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1.1 ! root 1: .\" Copyright (c) 1980 Regents of the University of California. ! 2: .\" All rights reserved. The Berkeley software License Agreement ! 3: .\" specifies the terms and conditions for redistribution. ! 4: .\" ! 5: .\" @(#)manCh2.rno 6.1 (Berkeley) 4/29/86 ! 6: .\" ! 7: .NS 1 "System Description" ! 8: .sp ! 9: .NS 2 "Objects" ! 10: .sp ! 11: .pp ! 12: The set of objects \(*W consists of ! 13: the atoms and sequences $fs$ (where the $x sub i memberOf OMEGA$). ! 14: (Lisp users should note the similarity to the list structure syntax, ! 15: just replace the parenthesis ! 16: by angle brackets and commas by blanks. There are no 'quoted' objects, ! 17: \*(IE 'abc). ! 18: The atoms uniquely determine the set of valid objects and consist of the ! 19: numbers (of the type found in \s-2FRANZ LISP\s+2 [Fod80]), quoted ascii strings ! 20: ("abcd"), and unquoted alphanumeric strings (abc3). ! 21: There are three predefined atoms, $T$ and $F$, that correspond ! 22: to the logical values 'true' and 'false', and the undefined atom $bold "?"$, ! 23: \fIbottom\fP. ! 24: \fIBottom\fP denotes the value returned as the result of an ! 25: undefined operation, \*(EG division by zero. ! 26: The empty sequence, $<>$ is also an atom. The following are examples ! 27: of valid FP objects: ! 28: .sp 2 ! 29: .(b C ! 30: .TS ! 31: center; ! 32: l l l. ! 33: $?$ $1.47$ $3888888888888$ ! 34: $ab$ "$CD$" $<1,<2,3>>$ ! 35: $<>$ $T$ $<a,<>>$ ! 36: .TE ! 37: .)b ! 38: .sp 2 ! 39: There is one restriction on object construction: no object may contain ! 40: the undefined atom, such an object is itself undefined, \*(EG ! 41: $<1,?>~==~?$. This property is the ! 42: so-called \*(lqbottom preserving property\*(rq [Ba78]. ! 43: .sp ! 44: .NS 2 "Application" ! 45: .sp ! 46: .pp ! 47: This is the single FP operation and is designated by the colon (":"). ! 48: For a function $sigma$ and an object $x$, $sigma : x$ is an application ! 49: and its meaning is the object that results from applying $sigma$ to ! 50: $x$ (\*(IE evaluating $sigma (x)$). ! 51: We say that $sigma$ is the \fIoperator\fP and that $x$ is ! 52: the \fIoperand\fP. ! 53: The following are examples of applications: ! 54: .sp 2 ! 55: .TS ! 56: center; ! 57: l c l l c l. ! 58: $bold + : <7,8>$ $==$ $15$ $bold tl :<1,2,3>$ $==$ $<2,3>$ ! 59: \fB1\fP $: <a,b,c,d>$ $==$ $a$ \fB2\fP $: <a,b,c,d>$ $==$ $b$ ! 60: .TE ! 61: .sp 2 ! 62: .nr ii 0.35i ! 63: .NS 2 "Functions" ! 64: .sp ! 65: .pp ! 66: All functions (\fIF\fP) map objects into objects, moreover, they are ! 67: \fIstrict\fP: ! 68: .sp 6p ! 69: .EQ I (2.1) ! 70: sigma^:^? equiv ?,~~ forAll ^sigma^ memberOf F ! 71: .EN ! 72: .sp 6p ! 73: To formally characterize the primitive functions, ! 74: we use a modification of McCarthy's conditional expressions [Mc60]: ! 75: .sp 6p ! 76: .EQ I (2.2) ! 77: p sub 1~->~ e sub 1~; ... ;~p sub n~->~ e sub n~;~e sub {n + 1} ! 78: .EN ! 79: .sp 6p ! 80: This statement is interpreted as follows: ! 81: return function $e sub 1$ if the predicate '$p sub 1$' is ! 82: true $,...,~e sub n$ if '$p sub n$' is true. ! 83: If none of the predicates are satisfied then default to $e sub {n + 1}$. ! 84: It is assumed that $x ,~x sub i ,~y ,~y sub i ,~z sub i memberOf OMEGA$. ! 85: .sp ! 86: .NS 3 "Structural" ! 87: .sp ! 88: .b "Selector Functions" ! 89: .(b M F ! 90: .sp ! 91: For a nonzero integer $mu$, ! 92: .sp ! 93: .nf ! 94: .ip "$bold mu~ : ~x~ ==$" ! 95: .sp 4p ! 96: \&$x = fs ~ andsign ~0~<~mu~<=~k ~->~ x sub mu ;$ ! 97: .sp 4p ! 98: \&$x = fs ~ andsign ~-k <= mu < 0 ~ -> ~ x sub {k + mu + 1}; ~ ?$ ! 99: .fi ! 100: .)b ! 101: .sp ! 102: .(b ! 103: .nf ! 104: .ip "$bold pick~ : ~<n,x>~ ==$" ! 105: .sp 4p ! 106: \&$x = fs ~ andsign ~0~<~n~<=~k ~->~ x sub n ;$ ! 107: .sp 4p ! 108: \&$x = fs ~ andsign ~-k <= n < 0 ~ -> ~ x sub {k + n + 1}; ~ ?$ ! 109: .fi ! 110: .)b ! 111: .sp 2 ! 112: .pp ! 113: The user should note that the function ! 114: symbols \fB1\fP,\fB2\fP,\fB3\fP$,...$ are ! 115: to be distinguished from the atoms $1,2,3,...$. ! 116: .(b M F ! 117: .sp ! 118: .ip "$bold last ~ : ~x~ ==$" ! 119: .sp 4p ! 120: \&$x = <> ~ -> ~ <> ~ ;$ ! 121: .sp 4p ! 122: \&$x = fs ~ andsign ~ k >= 1 ~ -> ~ x sub k ;~?$ ! 123: .)b ! 124: .(b M F ! 125: .sp ! 126: .ip "$bold first ~ : ~x~ ==$" ! 127: .sp 4p ! 128: \&$x = <> ~ -> ~ <> ~ ;$ ! 129: .sp 4p ! 130: \&$x = fs ~ andsign ~ k >= 1 ~ -> ~ x sub 1 ;~?$ ! 131: .)b ! 132: .sp ! 133: .(b M F ! 134: .b "Tail Functions" ! 135: .sp ! 136: .ip "$bold tl ~:~x ~==$" ! 137: .sp 4p ! 138: \&$x = <x sub 1 > ~ -> ~ <> ~ ;$ ! 139: .sp 4p ! 140: \&$x = fs ~ andsign ~ k >= ~ 2 ~ -> ~<x sub 2 ~,..., ~x sub k > ~ ;~ ?$ ! 141: .)b ! 142: .sp 6p ! 143: .(b M F ! 144: .ip "$bold tlr ~ : ~ x~==$" ! 145: .sp 4p ! 146: \&$x = <x sub 1 > ~ -> ~ <> ~ ;$ ! 147: .sp 4p ! 148: \&$x = fs ~ andsign ~ k >= ~ 2 ~ -> ~ <x sub 1 ~ ,..., ~x sub {k - 1} > ~ ; ~ ?$ ! 149: .)b ! 150: .sp ! 151: .pp ! 152: Note: There is also a function ! 153: \fBfront\fP that is equivalent to \fBtlr\fP. ! 154: .sp ! 155: .(b M F ! 156: .b "Distribute from left and right" ! 157: .sp ! 158: .ip "$bold distl ~:~x~==$" ! 159: .sp 4p ! 160: \&$x = <y,<>>~->~<>;$ ! 161: .sp 4p ! 162: \&$x = <y, qz >~->~<<y,z sub 1 >,...,<y,z sub k >>;~?$ ! 163: .)b ! 164: .sp 6p ! 165: .(b M F ! 166: .ip "$bold distr ~:~x~==$" ! 167: .sp 4p ! 168: \&$x = <<>,y>~->~<>;$ ! 169: .sp 4p ! 170: \&$x = < qy , z >~->~<<y sub 1 , z >,...,<y sub k , z >>;~?$ ! 171: .)b ! 172: .sp 6p ! 173: .(b M F ! 174: .b "Identity" ! 175: .sp 6p ! 176: $bold id ~:~x~==~x$ ! 177: .)b ! 178: .sp 6p ! 179: $bold out ~:~x~==~x$ ! 180: .pp ! 181: \fBOut\fP is similar to \fPid\fP. ! 182: Like \fBid\fP it returns its argument as the result, ! 183: unlike \fPid\fP ! 184: it prints its ! 185: result on \fIstdout\fP \(mi ! 186: It is the only function with a side effect. ! 187: \fPOut\fP is intended to be used for debugging only. ! 188: .sp ! 189: .(b M F ! 190: .b "Append left and right" ! 191: .sp ! 192: .nf ! 193: .ip "$bold apndl ~:~x~==$" ! 194: .sp 4p ! 195: \&$x = <y,<>>~->~<y>;$ ! 196: .sp 4p ! 197: \&$x = <y, qz >~->~<y, z sub 1 ,~ z sub 2 ,...,~ z sub k >;~?$ ! 198: .)b ! 199: .sp 6p ! 200: .(b M F ! 201: .ip "$bold apndr ~:~x~==$" ! 202: .sp 4p ! 203: \&$x = <<>,z>~->~<z>;$ ! 204: .sp 4p ! 205: \&$x = < qy , z >~->~< y sub 1 ,~ y sub 2 ,...,~ y sub k ,~z >;~?$ ! 206: .fi ! 207: .)b ! 208: .sp ! 209: .(b M F ! 210: .b "Transpose" ! 211: .sp ! 212: .nf ! 213: .ip "$bold trans~:~x~==$" ! 214: .sp 4p ! 215: \&$x = <<>,...,<>>~->~<>;$ ! 216: .sp 4p ! 217: \&$x = fs ~->~<y sub 1 ,..., y sub m >;~?$ ! 218: .sp 8p ! 219: \&where $x sub i ~=~<x sub i1 ,..., x sub im >~andsign ! 220: ~ y sub j ~=~<x sub 1j ,..., x sub kj >,$ ! 221: .sp 4p ! 222: \&$1<=i<=k ~,~ 1<=j<=m.$ ! 223: .)b ! 224: .sp ! 225: .fi ! 226: .(b M F ! 227: .ip "$bold reverse~:~x~==~$" ! 228: .sp 4p ! 229: \&$x =<>~ ->;$ ! 230: .sp 4p ! 231: \&$x = fs ~->~ <x sub k ,..., x sub 1 >;~?$ ! 232: .)b ! 233: .sp ! 234: .(b M F ! 235: .b "Rotate Left and Right" ! 236: .sp ! 237: .nf ! 238: .ip "$bold rotl~:~x~==$" ! 239: .sp 4p ! 240: \&$x = <>~ -> ~ <>;~x = <x sub 1 >~->~<x sub 1 >;$ ! 241: .sp 4p ! 242: \&$x = fs ~ andsign ~k >= 2~ -> ~ <x sub 2 ,..., x sub k , x sub 1 >;~?$ ! 243: .)b ! 244: .sp ! 245: .(b M F ! 246: .ip "$bold rotr~:x~==$" ! 247: .sp 4p ! 248: \&$x = <>~ -> ~ <>;~x = <x sub 1 >~->~<x sub 1 >;$ ! 249: .sp 4p ! 250: \&$x = fs ~ andsign ~k >= 2~ -> ~ ! 251: <x sub k ,~ x sub 1 ,..., x sub {k - 2},~ x sub {k - 1} >;~?$ ! 252: .)b ! 253: .fi ! 254: .sp 2 ! 255: .(b M F ! 256: .nf ! 257: .ip "$bold concat~:~x~==$" ! 258: .sp 4p ! 259: \&$x = <<x sub 11 ,..., x sub 1k >,<x sub 21 ,..., x sub 2n > ! 260: ,...,<x sub m1 ,..., x sub mp >> ~ andsign ~ k, ~m, ~n, ~p ~>~ 0 ~->$ ! 261: \&$<x sub 11 ,..., x sub 1k , x sub 21 ,..., x sub 2n ,..., x sub m1 ,..., x sub mp >; ?$ ! 262: .)b ! 263: .sp ! 264: .(b M F ! 265: .pp ! 266: Concatenate removes all occurrences of the null sequence: ! 267: .sp ! 268: .EQ I (2.3) ! 269: bold concat ~ :~ <<1,3>,<>,<2,4>,<>,<5>> ~==~ <1,3,2,4,5> ! 270: .EN ! 271: .)b ! 272: .sp ! 273: .(b M F ! 274: .ip "$bold pair~:~x~==$" ! 275: .sp 4p ! 276: \&$x = fs ~ andsign ~ k >0 ~ andsign ~ k~is~even~->~ <<x sub 1 , x sub 2 > ! 277: ,..., <x sub k-1 , x sub k >>;$ ! 278: .sp 4p ! 279: \&$x = fs ~ andsign ~ k >0 ~ andsign ~ k~is~odd~->~ <<x sub 1 , x sub 2 > ! 280: ,..., <x sub k >>;~?$ ! 281: .)b ! 282: .sp ! 283: .(b M F ! 284: .ip "$bold split~:~x~==$" ! 285: .sp 4p ! 286: \&$x = <x sub 1 > ~->~<<x sub 1 > , <>>;$ ! 287: .sp 4p ! 288: \&$x = fs ~ andsign ~ k > 1 ~ -> ! 289: ~<<x sub 1 ,..., x sub {left ceiling k/2 right ceiling} >, ! 290: <x sub {left ceiling k/2 right ceiling + 1} ,..., x sub k >>; ?$ ! 291: .)b ! 292: .sp ! 293: .(b M F ! 294: .ip "\fBiota\fP$~:x~==$" ! 295: .sp 4p ! 296: \&$x = 0 ~->~ <>;$ ! 297: .sp 4p ! 298: \&$x memberOf bold {N sup +} ~ ->~<1,2,...,x>;~?$ ! 299: .)b ! 300: .sp ! 301: .NS 3 "Predicate (Test) Functions" ! 302: .sp ! 303: $bold atom~:~x~==~x~ memberOf atoms~-> ~ T ; ~ x$\(!=$? -> ~ F ;~ ?$ ! 304: .sp ! 305: $bold eq ~:~x~==~x~=<y,z> ~ andsign ~ y=z ~ -> ~ T ;~x= <y,z> ~ andsign ~y$ \(!= $z ~->~ F ;~?$ ! 306: .sp ! 307: .pp ! 308: Also less than ($bold <$), greater than ($bold >$), ! 309: greater than or equal (\fB>=\fP), ! 310: less than or equal (\fB<=\fP), not equal (\fB~=\fP); ! 311: \&'$bold =$' is a synonym for \fBeq\fP. ! 312: .sp ! 313: $bold null ~ : x~==~x = <> ~ -> ~ T ;~x$\(!=$?~ -> ~ F ;~?$ ! 314: .sp 6p ! 315: $bold length ~ :~ x~==~x~= ~ fs ~ -> ~ k; ~ x = <> ~ -> ~0; ~ ?$ ! 316: .sp ! 317: .NS 3 "Arithmetic/Logical" ! 318: .sp ! 319: $bold +~:~ x ~ ==~ x = <y,z> ~ andsign ~y,z ~ are ~ numbers ~ ->~ y+z; ~ ?$ ! 320: $bold -~:~ x ~ ==~ x = <y,z> ~ andsign ~y,z ~ are ~ numbers ~ -> ~y-z; ~ ?$ ! 321: $bold *~:~ x ~ ==~ x = <y,z> ~ andsign ~y,z ~ are ~ numbers ~ -> ~y$\(mu$z; ~ ?$ ! 322: \fB/\fP$~:~ x ~ ==~ x = <y,z> ~ andsign ~y,z ~ are ~ numbers ~ andsign ! 323: ~z$\(!= $0 ~ ->~y$\(di$z ;~?$ ! 324: .sp ! 325: .b "And, or, not, xor" ! 326: .sp ! 327: $nd~:~<x,y>~==~x = T ~->~ y;~x= F ~->~ F ;~?$ ! 328: .sp 8p ! 329: $rr~:~<x,y>~==~x = F ~->~ y;~x= T ~->~ T ;~?$ ! 330: .sp 8p ! 331: .(b M F ! 332: .nf ! 333: .ip "$bold xor~:~<x,y>~==$" ! 334: .sp 4p ! 335: \&$x = T ~ andsign ~ y = T ~->~ F ;~ x = F ~ andsign ~ y = F ~->~ F ;$ ! 336: .sp 4p ! 337: \&$x = T ~ andsign ~ y = F ~->~ T ;~ x = F ~ andsign ~ y = T ~->~ T ;~?$ ! 338: .fi ! 339: .)b ! 340: .sp 6p ! 341: $bold not~:~x~==~x= T ~->~F;~x= F~->~T ;~?$ ! 342: .NS 3 "Library Routines" ! 343: .sp ! 344: $bold "sin"~:~x~==~x roman {~is~a~number} ~->~sin(x);~?$ ! 345: .sp 8p ! 346: $bold "asin"~:~x~==~x roman {~is~a~number} ! 347: ~andsign~|x|~<=~1 ~->~sin sup {-1} (x);~?$ ! 348: .sp 8p ! 349: $bold "cos"~:~x~==~x roman {~is~a~number} ~->~cos(x);~?$ ! 350: .sp 8p ! 351: $bold "acos"~:~x~==~x roman {~is~a~number} ! 352: ~andsign~|x|~<=~1 ~->~cos sup {-1} (x);~?$ ! 353: .sp 8p ! 354: $bold "exp"~:~x~==~x roman {~is~a~number} ~->~ e sup x;~?$ ! 355: .sp 8p ! 356: $bold "log"~:~x~==~x roman {~is~a~positive~number} ~->~ ln(x);~?$ ! 357: .sp 8p ! 358: $bold "mod"~:~<x,y>~==~ x nd y roman {~are~numbers} ~->~ x ~-~ y times ! 359: left floor x over y right floor ~;~?$ ! 360: .sx 2 ! 361: .sp ! 362: .NS 2 "Functional Forms" ! 363: .pp ! 364: Functional forms define new \fIfunctions\fP ! 365: by operating on ! 366: function and object \fIparameters\fP of the form. ! 367: The resultant expressions ! 368: can be compared and contrasted to the \fIvalue\fP-oriented ! 369: expressions of traditional programming languages. ! 370: The distinction lies in the domain of ! 371: the operators; functional forms manipulate functions, while traditional ! 372: operators manipulate values. ! 373: .pp ! 374: One functional form is \fIcomposition\fP. For two functions $phi$ and ! 375: $psi$ the form $phi ~ @ ~psi$ denotes their composition $phi ~$\*(cm$~ psi$: ! 376: .sp 4p ! 377: .EQ I (2.4) ! 378: ( phi ~@~ psi )~:~x~==~phi :( psi :x),~~ forAll ~~x memberOf OMEGA ! 379: .EN ! 380: .sp ! 381: The \fIconstant\fP function takes an object parameter: ! 382: .sp ! 383: .EQ I (2.5) ! 384: %x:y~==~y=?~->~?;~x,~~~ forAll ~~x,y~ memberOf OMEGA ! 385: .EN ! 386: .sp ! 387: The function $%?$ always returns ?. ! 388: .pp ! 389: In the following description of the functional forms, we assume that ! 390: $xi , ~xi sub i ,~sigma , ~sigma sub i ,~tau ,$ and $tau sub i$ ! 391: are functions and that ! 392: $x,~x sub i ,~ y$ are objects. ! 393: .sp 2 ! 394: .b "Composition" ! 395: .sp ! 396: $( sigma ~@~ tau ):x~==~sigma : ( tau : x)$ ! 397: .sp 2 ! 398: .b "Construction" ! 399: .sp ! 400: $[ sigma sub 1 ,..., sigma sub n ]:x~==~< sigma sub 1 : x,..., sigma sub n : x>$ ! 401: .sp ! 402: Note that construction is also bottom-preserving, \*(EG ! 403: .sp 6p ! 404: .EQ I (2.6) ! 405: [ bold +, bold /]^:^<3,0>~=~<3,?>~=~? ! 406: .EN ! 407: .sp 2 ! 408: .(b M F ! 409: .b "Condition" ! 410: .sp ! 411: .ip "$( xi~"->" ~sigma ;~tau ):x~==~$" ! 412: .sp 4p ! 413: \&$( xi : x) = T~->~sigma : x;$ ! 414: .sp 4p ! 415: \&$~ ( xi : x)= F~ ->~tau :x;~?$ ! 416: .)b ! 417: .sp 8p ! 418: .pp ! 419: The reader should be aware of the distinction between ! 420: \fIfunctional expressions\fP, in the variant of McCarthy's conditional ! 421: expression, ! 422: and the \fIfunctional form\fP introduced here. ! 423: In the former case the result is a \fIvalue\fP, while in the latter case ! 424: the result is a \fIfunction\fP. Unlike Backus' FP, the conditional form ! 425: \fImust\fP be enclosed in parenthesis, \*(EG ! 426: .sp 6p ! 427: .EQ I (2.7) ! 428: roman {"(isNegative -> - @ [%0,id] ; id")} ! 429: .EN ! 430: .sp ! 431: .(b M F ! 432: .b "Constant" ! 433: .sp ! 434: $%x:y~==~y=?~->~?;~x,~~~~~~ forAll ~x memberOf OMEGA$ ! 435: .sp ! 436: .)b ! 437: This function returns its object parameter as its result. ! 438: .sp ! 439: .(b M F ! 440: .b "Right Insert" ! 441: .sp 6p ! 442: .nf ! 443: .ip "$! bold sigma~:x~==$" ! 444: .sp 4p ! 445: \&$x = <>~->~e sub f : x;$ ! 446: .sp 4p ! 447: \&$x=<x sub 1 >~->~x sub 1 ;$ ! 448: .sp 4p ! 449: \&$x= fs ~ andsign ~k>2~->~ sigma :<x sub 1 ,~! sigma :<x sub 2 ,...,~x sub k >>;~?$ ! 450: .sp 10p ! 451: \&\*(EG $!+:<4,5,6> =15$. ! 452: .)b ! 453: .pp ! 454: If $sigma$ has a right identity element $e sub f$, ! 455: then $! sigma :<>~=~ e sub f$, \*(EG ! 456: .sp 6p ! 457: .EQ I (2.8) ! 458: !+^:^<> =0 ~roman "and" ~!*^:^<> =1 ! 459: .EN ! 460: .sp ! 461: Currently, ! 462: identity functions are defined for $+ \ (0),\ - \ (0), \ * \ (1), ! 463: \ / \ (1)$, ! 464: also for \fBand\fP (T), \fBor\fP (F), \fBxor\fP (F). All other unit functions ! 465: default to bottom (?). ! 466: .sp ! 467: .(b M F ! 468: .b "Tree Insert" ! 469: .sp 6p ! 470: .ip " \fB|\fP $sigma~:~x~==$" ! 471: .sp 4p ! 472: \&$x = <>~->~e sub f : x;$ ! 473: .sp 4p ! 474: \&$x=<x sub 1 >~->~x sub 1 ;$ ! 475: .sp 4p ! 476: .nf ! 477: \&$x= fs ~ andsign ~k>1~->$ ! 478: .sp 4p ! 479: \&$bold sigma ~:~< $\fB|\fP$~ sigma~:~ ! 480: <x sub 1 ,..., x sub {left ceiling k/2 right ceiling} >~,~ ! 481: "\fB|\fP" ~ sigma ~:~<x sub {left ceiling k/2 right ceiling + 1} ! 482: ,..., x sub k >>; ?$ ! 483: .sp 10p ! 484: \&\*(EG ! 485: .EQ I (2.9) ! 486: "\fB|\fP" +:<4,5,6,7> ~==~ +:<+:<4,5> , +:<6,7>> ~==~ 15 ! 487: .EN ! 488: .sp ! 489: .fi ! 490: .)b ! 491: .sp ! 492: .pp ! 493: Tree insert uses the same identity functions as right ! 494: insert. ! 495: .sp ! 496: .b "Apply to All" ! 497: .sp ! 498: .ip "\fB&\fP$^sigma :~x~==$" ! 499: .sp 4p ! 500: \&$x=<>~-><>;$ ! 501: .sp 4p ! 502: \&$x= fs~->~< sigma : x sub 1 ,...,~sigma : x sub k >;~?$ ! 503: .sp ! 504: .(b M F ! 505: .b "While" ! 506: .sp ! 507: .ip "(\fBwhile\fP$ ~xi~sigma ):x~==$" ! 508: \&$xi : x= T~->~($\fBwhile\fP$ ~xi~sigma ):( sigma : x);$ ! 509: .sp 4p ! 510: \&$xi : x= F~->~x;~?$ ! 511: .)b ! 512: .sp ! 513: .NS 2 "User Defined Functions" ! 514: .pp ! 515: An FP definition is entered as follows: ! 516: .sp 6p ! 517: .EQ I (2.10) ! 518: "{fn-name fn-form}," ! 519: .EN ! 520: .sp ! 521: where \fIfn-name\fP is an ascii string consisting of letters, numbers and the ! 522: underline symbol, and \fIfn-form\fP is any valid functional form, including ! 523: a single primitive or defined function. ! 524: For example the function ! 525: .sp ! 526: .EQ I (2.11) ! 527: "{factorial !* @ iota}" ! 528: .EN ! 529: .sp ! 530: .pp ! 531: is the non-recursive definition of ! 532: the factorial function. Since FP systems are applicative it is permissible ! 533: to substitute the actual definition of a function for any reference to it ! 534: in a functional form: if $f ~==~ 1 @ 2$ then $f~:~x~==~1@2~:~x,~~~ ! 535: forAll ~x memberOf OMEGA$. ! 536: .pp ! 537: References to undefined functions bottom out: ! 538: .sp ! 539: .EQ I (2.12) ! 540: f:x~==~? forAll x memberOf OMEGA ,~f^ notmemberof F ! 541: .EN ! 542: .sx 1 ! 543: .bp ! 544:
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