ul>1057 <p>performed. In fact, they <i>must</i> be performed in this1058 order. Continuation- passing removes the need for the rule about1059 left-to-right argument evaluation{Note Evalorder)1060 </p><ul>1061 <li><sup><a class="footref" name="fnr.2" href="#fn.2">2</a></sup> In the usual applicative expression there are two implicit1062 temporary values: those of <code>(* B B)</code> and <code>(* 4 A C)</code>. The first of1063 these values must be preserved over the computation of the second,1064 whereas the second is used as soon as it is produced. These facts1065 are <i>manifest</i> in the appearance and use of the variable X and Y in1066 the continuation-passing version.1067 </li>1068 </ul>1071 <p>1072 In short, the continuation-passing version specifies <i>exactly</i> and1073 explicitly what steps are necessary to compute the value of the1074 expression. One can think of conventional functional application1075 for value as being an abbreviation for the more explicit1076 continuation-passing style. Alternatively, one can think of the1077 interpreter as supplying to each function an implicit default1078 continuation of one argument. This continuation will receive the1079 value of the function as its argument, and then carry on the1080 computation. In an interpreter this implicit continuation is1081 represented by the control stack mechanism for function returns.1082 Now consider what computational steps are implied by:1083 </p>1084 <p>1085 <code>(LAMBDA (A B C ...) (F X Y Z ...))</code> when we "apply" the LAMBDA1086 expression we have some values to apply it to; we let the names A,1087 B, C … refer to these values. We then determine the values of X,1088 Y, Z … and pass these values (along with "the buck",1089 i.e. control!) to the lambda expression F (F is either a lambda1090 expression or a name for one). Passing control to F is an1091 unconditional transfer. (Note Jrsthack) {Note Hewitthack) Note that1092 we want values from X, Y, Z, … If these are simple expressions,1093 such as variables, constants, or LAMBDA expressions, the evaluation1094 process is trivial, in that no temporary storage is required. In1095 pure continuation-passing style, all evaluations are trivial: no1096 combination is nested within another, and therefore no ‘hidden1097 temporaries" are required. But if X is a combination, say (G P Q),1098 then we want to think of G as a function, because we want a value1099 from it, and we will need an implicit temporary place to keep the1100 result while evaluating Y and Z. (An interpreter usually keeps these1101 temporary places in the control stack!) On the other hand, we do not1102 necessarily need a value from F. This is what we mean by tail-1103 recursion: F is called tail-recursively, whereas G is not. A better1104 name for tail-recursion would be "tail-transfer", since no real1105 recursion is implied. This is why we have made such a fuss about1106 tail-recursion: it can be used for transfer of control without1107 making any commitment about whether the expression expected to1108 return a value. Thus it can be used to model statement-like control1109 structures. Put another way, tail—recursion does not require a1110 control stack as nested recursion does. In our models of iteration1111 and imperative style all the LAMBDA expressions used for control (to1112 simulate GO statements, for example) are called in tail-recursive1113 fashion.1114 </p>1115 </div>1117 </div>1119 <div id="outline-container-4-2" class="outline-3">1120 <h3 id="sec-4-2"><span class="section-number-3">4.2</span> Escape Expressions </h3>1121 <div class="outline-text-3" id="text-4-2">1123 <p>Reynolds [Reynolds 72] defines the construction1124 = escape x in r =1125 </p>1126 <p>1127 to evaluate the expression \(r\) in an environment such that the variable \(x\) is1128 bound to an escape function. If the escape function is never applied, then1129 the value of the escape expression is the value of \(r\). If the escape function1130 is applied to an argument \(a\), however, then evaluation of \(r\) is aborted and the1131 escape expression returns \(a\). {Note J-operator} (Reynolds points out that1132 this definition is not quite accurate, since the escape function may be called1133 even after the escape expression has returned a value; if this happens, it1134 “returns again"!)1135 </p>1136 <p>1137 As an example of the use of an escape expression, consider this1138 procedure to compute the harmonic mean of an array of numbers. If any of the1139 numbers is zero, we want the answer to be zero. We have a function hannaunl1140 which will sum the reciprocals of numbers in an array, or call an escape1141 function with zero if any of the numbers is zero. (The implementation shown1142 here is awkward because ALGOL requires that a function return its value by1143 assignment.)1144 </p>1147 <pre class="src src-algol">begin1148 real procedure Imrmsum(a, n. escfun)§ p1149 real array at integer n; real procedure esciun(real);1150 begin1151 real sum;1152 sum := 0;1153 for i :1 0 until n-l do1154 begin1155 if a[i]=0 then cscfun(0);1156 sum :1 sum ¢ I/a[i];1157 enm1158 harmsum SI sum;1159 end: .1160 real array b[0:99]:1161 print(escape x in I00/hm-m.mm(b, 100, x));1162 end1164 </pre>1169 <p>1170 If harmsum exits normally, the number of elements is divided by the sum and1171 printed. Otherwise, zero is returned from the escape expression and printed1172 without the division ever occurring.1173 This program can be written in SCHEME using the built-in escape1174 operator <code>CATCH</code>:1175 </p>1179 <pre class="src src-clojure">abc1181 </pre>1185 <div id="footnotes">1186 <h2 class="footnotes">Footnotes: </h2>1187 <div id="text-footnotes">1188 <p class="footnote"><sup><a class="footnum" name="fn.1" href="#fnr.1">1</a></sup> DEFINITION NOT FOUND: 11189 </p>1191 <p class="footnote"><sup><a class="footnum" name="fn.2" href="#fnr.2">2</a></sup> DEFINITION NOT FOUND: 21192 </p>1193 </div>1194 </div>1195 </div>1197 </div>1198 </div>1199 <div id="postamble">1200 <p class="date">Date: 2011-07-15 02:49:04 EDT</p>1201 <p class="author">Author: Guy Lewis Steele Jr. and Gerald Jay Sussman</p>1202 <p class="creator">Org version 7.6 with Emacs version 23</p>1203 <a href="http://validator.w3.org/check?uri=referer">Validate XHTML 1.0</a>1204 </div>1205 </div>1206 </body>1207 </html>