%%Creator: %!PS-Adobe-2.0 dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: v1c5.dvi %%Pages: 34 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%EndComments %DVIPSCommandLine: dvips -o v1c5.ps v1c5.dvi %DVIPSParameters: dpi=300, comments removed %DVIPSSource: TeX output 1999.12.01:0821 %%BeginProcSet: tex.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[matrix currentmatrix{dup dup round sub abs 0.00001 lt{round}if} forall round exch round exch]setmatrix}N /@landscape{/isls true N}B /@manualfeed{statusdict /manualfeed true put}B /@copies{/#copies X}B /FMat[1 0 0 -1 0 0]N /FBB[0 0 0 0]N /nn 0 N /IE 0 N /ctr 0 N /df-tail{ /nn 8 dict N nn begin /FontType 3 N /FontMatrix fntrx N /FontBBox FBB N string /base X array /BitMaps X /BuildChar{CharBuilder}N /Encoding IE N end dup{/foo setfont}2 array copy cvx N load 0 nn put /ctr 0 N[}B /df{ /sf 1 N /fntrx FMat N df-tail}B /dfs{div /sf X /fntrx[sf 0 0 sf neg 0 0] N df-tail}B /E{pop nn dup definefont setfont}B /ch-width{ch-data dup length 5 sub get}B /ch-height{ch-data dup length 4 sub get}B /ch-xoff{ 128 ch-data dup length 3 sub get sub}B /ch-yoff{ch-data dup length 2 sub get 127 sub}B /ch-dx{ch-data dup length 1 sub get}B /ch-image{ch-data dup type /stringtype ne{ctr get /ctr ctr 1 add N}if}B /id 0 N /rw 0 N /rc 0 N /gp 0 N /cp 0 N /G 0 N /sf 0 N /CharBuilder{save 3 1 roll S dup /base get 2 index get S /BitMaps get S get /ch-data X pop /ctr 0 N ch-dx 0 ch-xoff ch-yoff ch-height sub ch-xoff ch-width add ch-yoff setcachedevice ch-width ch-height true[1 0 0 -1 -.1 ch-xoff sub ch-yoff .1 sub]{ch-image}imagemask restore}B /D{/cc X dup type /stringtype ne{]} if nn /base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{dup dup length 1 sub dup 2 index S get sf div put}if put /ctr ctr 1 add N}B /I{ cc 1 add D}B /bop{userdict /bop-hook known{bop-hook}if /SI save N @rigin 0 0 moveto /V matrix currentmatrix dup 1 get dup mul exch 0 get dup mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N /eop{SI restore userdict /eop-hook known{eop-hook}if showpage}N /@start{userdict /start-hook known{start-hook}if pop /VResolution X /Resolution X 1000 div /DVImag X /IE 256 array N 0 1 255{IE S 1 string dup 0 3 index put cvn put}for 65781.76 div /vsize X 65781.76 div /hsize X}N /p{show}N /RMat[1 0 0 -1 0 0]N /BDot 260 string N /rulex 0 N /ruley 0 N /v{/ruley X /rulex X V}B /V {}B /RV statusdict begin /product where{pop product dup length 7 ge{0 7 getinterval dup(Display)eq exch 0 4 getinterval(NeXT)eq or}{pop false} ifelse}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale rulex ruley false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR rulex ruley scale 1 1 false RMat{BDot}imagemask grestore}}ifelse B /QV{gsave newpath transform round exch round exch itransform moveto rulex 0 rlineto 0 ruley neg rlineto rulex neg 0 rlineto fill grestore}B /a{moveto}B /delta 0 N /tail {dup /delta X 0 rmoveto}B /M{S p delta add tail}B /b{S p tail}B /c{-4 M} B /d{-3 M}B /e{-2 M}B /f{-1 M}B /g{0 M}B /h{1 M}B /i{2 M}B /j{3 M}B /k{ 4 M}B /w{0 rmoveto}B /l{p -4 w}B /m{p -3 w}B /n{p -2 w}B /o{p -1 w}B /q{ p 1 w}B /r{p 2 w}B /s{p 3 w}B /t{p 4 w}B /x{0 S rmoveto}B /y{3 2 roll p a}B /bos{/SS save N}B /eos{SS restore}B end %%EndProcSet TeXDict begin 40258431 52099146 1000 300 300 (v1c5.dvi)
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Fj(L)p Fw(\()p Fj(A)1208 228 y Fg(0)1219 243 y Fw(\))e(=)h Fj(L)p Fw(\()p Fj(A)p Fw(\).)24 b(The)17 b(construction)300 293 y(for)e Fi(V)398 278 y Fg(0)389 304 y Fh(T)415 293 y Fw(,)g Fi(V)475 278 y Fg(0)466 304 y Fh(N)497 293 y Fw(,)g(G)557 278 y Fg(0)583 293 y Fw(and)g(\005)696 278 y Fg(0)723 293 y Fw(follo)o(ws.)k(The)d(cen)o(tral)f(idea)g(is)g(that)g (the)g Fv(set)h(of)g(states)i Fw(of)c(the)300 343 y Fu(nf)l(a)j Fw(whic)o(h)f(can)g(b)q(e)g(reac)o(hed)h(b)o(y)f(some)f(string)h Fi(\013)f Fw(are)i(all)d(mapp)q(ed)h(in)o(to)g(a)h Fv(single)g(state) 300 392 y Fw(of)h(the)i Fu(df)l(a)q Fw(.)30 b(Once)19 b(the)g(construction)g(is)f(complete,)g(the)g(states)h(of)f(the)g Fu(df)l(a)h Fw(can)f(b)q(e)300 442 y(relab)q(elled)12 b(to)g(simplify)d(the)k(description)g(of)e(the)i Fu(df)l(a)q Fw(.)k(The)c(construction)g(of)f(\005)1582 427 y Fg(0)1605 442 y Fw(and)g Fi(V)1718 427 y Fg(0)1708 454 y Fh(N)300 492 y Fw(is)f(m)o(utually)e(dep)q(enden)o(t,)14 b(as)d(represen)o(ted)k (b)o(y)c(the)h(m)o(utually)d(recursiv)o(e)k(form)o(ulas)d(b)q(elo)o(w.) 362 542 y(The)18 b(construction)h(is)e(based)i(on)e(a)g(function)h Fi(S)i Fw(whic)o(h)e(collects)g(the)g(set)h(of)e(states)300 592 y Fi(S)r Fw(\(A)374 577 y Fg(0)387 592 y Fi(;)7 b Fw(a\))j(that)g(can)h(b)q(e)h(reac)o(hed)g(from)d(the)i(set)g(of)g (states)g(A)1228 577 y Fg(0)1251 592 y Fw(via)f(a)g(transition)g(on)h (terminal)300 642 y(sym)o(b)q(ol)g(a.)17 b(The)c(v)n(alue)e(of)h Fi(S)j Fw(is)d(often)h(the)g(empt)o(y)e(set,)i(corresp)q(onding)h(to)e (an)g Fv(err)n(or)g(state)300 691 y Fw(\(rejection)j(state\))g(in)e (the)i(resulting)f Fu(df)l(a)p Fw(.)615 795 y Fi(C)s Fw(\(A)695 778 y Fg(0)707 795 y Fw(\))765 772 y Fp(def)771 795 y Fw(=)49 b Fj(f)p Fw(B)14 b Fj(j)f Fw(A)f Fj(2)f Fw(A)1054 778 y Fg(0)1075 795 y Fj(^)e Fw(B)j Fj(2)f Fi(V)1216 801 y Fh(N)1257 795 y Fj(^)e Fw(A)j Fj(!)1379 778 y Fg(\003)1409 795 y Fw(B)p Fj(g)175 b Fw(\(5.20\))581 870 y Fi(S)r Fw(\(A)655 853 y Fg(0)668 870 y Fi(;)7 b Fw(a\))765 847 y Fp(def)771 870 y Fw(=)49 b Fj(f)p Fw(B)14 b Fj(j)f Fw(A)f Fj(2)f Fi(C)s Fw(\(A)1103 853 y Fg(0)1115 870 y Fw(\))e Fj(^)g Fw(A)j Fj(!)f Fw(aB)g Fj(2)h Fw(\005)p Fj(g)208 b Fw(\(5.21\))362 962 y(Using)14 b Fi(S)j Fw(and)c Fi(C)k Fw(w)o(e)d(can)g(compute)f Fj(A)986 947 y Fg(0)998 962 y Fw(.)561 1064 y Fi(V)595 1046 y Fg(0)585 1074 y Fh(T)653 1040 y Fp(def)660 1064 y Fw(=)48 b Fi(V)764 1070 y Fh(T)567 1138 y Fw(G)600 1121 y Fg(0)653 1115 y Fp(def)660 1138 y Fw(=)g Fi(C)s Fw(\()p Fj(f)p Fw(G)p Fj(g)p Fw(\))556 1213 y Fi(V)589 1196 y Fg(0)580 1223 y Fh(N)653 1190 y Fp(def)660 1213 y Fw(=)g Fj(D)q Fw(\(\005)820 1196 y Fg(0)832 1213 y Fw(\))560 1288 y Fi(V)593 1271 y Fg(0)584 1298 y Fh(F)653 1264 y Fp(def)660 1288 y Fw(=)g Fj(f)p Fw(A)792 1271 y Fg(0)818 1288 y Fj(j)13 b Fw(A)874 1271 y Fg(0)897 1288 y Fj(2)f Fi(V)970 1271 y Fg(0)961 1298 y Fh(N)1002 1288 y Fj(^)c Fw(A)1069 1271 y Fg(0)1090 1288 y Fj(\\)h Fi(V)1151 1294 y Fh(F)1191 1288 y Fj(6)p Fw(=)j Fj(fgg)569 1363 y Fw(\005)600 1346 y Fg(0)653 1339 y Fp(def)660 1363 y Fw(=)48 b Fj(f)p Fw(A)792 1346 y Fg(0)815 1363 y Fj(!)11 b Fw(aB)919 1345 y Fg(0)944 1363 y Fj(j)i Fw(A)1000 1346 y Fg(0)1024 1363 y Fj(2)e Fi(V)1096 1346 y Fg(0)1087 1373 y Fh(N)1128 1363 y Fj(^)e Fw(B)1194 1346 y Fg(0)1217 1363 y Fw(=)j Fi(S)r Fw(\(A)1335 1346 y Fg(0)1348 1363 y Fi(;)7 b Fw(a)o(\))p Fj(g)23 b([)740 1425 y(f)p Fw(A)792 1408 y Fg(0)815 1425 y Fj(!)11 b Fi(\025)k Fj(j)e Fw(A)963 1408 y Fg(0)986 1425 y Fj(2)e Fi(V)1059 1408 y Fg(0)1049 1435 y Fh(F)1077 1425 y Fj(g)567
1500 y(A)600 1483 y Fg(0)653 1476 y Fp(def)660 1500 y Fw(=)48 b Fj(h)p Fi(V)790 1483 y Fg(0)780 1510 y Fh(T)807 1500 y Fi(;)7 b(V)859 1483 y Fg(0)850 1510 y Fh(N)881 1500 y Fi(;)g Fw(G)933 1483 y Fg(0)944 1500 y Fi(;)g Fw(\005)994 1483 y Fg(0)1005 1500 y Fj(i)673 1591 y Fw(T)m(able)13 b(5.3:)k Fu(nf)l(a)e Fw(to)f Fu(df)l(a)h Fw(T)m(ransformation)362 1675 y(T)m(o)e(start)i(the)f(m)o(utual)e(recursion)j(o\013,)f(the)g (start)h(sym)o(b)q(ol)d(G)1331 1660 y Fg(0)1356 1675 y Fw(ma)o(y)g(b)q(e)i(put)g(in)o(to)f Fi(V)1696 1660 y Fg(0)1686 1686 y Fh(N)1718 1675 y Fw(.)362 1724 y(The)19 b(error)g(state)g(is)f(a)f(mem)o(b)q(er)g(of)g Fi(V)997 1709 y Fg(0)987 1736 y Fh(N)1037 1724 y Fw(in)g(an)o(y)h(constructed)i Fu(df)l(a)f Fw(that)f(do)q(es)h(not)300 1774 y(accept)f(all)d(strings)i (\(i.e.,)f Fj(L)p Fw(\()p Fj(A)805 1759 y Fg(0)817 1774 y Fw(\))g Fj(6)p Fw(=)h Fi(V)931 1759 y Fg(\003)922 1786 y Fh(T)950 1774 y Fw(\).)26 b(This)17 b(happ)q(ens)g(b)q(ecause)i Fi(S)r Fw(\(A)1498 1759 y Fg(0)1511 1774 y Fi(;)7 b Fw(a\))16 b(is)g(empt)o(y)300 1824 y(whenev)o(er)c(there)g(is)f(no)f(transition)h (on)f(`a')g(out)g(of)g(an)o(y)h(set)g(A)h Fj(2)f Fw(A)1335 1809 y Fg(0)1358 1824 y Fw(in)f(the)h Fu(nf)l(a)r Fw(.)17 b(The)11 b(error)300 1874 y(state)17 b(is)e(the)i(unique)f(place)g(to)g (whic)o(h)g(all)e(rejected)k(strings)e(tak)o(e)g(the)h Fu(df)l(a)q Fw(.)24 b(The)16 b(error)300 1924 y(state)e(has)f(man)o(y)e (en)o(tries)j(and)f(no)f(exits)i(|)e(it)h(is)f(a)h(trap)g(from)e(whic)o (h)i(no)g(string)g(returns.)300 1974 y(The)g(existence)h(of)d(the)i (error)g(state)g(insures)h(that)e(the)h(state-transition)f(function)g (for)g(the)300 2023 y Fu(df)l(a)j Fw(is)f(a)f(single-v)n(alued)g (function.)300 2132 y Fo(Exercises)330 2208 y Fw(13.)20 b([1,1])12 b(Sho)o(w)h(A)f Fj(2)f Fi(C)s Fw(\()p Fj(f)p Fw(A)p Fj(g)p Fw(\).)330 2292 y(14.)20 b([1,1])12 b(Sho)o(w)h(A)634 2276 y Fg(0)658 2292 y Fj(\022)e Fi(C)s Fw(\(A)781 2276 y Fg(0)793 2292 y Fw(\).)330 2375 y(15.)20 b([1,1])12 b(Sho)o(w)h(that)h(the)h(error)g(state)f(is)g(not)g(a)f(\014nal)h (state.)p eop %%Page: 187 11 187 10 bop 300 50 a Fr(5.5.)31 b(A)o(UTOMA)m(TIC)14 b(P)m(ARSING)814 b Fw(187)330 166 y(16.)20 b([1,1])12 b(Sho)o(w)h Fj(fg)e(2)h Fi(V)729 151 y Fg(0)720 178 y Fh(N)765 166 y Fw(i\013)17 b Fj(L)p Fw(\()p Fj(A)896 151 y Fg(0)907 166 y Fw(\))12 b Fj(\032)g Fi(V)1012 151 y Fg(\003)1003 178 y Fh(T)1031 166 y Fw(.)330 248 y(17.)20 b([1,1])12 b(Sho)o(w)h(that)h(for)g(an)o(y) f(constructed)j Fu(df)l(a)f Fj(A)1176 233 y Fg(0)699 336 y Fw(size)q(\(\005)811 319 y Fg(0)823 336 y Fw(\))c(=)h(size)q(\()p Fi(V)1009 319 y Fg(0)999 347 y Fh(N)1031 336 y Fw(\))d Fj(\002)h Fw(size)q(\()p Fi(V)1212 319 y Fg(0)1203 347 y Fh(T)1229 336 y Fw(\))f(+)h(size)q(\()p Fi(V)1410 319 y Fg(0)1401 347 y Fh(F)1429 336 y Fw(\))330 440 y(18.)20 b([1,1])13 b(Supp)q(ose)i(the)h Fu(nf)l(a)q Fw({)p Fu(df)l(a)g Fw(transformation)d(w)o(ere)i(applied)g(to)f(a)h Fu(df)l(a)q Fw(.)20 b(W)m(ould)404 490 y(the)14 b(input)g(and)g(output)g(of)f(the)i (transformation)c(necessarily)k(b)q(e)g(the)f(same?)330 572 y(19.)20 b([1,1])c(Supp)q(ose)i(the)h Fu(nf)l(a)q Fw({)p Fu(df)l(a)g Fw(transformation)d(w)o(ere)i(applied)g(t)o(wice,)g (to)g(a)g Fu(nf)l(a)404 622 y Fw(and)13 b(then)i(to)e(the)i(resulting)f Fu(df)l(a)q Fw(.)k(W)m(ould)12 b(the)i(input)g(and)f(output)h(of)f(the) i(second)404 672 y(transformation)d(necessarily)j(b)q(e)f(the)h(same?) 300 779 y Fo(Example)300 856 y Fw(The)c(follo)o(wing)d Fu(nf)l(a)13 b Fw(is)d(an)h(answ)o(er)g(to)g(Exercise)h(5,)f(to)g (build)f(an)g(automaton)f(to)i(recognize)300 905 y(a)321
890 y Fg(\003)340 905 y Fw(b)363 890 y Fg(\003)382 905 y Fw(c.)18 b(It)c(is)g(a)g Fu(nf)l(a)q Fw(.)542 1043 y Fi(V)566 1049 y Fh(T)634 1043 y Fw(=)42 b Fj(f)p Fw(a)p Fi(;)7 b Fw(b)p Fi(;)g Fw(c)p Fj(g)537 1106 y Fi(V)561 1112 y Fh(N)634 1106 y Fw(=)42 b Fj(f)p Fw(A)p Fi(;)7 b Fw(B)p Fi(;)g Fw(C)o Fj(g)560 1168 y Fw(G)41 b(=)h(A)541 1230 y Fi(V)565 1236 y Fh(F)634 1230 y Fw(=)g Fj(f)p Fw(C)p Fj(g)561 1293 y Fw(\005)g(=)g Fj(f)p Fw(A)11 b Fj(!)g Fw(aA)p Fi(;)c Fw(A)k Fj(!)g Fw(B)q Fi(;)c Fw(B)k Fj(!)g Fw(bB)p Fi(;)c Fw(B)12 b Fj(!)f Fw(cC)p Fi(;)c Fw(C)k Fj(!)g Fi(\025)q Fj(g)559 1355 y(A)42 b Fw(=)g Fj(h)p Fi(V)748 1361 y Fh(T)774 1355 y Fi(;)7 b(V)817 1361 y Fh(N)848 1355 y Fi(;)g Fw(G)p Fi(;)g Fw(\005)p Fj(i)300 1443 y Fw(The)14 b(corresp)q(onding)h Fu(df)l(a)g Fw(for)e(a)820 1428 y Fg(\003)839 1443 y Fw(b)862 1428 y Fg(\003)881 1443 y Fw(c)i(follo)o(ws.)739 1531 y Fi(V)773 1514 y Fg(0)763 1541 y Fh(T)831 1531 y Fw(=)42 b Fj(f)p Fw(a)p Fi(;)7 b Fw(b)p Fi(;)g Fw(c)p Fj(g)734 1593 y Fi(V)767 1576 y Fg(0)758 1604 y Fh(N)831 1593 y Fw(=)42 b Fj(ff)p Fw(A)p Fi(;)7 b Fw(B)o Fj(g)p Fi(;)g Fj(f)p Fw(B)p Fj(g)p Fi(;)g Fj(f)p Fw(C)p Fj(g)p Fi(;)g Fj(fgg)745 1656 y Fw(G)778 1639 y Fg(0)831 1656 y Fw(=)42 b Fj(f)p Fw(A)p Fi(;)7 b Fw(B)p Fj(g)738 1718 y Fi(V)771 1701 y Fg(0)762 1728 y Fh(F)831 1718 y Fw(=)42 b Fj(ff)p Fw(C)p Fj(gg)332 1971 y Fw(\005)363 1954 y Fg(0)416 1971 y Fw(=)490 1861 y Ff(8)490 1899 y(>)490 1911 y(>)490 1923 y(<)490 1998 y(>)490 2011 y(>)490 2023 y(:)548 1896 y Fj(f)p Fw(A)p Fi(;)7 b Fw(B)o Fj(g)12 b(!)f Fw(a)p Fj(f)p Fw(A)p Fi(;)c Fw(B)o Fj(g)p Fi(;)41 b Fj(f)p Fw(B)p Fj(g)12 b(!)f Fw(a)p Fj(fg)p Fi(;)72 b Fj(f)p Fw(C)p Fj(g)11 b(!)g Fw(a)p Fj(fg)p Fi(;)42 b Fj(fg)11 b(!)g Fw(a)p Fj(fg)p Fi(;)548 1946 y Fj(f)p Fw(A)p Fi(;)c Fw(B)o Fj(g)12 b(!)f Fw(b)p Fj(f)p Fw(B)p Fj(g)p Fi(;)88 b Fj(f)p Fw(B)p Fj(g)12 b(!)f Fw(b)p Fj(f)p Fw(B)p Fj(g)p Fi(;)41 b Fj(f)p Fw(C)p Fj(g)11 b(!)g Fw(b)p Fj(fg)p Fi(;)40 b Fj(fg)11 b(!)g Fw(b)p Fj(fg)p Fi(;)548 1996 y Fj(f)p Fw(A)p Fi(;)c Fw(B)o Fj(g)12 b(!)f Fw(c)p Fj(f)p Fw(C)p Fj(g)p Fi(;)92 b Fj(f)p Fw(B)p Fj(g)12 b(!)f Fw(c)p Fj(f)p Fw(C)p Fj(g)p Fi(;)45 b Fj(f)p Fw(C)p Fj(g)11 b(!)g Fw(c)p Fj(fg)p Fi(;)45 b Fj(fg)11 b(!)g Fw(c)p Fj(fg)p Fi(;)1210 2045 y Fj(f)p Fw(C)p Fj(g)g(!)g Fi(\025)1666 1861 y Ff(9)1666 1899 y(>)1666 1911 y(>)1666 1923 y(=)1666 1998 y(>)1666 2011 y(>)1666 2023 y(;)330 2110 y Fj(A)363 2093 y Fg(0)416 2110 y Fw(=)42 b Fj(h)p Fi(V)540 2093 y Fg(0)530 2121 y Fh(T)557 2110 y Fi(;)7 b(V)608 2093 y Fg(0)600 2121 y Fh(N)631 2110 y Fi(;)g Fw(G)683 2093 y Fg(0)693 2110 y Fi(;)g Fw(\005)743 2093 y Fg(0)755 2110 y Fj(i)300 2218 y Fo(Exercises)300 2294 y Fw(Using)14 b(the)g(example)f(ab)q(o)o (v)o(e:)330 2375 y(20.)20 b([1,1])12 b(V)m(erify)h(the)i Fu(nf)l(a)q Fw({)p Fu(df)l(a)g Fw(transformation.)p eop %%Page: 188 12 188 11 bop 300 50 a Fw(188)677 b Fr(CHAPTER)14 b(5.)31 b(AD)o(V)-5 b(ANCED)14 b(TOPICS)330 166 y Fw(21.)20 b([1,1])12 b(V)m(erify)h(the)i(form)o(ula)c(in)i(Exercise)j(17.)330 254 y(22.)k([1,1])12 b(Lab)q(el)h(the)i(states)g(of)e(the)i Fu(df)l(a)g Fw(for)e(a)1085 239 y Fg(\003)1104 254 y Fw(b)1127 239 y Fg(\003)1146 254 y Fw(c)h(and)g(dra)o(w)g(the)g (diagram)e(for)h(it.)330 341 y(23.)20 b([1,1])12 b(Rep)q(eat)i(the)h (previous)f(three)h(exercises)h(for)e(a)1234 326 y Fg(\003)1253
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Fw(=)49 b Fj(f)p Fw([A)11 b Fj(!)g Fi(\013)p Fw(])i Fj(j)g Fw(A)f Fj(!)f Fi(\013\014)j Fj(2)d Fw(\005)p Fj(g)394 406 y Fw(G)427 389 y Fg(0)480 383 y Fp(def)486 406 y Fw(=)49 b([G)10 b Fj(!)h Fi(\025)p Fw(])395 481 y(\005)426 464 y Fg(0)480 458 y Fp(def)486 481 y Fw(=)49 b Fj(f)p Fw([A)11 b Fj(!)g Fi(\013)p Fw(])g Fj(!)g Fw(B[A)g Fj(!)g Fi(\013)p Fw(B])j Fj(j)f Fw(B)f Fj(2)f Fi(V)1145 487 y Fh(T)1181 481 y Fj([)e Fi(V)1242 487 y Fh(N)1283 481 y Fj(^)f Fw(A)k Fj(!)f Fi(\013)p Fw(B)p Fi(\015)k Fj(2)c Fw(\005)p Fj(g)23 b([)567 543 y(f)p Fw([A)11 b Fj(!)g Fi(\013)p Fw(])g Fj(!)g Fw([B)g Fj(!)g Fi(\025)p Fw(])j Fj(j)f Fw(B)f Fj(2)f Fi(V)1082 549 y Fh(N)1123 543 y Fj(^)e Fw(A)j Fj(!)f Fi(\013)p Fw(B)p Fi(\015)j Fj(2)d Fw(\005)p Fj(g)23 b([)567 606 y(f)p Fw([A)11 b Fj(!)g Fi(\013)p Fw(])g Fj(!)g Fi(\025)j Fj(j)f Fw(A)f Fj(!)f Fi(\013)g Fj(2)g Fw(\005)p Fj(g)393 680 y(A)426 663 y Fg(0)480 657 y Fp(def)486 680 y Fw(=)49 b Fj(h)p Fi(V)616 663 y Fg(0)607 691 y Fh(T)633 680 y Fi(;)7 b(V)685 663 y Fg(0)676 691 y Fh(N)707 680 y Fi(;)g Fw(G)759 663 y Fg(0)770 680 y Fi(;)g Fw(\005)820 663 y Fg(0)831 680 y Fj(i)637 771 y Fw(T)m(able)13 b(5.4:)k Fu(cf)o(g)c Fw(to)h Fu(lr)p Fw(\(0\))g Fu(nf)l(a)i Fw(Construction)362 853 y(Since)g(the)g(parse)h(stac)o(k)f(can)g(ha)o(v)o(e)f(b)q(oth)h (the)g(terminal)e(and)h(non)o(terminal)e(sym)o(b)q(ols)300 903 y(from)18 b(the)i Fu(cf)o(g)p Fw(,)g(the)h Fu(nf)l(a)g Fw(m)o(ust)d(ha)o(v)o(e)i(transitions)g(de\014ned)g(for)g(all)e(of)h (them)g(|)g(the)300 953 y(terminal)14 b(v)o(o)q(cabulary)i Fi(V)716 938 y Fg(0)707 964 y Fh(T)749 953 y Fw(is)h(the)f(whole)g(v)o (o)q(cabulary)g(of)f(the)i Fu(cf)o(g)p Fw(.)24 b(The)17 b(non)o(terminal)300 1002 y(v)o(o)q(cabulary)f(is)g(partially)f (complete)h(rules)h(from)e(the)i Fu(cf)o(g)o Fw(.)26 b(Eac)o(h)17 b(of)f(them)f(represen)o(ts)300 1052 y(the)f(state)h(of)e (ha)o(ving)g(recognized)i(part)f(of)g(that)g(rule.)362 1102 y(The)j(canonical)f(parse)i(in)e(the)h Fu(cf)o(g)f Fw(requires)i(that)e(the)i(righ)o(tmost)d(non)o(terminal)f(is)300 1152 y(expanded.)39 b(Here)22 b(is)f(the)g(source)h(of)e(the)i (surprising)f(fact)f(that)h(the)g(parse)h(stac)o(k)f(is)300 1202 y(describable)d(b)o(y)f(a)g Fu(f)l(a)q Fw(.)28 b(As)18 b(the)f Fu(f)l(a)i Fw(w)o(alks)d(the)i(stac)o(k,)g(it)f(is)g(w)o (alking)e(across)k(partially)300 1252 y(complete)e(righ)o(t-hand)g (sides)h(of)f(rules.)30 b(Whenev)o(er)19 b(the)f(next)h(sym)o(b)q(ol)c (in)j(a)f(grammar)300 1301 y(rule)d(is)g(a)f(non)o(terminal)f(in)i(the) g Fu(cf)o(g)p Fw(,)f(there)i(are)g(t)o(w)o(o)e(p)q(ossibilities.)k (Either)362 1384 y Fj(\017)k Fw(the)14 b(non)o(terminal)e(is)i(already) f(there)j(in)d(the)i(stac)o(k)362 1466 y Fj(\017)21 b Fw(part)14 b(of)f(the)i(righ)o(t-hand)e(side)h(of)f(one)h(of)g(its)g (rules)g(is)g(already)f(in)h(the)g(stac)o(k)300 1548 y(The)d(\014rst)h(t)o(w)o(o)f(sc)o(hema)f(for)h(\014lling)e(\005)886 1533 y Fg(0)909 1548 y Fw(in)h(T)m(able)h(5.4)e(re\015ect)k(these)g(t)o (w)o(o)d(p)q(ossibilities.)17 b(The)300 1598 y(\014rst)e(sc)o(hema)e (steps)i(to)f(a)g(state)g(including)f(the)i(sym)o(b)q(ol)d(\(terminal)g (or)i(non)o(terminal\))e(on)300 1648 y(the)17 b(stac)o(k.)27 b(The)17 b(second)h(sc)o(hema)e(starts)h(expanding)f(the)i(non)o (terminal)c(b)o(y)j(an)f(empt)o(y)300 1698 y(transition)c(to)f(a)h (state)h(where)g(the)g(new)f(righ)o(t-hand)f(side)i(is)e(empt)o(y)g (and)h(therefore)i(ready)300 1748 y(to)g(start)g(b)q(eing)g(built)f (from)f(the)j(input.)362 1798 y(The)i(third)f(sc)o(hema)f(terminates)h (the)h Fu(f)l(a)g Fw(b)q(ecause)h(a)d(rule)i(has)f(b)q(een)h(fully)e (built)g(on)300 1847 y(the)i(parse)h(stac)o(k.)28 b(There)18
b(are)f(terminating)e(sc)o(hema)i(for)f(eac)o(h)h(rule)h(in)e(the)i (gramm)o(ar.)300 1897 y(The)12 b(application)f(of)g(the)i Fu(lr)p Fw(\(0\))f(mac)o(hine)f(will)g(rep)q(eatedly)i(arriv)o(e)f(in)f (these)j(terminating)300 1947 y(states)h(and)f(then)g(b)q(e)h (restarted)h(to)d(do)h(the)g(next)h(step)g(in)e(the)i(canonical)e (parse.)362 1997 y(The)18 b(start)f(state)h(of)f(the)g(automaton)e(is)i (a)g(rule)g(for)g(the)h(goal)d(sym)o(b)q(ol)g(of)i(the)h Fu(cf)o(g)300 2047 y Fw(with)c(none)g(of)f(the)h(righ)o(t-hand)g(side)g (built.)362 2096 y(The)h Fu(lr)p Fw(\(0\))f Fu(df)l(a)h Fw(is)e(then)i(computed)e(from)f(the)j Fu(lr)p Fw(\(0\))f Fu(nf)l(a)r Fw(.)300 2204 y Fo(Exercises)330 2281 y Fw(24.)20 b([1,1])12 b(Sho)o(w)h(that)h(the)h(set)f(of)g(parse)h(stac)o(ks)f Fi(\032)g Fw(is)g(a)g Fu(f)l(a)h Fw(language.)330 2363 y(25.)20 b([1,1])12 b(Sho)o(w)h(that)h(for)g(an)f Fu(lr)q Fw(\(0\))h(automaton,)d Fi(V)1182 2348 y Fg(0)1173 2375 y Fh(F)1212 2363 y Fw(=)h(\005.)p eop %%Page: 190 14 190 13 bop 300 50 a Fw(190)677 b Fr(CHAPTER)14 b(5.)31 b(AD)o(V)-5 b(ANCED)14 b(TOPICS)300 166 y Fo(A)i(W)l(ork)o(ed)f (Example:)21 b(LR\(0\))14 b(Construction)300 243 y Fw(Consider)g(the)h (follo)o(wing)c(grammar)g Fj(G)16 b Fw(where)781 379 y Fi(V)805 385 y Fh(T)872 379 y Fw(=)42 b Fj(f)p Fw(f)s Fi(;)7 b Fw(\()p Fi(;)g Fw(\))p Fi(;)g Fj(_)p Fi(;)g Fj(?g)775 441 y Fi(V)799 447 y Fh(N)872 441 y Fw(=)42 b Fj(f)p Fw(P)p Fi(;)7 b Fw(D)p Fi(;)g Fw(C)p Fj(g)798 503 y Fw(G)41 b(=)h(P)800 566 y(\005)f(=)h Fj(f)p Fi(r)q Fw(0)p Fi(;)7 b(r)q Fw(1)p Fi(;)g(r)q Fw(2)p Fi(;)g(r)q Fw(3)p Fi(;)f(r)q Fw(4)p Fj(g)300 652 y Fw(where)812 738 y Fi(r)q Fw(0)69 b(=)h(P)11 b Fj(!)g Fw(D)p Fj(?)812 800 y Fi(r)q Fw(1)69 b(=)h(D)11 b Fj(!)g Fw(D)e Fj(_)g Fw(C)812 862 y Fi(r)q Fw(2)69 b(=)h(D)11 b Fj(!)g Fw(C)812 924 y Fi(r)q Fw(3)69 b(=)h(C)11 b Fj(!)g Fw(f)812 987 y Fi(r)q Fw(4)69 b(=)h(C)11 b Fj(!)g Fw(\()j(D)g(\))362 1073 y(The)20 b(gramma)o(r)d(consists)i(of)g(\014v)o(e)g(rules,)h (including)e(one)i(ending)e(in)h Fj(?)p Fw(,)g(signifying)300 1123 y(end-of-input.)31 b(The)19 b(language)f(is)h(a)f(com)o(bination)e (of)i(b)q(o)q(olean)g(v)n(alue)g(f,)h(logical)d(\\or")300 1172 y(op)q(erations)e(and)g(paren)o(theses.)20 b(It)15 b(is)e(a)h(subset)h(of)f(the)g(language)f(for)h(whic)o(h)g(a)f (recursiv)o(e)300 1222 y(descen)o(t)j(parser)f(w)o(as)e(written)i(in)e (Chapter)i(2.)362 1272 y(This)20 b(language)g(is)g(pic)o(k)o(ed)g(for)g (the)h(example)e(b)q(ecause)i(it)f(do)q(es)h(not)f(require)h(an)o(y)300 1322 y(lo)q(ok)n(ahead.)33 b(This)20 b(is)f(\014ne)h(for)f(purp)q(oses) i(of)e(illustrating)f(the)i(construction)g(and)f(use)300 1372 y(of)e(the)g Fu(lr)q Fw(\(0\))g(mac)o(hine,)f(but)i(sligh)o(tly)d (misleading)g(since)j(ev)o(ery)g(useful)g(programmi)o(ng)300 1421 y(language)13 b(do)q(es)i(require)f(lo)q(ok)n(ahead.)362 1471 y(An)f Fu(lr)p Fw(\(0\))f(parser)h(for)f(this)g(language)g(follo)o (ws.)j(The)e(\014rst)g(step)g(is)f(to)g(write)h(do)o(wn)e(the)300 1521 y Fu(nf)l(a)k Fw(for)f(the)g Fu(lr)q Fw(\(0\))g(mac)o(hine.)300 1628 y Fo(The)i(LR\(0\))e(NF)-5 b(A)16 b(\(example\))405 1741 y Fi(V)439 1724 y Fg(0)429 1751 y Fh(T)497 1741 y Fw(=)42 b Fj(f)p Fw(P)p Fi(;)7 b Fw(D)p Fi(;)g Fw(C)p Fi(;)g Fj(?)p Fi(;)g Fj(_)p Fi(;)f Fw(f)s Fi(;)g Fw(\()p Fi(;)h Fw(\))p Fj(g)400 1803 y Fi(V)434 1786 y Fg(0)424 1813 y Fh(N)497 1803 y Fw(=)42 b Fj(f)p Fi(m)p Fw(0)p Fi(;)7 b(m)p Fw(1)p Fi(;)g(m)p Fw(2)p Fi(;)g(m)p Fw(3)p Fi(;)g(m)p Fw(4)p Fi(;)g(m)p Fw(5)p Fi(;)g(m)p Fw(6)p Fi(;)g(m)p Fw(7)p Fi(;)g(m)p Fw(8)p Fi(;)g(m)p Fw(9)p
Fi(;)f(m)p Fw(10)p Fi(;)h(m)p Fw(11)p Fi(;)f(m)p Fw(12)p Fj(g)412 1865 y Fw(G)445 1848 y Fg(0)497 1865 y Fw(=)42 b Fi(m)p Fw(0)404 1928 y Fi(V)437 1911 y Fg(0)428 1938 y Fh(F)497 1928 y Fw(=)g Fj(f)p Fi(m)p Fw(2)p Fi(;)7 b(m)p Fw(6)p Fi(;)g(m)p Fw(7)p Fi(;)g(m)p Fw(9)p Fi(;)g(m)p Fw(12)p Fj(g)413 1990 y Fw(\005)444 1973 y Fg(0)497 1990 y Fw(=)42 b Fj(f)p Fi(t)p Fw(0)p Fi(;)7 b(t)p Fw(1)p Fi(;)g(t)p Fw(2)p Fi(;)g(t)p Fw(3)p Fi(;)g(t)p Fw(4)o Fi(;)g(t)p Fw(5)p Fi(;)f(t)p Fw(6)p Fi(;)g(t)p Fw(7)p Fi(;)g(t)p Fw(8)p Fi(;)h(t)p Fw(9)o Fi(;)g(t)p Fw(10)o Fi(;)g(t)p Fw(11)o Fi(;)g(t)p Fw(1)o(2)p Fi(;)g(:)f(:)h(:)s(;)g(t)p Fw(19)p Fj(g)411 2052 y(A)444 2035 y Fg(0)497 2052 y Fw(=)42 b Fj(h)p Fi(V)621 2035 y Fg(0)611 2062 y Fh(T)638 2052 y Fi(;)7 b(V)689 2035 y Fg(0)681 2062 y Fh(N)712 2052 y Fi(;)g Fw(G)764 2035 y Fg(0)774 2052 y Fi(;)g Fw(\005)824 2035 y Fg(0)836 2052 y Fj(i)300 2138 y Fw(where)383 2177 y Fi(m)p Fw(0)42 b(=)g([P)11 b Fj(!)g Fi(\025)p Fw(])383 2227 y Fi(m)p Fw(1)42 b(=)g([P)11 b Fj(!)g Fw(D])383 2277 y Fi(m)p Fw(2)42 b(=)g([P)11 b Fj(!)g Fw(D)p Fj(?)p Fw(])383 2327 y Fi(m)p Fw(3)42 b(=)g([D)10 b Fj(!)h Fi(\025)p Fw(])383 2376 y Fi(m)p Fw(4)42 b(=)g([D)10 b Fj(!)h Fw(D])777 2202 y Fi(m)p Fw(5)42 b(=)f([D)11 b Fj(!)g Fw(D)p Fj(_)p Fw(])777 2252 y Fi(m)p Fw(6)42 b(=)f([D)11 b Fj(!)g Fw(D)e Fj(_)g Fw(C])777 2302 y Fi(m)p Fw(7)42 b(=)f([D)11 b Fj(!)g Fw(C])777 2352 y Fi(m)p Fw(8)42 b(=)f([C)11 b Fj(!)h Fi(\025)p Fw(])1218 2202 y Fi(m)p Fw(9)63 b(=)41 b([C)11 b Fj(!)g Fw(f)s(])1218 2252 y Fi(m)p Fw(10)42 b(=)f([C)11 b Fj(!)g Fw(\(])1218 2302 y Fi(m)p Fw(11)42 b(=)f([C)11 b Fj(!)g Fw(\(D])1218 2352 y Fi(m)p Fw(12)42 b(=)f([C)11 b Fj(!)g Fw(\(D\)])p eop %%Page: 191 15 191 14 bop 300 50 a Fr(5.5.)31 b(A)o(UTOMA)m(TIC)14 b(P)m(ARSING)814 b Fw(191)300 166 y(and)383 207 y Fi(t)p Fw(0)41 b(=)h Fi(m)p Fw(0)12 b Fj(!)f Fw(D)c Fi(m)p Fw(1)383 257 y Fi(t)p Fw(1)41 b(=)h Fi(m)p Fw(1)12 b Fj(!)f(?)c Fi(m)p Fw(2)383 307 y Fi(t)p Fw(2)41 b(=)h Fi(m)p Fw(3)12 b Fj(!)f Fw(D)c Fi(m)p Fw(4)383 356 y Fi(t)p Fw(3)41 b(=)h Fi(m)p Fw(4)12 b Fj(!)f(_)c Fi(m)p Fw(5)383 406 y Fi(t)p Fw(4)41 b(=)h Fi(m)p Fw(5)12 b Fj(!)f Fw(C)c Fi(m)p Fw(6)383 456 y Fi(t)p Fw(5)41 b(=)h Fi(m)p Fw(3)12 b Fj(!)f Fw(C)c Fi(m)p Fw(7)383 506 y Fi(t)p Fw(6)41 b(=)h Fi(m)p Fw(8)12 b Fj(!)f Fw(f)f Fi(m)p Fw(9)835 207 y Fi(t)p Fw(7)62 b(=)42 b Fi(m)p Fw(8)11 b Fj(!)g Fw(\()c Fi(m)p Fw(10)835 257 y Fi(t)p Fw(8)62 b(=)42 b Fi(m)p Fw(10)11 b Fj(!)g Fw(D)c Fi(m)p Fw(11)835 307 y Fi(t)p Fw(9)62 b(=)42 b Fi(m)p Fw(11)11 b Fj(!)g Fw(\))c Fi(m)p Fw(12)835 356 y Fi(t)p Fw(10)41 b(=)h Fi(m)p Fw(0)11 b Fj(!)g Fi(m)p Fw(3)835 406 y Fi(t)p Fw(11)41 b(=)h Fi(m)p Fw(3)11 b Fj(!)g Fi(m)p Fw(3)835 456 y Fi(t)p Fw(12)41 b(=)h Fi(m)p Fw(3)11 b Fj(!)g Fi(m)p Fw(8)835 506 y Fi(t)p Fw(13)41 b(=)h Fi(m)p Fw(5)11 b Fj(!)g Fi(m)p Fw(8)1307 232 y Fi(t)p Fw(14)41 b(=)h Fi(m)p Fw(10)11 b Fj(!)g Fi(m)p Fw(3)1307 282 y Fi(t)p Fw(15)41 b(=)h Fi(m)p Fw(2)12 b Fj(!)f Fi(\025)1307 332 y(t)p Fw(16)41 b(=)h Fi(m)p Fw(6)12 b Fj(!)f Fi(\025)1307 381 y(t)p Fw(17)41 b(=)h Fi(m)p Fw(7)12 b Fj(!)f Fi(\025)1307 431 y(t)p Fw(18)41 b(=)h Fi(m)p Fw(9)12 b Fj(!)f Fi(\025)1307 481 y(t)p Fw(19)41 b(=)h Fi(m)p Fw(12)11 b Fj(!)g Fi(\025)300 621 y Fo(Exercise)330 700 y Fw(26.)20 b([1,1])12 b(V)m(erify)h(the)i
(construction)f(of)g(the)g(example)f Fu(nf)l(a)q Fw(.)300 817 y Fo(The)j(LR\(0\))e(DF)-5 b(A)15 b(\(example\))300 897 y Fw(The)10 b(next)h(step)g(is)f(to)g(compute)f(the)i Fu(df)l(a)g Fj(A)985 881 y Fg(00)1016 897 y Fw(from)e(the)h Fu(nf)l(a)i Fj(A)1291 881 y Fg(0)1303 897 y Fw(.)k(The)11 b(ma)r(jorit)o(y)d(of)h(transi-)300 946 y(tions)14 b(in)o(v)o(olv)o(e)f (the)h(generated)h(error)g(state)g(|)f(something)e(that)j(is)e(relativ) o(ely)h(unsigh)o(tly)300 996 y(to)k(write)f(do)o(wn)h(and)f(also)g(not) h(v)o(ery)g(informativ)o(e.)27 b(W)m(e)17 b(establish)h(the)g(con)o(v)o (en)o(tion)g(of)300 1046 y(lea)o(ving)11 b(all)g(transitions)h(to)g (the)g(error)h(state,)g(and)f(the)h(error)g(state)f(itself,)g(out)g(of) g(the)g(dis-)300 1096 y(pla)o(y)o(ed)h(computations)f(and)i(also)f(the) h(ev)o(en)o(tual)f Fu(df)l(a)i Fw(diagram.)h(In)d(an)g(implemen)o (tation)300 1146 y(it)h(is,)f(on)g(the)i(other)f(hand,)g(of)f(no)h (cost)g(to)g(retain)g(the)g(error)h(state)g(and)f(its)g(transitions.) 420 1292 y Fi(V)453 1275 y Fg(00)444 1302 y Fh(T)516 1292 y Fw(=)42 b Fj(f)p Fw(P)p Fi(;)7 b Fw(D)p Fi(;)g Fw(C)p Fi(;)g Fj(?)p Fi(;)g Fj(_)o Fi(;)g Fw(f)r Fi(;)g Fw(\()p Fi(;)g Fw(\))p Fj(g)419 1354 y Fi(V)452 1337 y Fg(00)443 1364 y Fh(N)516 1354 y Fw(=)42 b Fj(f)p Fi(n)p Fw(0)p Fi(;)7 b(n)p Fw(1)p Fi(;)g(n)p Fw(2)p Fi(;)g(n)p Fw(3)p Fi(;)f(n)p Fw(4)p Fi(;)g(n)p Fw(5)p Fi(;)g(n)p Fw(6)p Fi(;)g(n)p Fw(7)p Fi(;)h(n)o Fw(8)p Fi(;)g(n)p Fw(9)o Fi(;)g(n)p Fw(1)o(0)p Fj(g)421 1416 y Fw(G)454 1399 y Fg(00)516 1416 y Fw(=)42 b Fi(n)p Fw(0)420 1478 y Fi(V)453 1461 y Fg(00)444 1489 y Fh(F)516 1478 y Fw(=)g Fj(f)p Fi(n)p Fw(2)p Fi(;)7 b(n)p Fw(3)p Fi(;)g(n)p Fw(5)p Fi(;)g(n)p Fw(8)p Fi(;)f(n)p Fw(9)p Fj(g)422 1541 y Fw(\005)453 1524 y Fg(00)516 1541 y Fw(=)42 b Fj(f)p Fi(v)q Fw(0)p Fi(;)7 b(v)q Fw(1)p Fi(;)g(v)q Fw(2)p Fi(;)g(v)q Fw(3)p Fi(;)g(v)q Fw(4)p Fi(;)g(v)q Fw(5)p Fi(;)g(v)q Fw(6)p Fi(;)g(v)q Fw(7)p Fi(;)g(v)q Fw(8)p Fi(;)g(v)q Fw(9)p Fi(;)g(v)q Fw(10)p Fi(;)g(v)q Fw(11)p Fi(;)g(v)q Fw(12)p Fi(;)f(:)h(:)f(:)t(;)h(v)q Fw(19)p Fj(g)420 1603 y(A)453 1586 y Fg(00)516 1603 y Fw(=)42 b Fj(h)p Fi(V)639 1586 y Fg(00)630 1613 y Fh(T)661 1603 y Fi(;)7 b(V)713 1586 y Fg(00)704 1613 y Fh(N)735 1603 y Fi(;)g Fw(G)787 1586 y Fg(00)807 1603 y Fi(;)g Fw(\005)857 1586 y Fg(00)878 1603 y Fj(i)300 1698 y Fw(where)383 1738 y Fi(n)p Fw(0)41 b(=)h Fj(f)p Fi(m)p Fw(0)p Fi(;)7 b(m)p Fw(3)p Fi(;)g(m)p Fw(8)p Fj(g)383 1788 y Fi(n)p Fw(1)41 b(=)h Fj(f)p Fi(m)p Fw(1)p Fi(;)7 b(m)p Fw(4)p Fj(g)383 1838 y Fi(n)p Fw(2)41 b(=)h Fj(f)p Fi(m)p Fw(7)p Fj(g)383 1888 y Fi(n)p Fw(3)f(=)h Fj(f)p Fi(m)p Fw(9)p Fj(g)383 1938 y Fi(n)p Fw(4)f(=)h Fj(f)p Fi(m)p Fw(3)p Fi(;)7 b(m)p Fw(8)p Fi(;)g(m)p Fw(10)p Fj(g)877 1738 y Fi(n)p Fw(5)41 b(=)h Fj(f)p Fi(m)p Fw(2)p Fj(g)877 1788 y Fi(n)p Fw(6)f(=)h Fj(f)p Fi(m)p Fw(5)p Fi(;)7 b(m)p Fw(8)p Fj(g)877 1838 y Fi(n)p Fw(7)41 b(=)h Fj(f)p Fi(m)p Fw(4)p Fi(;)7 b(m)p Fw(11)p Fj(g)877 1888 y Fi(n)p Fw(8)41 b(=)h Fj(f)p Fi(m)p Fw(6)p Fj(g)877 1938 y Fi(n)p Fw(9)f(=)h Fj(f)p Fi(m)p Fw(12)p Fj(g)300 1987 y Fw(and)383 2028 y Fi(v)q Fw(0)g(=)g Fi(n)p Fw(0)11 b Fj(!)g Fw(D)c Fi(n)p Fw(1)383 2078 y Fi(v)q Fw(1)42 b(=)g Fi(n)p Fw(0)11 b Fj(!)g Fw(C)c Fi(n)p Fw(2)383 2127 y Fi(v)q Fw(2)42 b(=)g Fi(n)p Fw(0)11 b Fj(!)g Fw(f)f Fi(n)p Fw(3)383 2177 y Fi(v)q Fw(3)42 b(=)g Fi(n)p Fw(0)11 b Fj(!)g Fw(\()c Fi(n)p Fw(4)383 2227 y Fi(v)q Fw(4)42 b(=)g Fi(n)p Fw(1)11 b Fj(!)g(?)c Fi(n)p Fw(5)383 2277 y Fi(v)q Fw(5)42 b(=)g Fi(n)p Fw(1)11 b Fj(!)g(_)c Fi(n)p
Fw(6)383 2327 y Fi(v)q Fw(6)42 b(=)g Fi(n)p Fw(4)11 b Fj(!)g Fw(D)c Fi(n)p Fw(7)383 2376 y Fi(v)q Fw(7)42 b(=)g Fi(n)p Fw(4)11 b Fj(!)g Fw(C)c Fi(n)p Fw(2)777 2053 y Fi(v)q Fw(8)63 b(=)42 b Fi(n)p Fw(4)11 b Fj(!)g Fw(f)f Fi(n)p Fw(3)777 2103 y Fi(v)q Fw(9)63 b(=)42 b Fi(n)p Fw(4)11 b Fj(!)g Fw(\()c Fi(n)p Fw(4)777 2152 y Fi(v)q Fw(10)42 b(=)g Fi(n)p Fw(6)11 b Fj(!)g Fw(C)c Fi(n)p Fw(8)777 2202 y Fi(v)q Fw(11)42 b(=)g Fi(n)p Fw(6)11 b Fj(!)g Fw(f)f Fi(n)p Fw(3)777 2252 y Fi(v)q Fw(12)42 b(=)g Fi(n)p Fw(6)11 b Fj(!)g Fw(\()c Fi(n)p Fw(4)777 2302 y Fi(v)q Fw(13)42 b(=)g Fi(n)p Fw(7)11 b Fj(!)g(_)c Fi(n)p Fw(6)777 2352 y Fi(v)q Fw(14)42 b(=)g Fi(n)p Fw(7)11 b Fj(!)g Fw(\))c Fi(n)p Fw(9)1190 2103 y Fi(v)q Fw(15)42 b(=)g Fi(n)p Fw(2)11 b Fj(!)g Fi(\025)1190 2152 y(v)q Fw(16)42 b(=)g Fi(n)p Fw(3)11 b Fj(!)g Fi(\025)1190 2202 y(v)q Fw(17)42 b(=)g Fi(n)p Fw(5)11 b Fj(!)g Fi(\025)1190 2252 y(v)q Fw(18)42 b(=)g Fi(n)p Fw(8)11 b Fj(!)g Fi(\025)1190 2302 y(v)q Fw(19)42 b(=)g Fi(n)p Fw(9)11 b Fj(!)g Fi(\025)p eop %%Page: 192 16 192 15 bop 300 50 a Fw(192)677 b Fr(CHAPTER)14 b(5.)31 b(AD)o(V)-5 b(ANCED)14 b(TOPICS)300 166 y Fo(Exercise)330 247 y Fw(27.)20 b([1,1])12 b(V)m(erify)h(the)i(construction)f(of)g(the) g(example)f Fu(df)l(a)q Fw(.)300 368 y Fo(Diagram)i(of)g(the)g(LR\(0\)) f(DF)-5 b(A)16 b(\(example\))300 449 y Fw(T)m(o)j(displa)o(y)f(the)i (LR\(0\))f(mac)o(hine,)g(it)g(is)g(con)o(v)o(enien)o(t)h(to)g(use)g (the)g(state)g(names)f(from)300 499 y(the)e(construction)f(for)g(the)h (non-\014nal)d(states,)k(and)d(the)i(rule)f(names)f(from)f(the)j (original)300 548 y(grammar)8 b(for)i(the)i(\014nal)e(states.)18 b(This)11 b(allo)o(ws)e(the)j(reader)g(to)e(readily)h(apply)f(the)h (diagram)300 598 y(to)18 b(arbitrary)g(input)g(texts.)32 b(The)19 b(mapping)d(from)g(states)k(to)e(rules)h(here)g(is)f Fi(n)p Fw(2)h(=)g Fi(r)q Fw(2,)300 648 y Fi(n)p Fw(3)11 b(=)h Fi(r)q Fw(3,)h Fi(n)p Fw(5)e(=)h Fi(r)q Fw(0,)h Fi(n)p Fw(8)e(=)h Fi(r)q Fw(1,)h Fi(n)p Fw(9)e(=)h Fi(r)q Fw(4.)p 279 735 74 2 v 279 787 2 52 v 293 775 a Fi(n)p Fw(0)p 351 787 V 279 789 74 2 v 425 746 a(D)p 354 759 175 2 v 487 758 a Fc(-)p 528 735 74 2 v 528 787 2 52 v 542 775 a Fi(n)p Fw(1)p 601 787 V 528 789 74 2 v 757 746 a Fj(?)p 603 759 258 2 v 819 758 a Fc(-)p 860 721 98 2 v 860 801 2 80 v 875 735 69 2 v 875 787 2 52 v 889 775 a Fi(r)q Fw(0)p 942 787 V 875 789 69 2 v 956 801 2 80 v 860 803 98 2 v 382 1256 2 499 v 425 912 a(C)p 383 925 146 2 v 487 924 a Fc(-)p 528 887 98 2 v 528 967 2 80 v 542 902 69 2 v 542 953 2 52 v 557 941 a Fi(r)q Fw(2)p 610 953 V 542 955 69 2 v 624 967 2 80 v 528 969 98 2 v 433 1078 a(f)p 383 1091 146 2 v 487 1090 a Fc(-)p 528 1053 98 2 v 528 1133 2 80 v 542 1068 69 2 v 542 1119 2 52 v 557 1107 a Fi(r)q Fw(3)p 610 1119 V 542 1121 69 2 v 624 1133 2 80 v 528 1135 98 2 v 383 1257 748 2 v 1089 1256 a Fc(-)p 714 924 2 167 v 757 912 a Fj(_)p 715 925 146 2 v 819 924 a Fc(-)p 860 902 74 2 v 860 953 2 52 v 875 941 a Fi(n)p Fw(6)p 933 953 V 860 955 74 2 v 1006 912 a(C)p 935 925 196 2 v 1089 924 a Fc(-)p 984 1256 2 333 v 1130 887 98 2 v 1130 967 2 80 v 1144 902 69 2 v 1144 953 2 52 v 1158 941 a Fi(r)q Fw(1)p 1212 953 V 1144 955 69 2 v 1226 967 2 80 v 1130 969 98 2 v 1014 1078 a(f)p 985 1091 146 2 v 1089 1090 a Fc(-)p 1130 1053 98 2 v 1130 1133 2 80 v 1144 1068 69 2 v 1144 1119
2 52 v 1158 1107 a Fi(r)q Fw(3)p 1212 1119 V 1144 1121 69 2 v 1226 1133 2 80 v 1130 1135 98 2 v 1006 1223 a Fa(\()p 1130 1234 74 2 v 1130 1285 2 52 v 1144 1273 a Fi(n)p Fw(4)p 1202 1285 V 1130 1287 74 2 v 1276 1244 a(D)p 1205 1257 175 2 v 1338 1256 a Fc(-)p 1379 1234 74 2 v 1379 1285 2 52 v 1393 1273 a Fi(n)p Fw(7)p 1452 1285 V 1379 1287 74 2 v 1587 1223 a Fa(\))p 1454 1257 196 2 v 1608 1256 a Fc(-)p 1649 1219 98 2 v 1649 1299 2 80 v 1663 1234 69 2 v 1663 1285 2 52 v 1677 1273 a Fi(r)q Fw(4)p 1730 1285 V 1663 1287 69 2 v 1745 1299 2 80 v 1649 1301 98 2 v 1544 1256 2 416 v 1462 829 a Fj(_)p 972 842 573 2 v 931 883 a Fc(\000)914 899 y(\000)-42 b(\011)p 1233 1605 2 349 v 1276 1410 a Fw(C)p 1234 1423 146 2 v 1338 1422 a Fc(-)p 1379 1386 98 2 v 1379 1465 2 80 v 1393 1400 69 2 v 1393 1451 2 52 v 1408 1439 a Fi(r)q Fw(2)p 1461 1451 V 1393 1453 69 2 v 1475 1465 2 80 v 1379 1467 98 2 v 1284 1576 a(f)p 1234 1589 146 2 v 1338 1588 a Fc(-)p 1379 1552 98 2 v 1379 1631 2 80 v 1393 1566 69 2 v 1393 1617 2 52 v 1408 1605 a Fi(r)q Fw(3)p 1461 1617 V 1393 1619 69 2 v 1475 1631 2 80 v 1379 1633 98 2 v 1089 1606 146 2 v 1139 1576 a(\()p 1088 1605 2 266 v 1089 1339 a Fc(\000)1097 1331 y(\000)g(\022)799 1710 y Fw(Figure)14 b(5.2.)j(A)d Fu(lr)p Fw(\(0\))g Fu(df)l(a)300 1830 y Fo(Exercises)300 1911 y Fw(Eac)o(h)f(of)f(the)i(follo)o(wing)c (grammars)g(p)q(oses)k(a)e(problem)g(for)g Fu(lr)h Fw(parsers.)19 b(Construct)14 b(eac)o(h)300 1961 y Fu(lr)p Fw(\(0\))g Fu(nf)l(a)h Fw(and)f Fu(df)l(a)q Fw(.)k(Y)m(our)c(results)h(will)d(b)q (e)j(used)f(in)g(the)g(discussion)h(on)e(lo)q(ok)n(ahead.)330 2062 y(28.)20 b([1,1])12 b Fu(lalr)q Fw(\(1\))i(|)f Fv(shift-r)n(e)n (duc)n(e)j Fw(con\015ict)e(for)g Fu(lr)p Fw(\(0\).)425 2133 y(G)41 b Fj(!)g Fw(E)14 b Fj(?)425 2182 y Fw(E)46 b Fj(!)41 b Fw(T)425 2232 y(T)j Fj(!)d Fw(T)13 b(x)425 2282 y(T)44 b Fj(!)d Fw(x)330 2375 y(29.)20 b([1,1])12 b Fu(lalr)q Fw(\(1\))i(|)f(reduce-reduce)k(con\015ict)d(for)g Fu(lr)p Fw(\(0\).)p eop %%Page: 193 17 193 16 bop 300 50 a Fr(5.5.)31 b(A)o(UTOMA)m(TIC)14 b(P)m(ARSING)814 b Fw(193)425 160 y(G)41 b Fj(!)g Fw(E)14 b Fj(?)425 209 y Fw(E)46 b Fj(!)41 b Fw(S)14 b(x)425 259 y(E)46 b Fj(!)41 b Fw(T)13 b(z)425 309 y(S)51 b Fj(!)41 b Fw(a)425 359 y(T)j Fj(!)d Fw(a)330 453 y(30.)20 b([1,1])12 b Fu(lalr)q Fw(\(1\))i(|)f Fv(r)n(e)n(duc)n(e-r)n(e)n(duc)n(e)k Fw(con\015ict)d (for)f Fu(slr)q Fw(\(1\))425 525 y(G)41 b Fj(!)g Fw(E)14 b Fj(?)425 574 y Fw(E)46 b Fj(!)41 b Fw(a)13 b(T)h(a)425 624 y(E)46 b Fj(!)41 b Fw(b)14 b(T)f(b)425 674 y(E)46 b Fj(!)41 b Fw(a)13 b(x)h(b)425 724 y(T)44 b Fj(!)d Fw(x)330 818 y(31.)20 b([1,1])12 b Fu(lalr)q Fw(\(1\))i(|)f(erasure)i(in)f(the)g (lo)q(ok)n(ahead.)425 889 y(G)41 b Fj(!)g Fw(E)14 b Fj(?)425 939 y Fw(E)46 b Fj(!)41 b Fw(S)14 b(x)425 989 y(E)46 b Fj(!)41 b Fw(T)13 b(U)h(y)425 1039 y(S)51 b Fj(!)41 b Fw(a)425 1089 y(T)j Fj(!)d Fw(a)425 1138 y(U)i Fj(!)e Fi(\025)330 1233 y Fw(32.)20 b([1,1])11 b Fu(lalr)q Fw(\(1\))j(|)e (defeats)j(simple)c(lo)q(ok)n(ahead)i(analysis)f(of)h Fu(lr)p Fw(\(0\))h Fu(nf)l(a)q Fw(|)f(a)g Fu(nqlr)404 1283 y Fw(example.)425 1352 y(G)41 b Fj(!)g Fw(E)14 b Fj(?)425 1401 y Fw(E)46 b Fj(!)41 b Fw(b)14 b(A)g(d)425 1451 y(E)46 b Fj(!)41 b Fw(a)13 b(A)h(c)425 1501 y(E)46 b Fj(!)41 b Fw(b)14 b(g)g(c)425 1551 y(E)46 b Fj(!)41 b Fw(a)13 b(g)h(d)425 1601 y(A)43 b Fj(!)e Fw(B)425 1650
y(B)k Fj(!)c Fw(g)330 1745 y(33.)20 b([1,1])12 b(not)i Fu(lalr)q Fw(\(1\))425 1816 y(G)41 b Fj(!)g Fw(E)14 b Fj(?)425 1866 y Fw(E)46 b Fj(!)41 b Fw(S)14 b(x)f(y)425 1916 y(E)46 b Fj(!)41 b Fw(T)13 b(x)h(z)425 1966 y(S)51 b Fj(!)41 b Fw(a)425 2015 y(T)j Fj(!)d Fw(a)330 2110 y(34.)20 b([1,1])12 b(Simple)g(am)o(biguous)f(grammar.)k(P)o(arse)g (xxx)p Fj(?)e Fw(t)o(w)o(o)g(w)o(a)o(ys.)425 2181 y(G)41 b Fj(!)g Fw(E)14 b Fj(?)425 2231 y Fw(E)46 b Fj(!)41 b Fw(E)14 b(E)425 2281 y(E)46 b Fj(!)41 b Fw(x)330 2375 y(35.)20 b([1,1])12 b(Classical)h(\\dangling)f(else")i(am)o(biguit)o(y) m(.)h(P)o(arse)g(iixtx)p Fj(?)d Fw(t)o(w)o(o)h(w)o(a)o(ys.)p eop %%Page: 194 18 194 17 bop 300 50 a Fw(194)677 b Fr(CHAPTER)14 b(5.)31 b(AD)o(V)-5 b(ANCED)14 b(TOPICS)425 160 y Fw(G)41 b Fj(!)g Fw(E)14 b Fj(?)425 209 y Fw(E)46 b Fj(!)41 b Fw(i)13 b(E)425 259 y(E)46 b Fj(!)41 b Fw(i)13 b(E)h(t)g(E)425 309 y(E)46 b Fj(!)41 b Fw(x)300 417 y Fo(Applying)13 b(the)i(LR\(0\))f(Mac)o(hine)300 493 y Fw(The)i(canonical)e(parse)j(is) e(a)g(sequence)i(of)e(grammar)d(rule)k(applications.)21 b(The)16 b(rules)g(are)300 543 y(applied)10 b(to)g(the)g(catenation)h (of)e(the)i(parse)g(stac)o(k)g Fi(\032)f Fw(and)g(the)h(remaining)d (input)i Fi(\016)r Fw(.)17 b(Initially)300 593 y Fi(\032)f Fw(is)g(empt)o(y)f(and)h(all)e(of)i Fi(\016)i Fw(is)d(a)o(v)n(ailable.) 22 b(A)o(t)16 b(eac)o(h)h(rule)f(application,)e(the)j(righ)o(t)e(side)i (of)300 643 y(a)i(rule)g(matc)o(hes)g(a)g(substring)h(of)e Fi(\032\016)r Fw(.)35 b(The)19 b(matc)o(hed)g(substring)h(is)f(remo)o (v)o(ed)f(and)h(is)300 693 y(replaced)c(b)o(y)f(the)g(left)g(side)g(of) f(the)i(rule.)362 743 y(The)j Fu(lr)p Fw(\(0\))g(mac)o(hine)e(is)h(rep) q(etitiv)o(ely)h(applied,)f(yielding)g(one)g(parse)i(step)f(p)q(er)g (ap-)300 792 y(plication.)k(As)16 b(one)f(migh)o(t)f(exp)q(ect,)j(the)f Fu(df)l(a)g Fw(is)g(started)h(in)e(its)g(initial)f(state.)24 b(When)15 b(a)300 842 y(transition)g(o)q(ccurs,)j(the)e(transition)g (sym)o(b)q(ol)e(is)h(tak)o(en)h(from)e Fi(\016)k Fw(pushed)f(on)o(to)e Fi(\032)p Fw(.)25 b(When)300 892 y(the)14 b Fu(lr)p Fw(\(0\))f(mac)o (hine)e(reac)o(hes)k(a)e(\014nal)f(state,)h(the)h(grammar)c(rule)j(to)g (b)q(e)g(applied)g(is)f(giv)o(en)300 942 y(b)o(y)g(the)g(state)h(lab)q (el,)e(and)h(the)g(righ)o(t)g(side)g(of)f(that)h(rule)h(is)e(on)h(the)h (top)e(of)h(the)g(parse)h(stac)o(k.)300 992 y(The)18 b(matc)o(hed)f(string)h(is)f(p)q(opp)q(ed)i(o\013)e Fi(\032)h Fw(and)g(replaced)h(b)o(y)e(the)h(left)g(side)g(of)f(the)h(rule.)300 1041 y(The)e(pro)q(cess)h(is)f(rep)q(eated,)h(starting)e(again)f(in)h (the)h(initial)e(state)i(and)g(at)f(the)h(left)f(of)g Fi(\032)p Fw(.)300 1091 y(One)h(of)e(t)o(w)o(o)g(things)h(\014nally)e (happ)q(ens:)21 b(the)16 b(goal)d(sym)o(b)q(ol)g(G)i(of)f(the)h (grammar)d(app)q(ears)300 1141 y(or)i(the)h Fu(lr)q Fw(\(0\))f(mac)o (hine)f(rejects)j(the)g(input.)j(In)14 b(the)h(former)f(case)h(the)g (sequence)i(of)d(rule)300 1191 y(applications)e(is)h(the)h(canonical)e (parse.)19 b(In)13 b(the)g(latter)h(case)g(an)f(error)h(diagnostic)e (can)h(b)q(e)300 1241 y(rep)q(orted.)393 1329 y Fv(p)n(arse)218 b(unr)n(e)n(ad)253 b(c)n(ommentary)393 1379 y(stack)222 b(input)435 1428 y Fi(\032)290 b(\016)705 1478 y Fw(\(f)12 b Fj(_)d Fw(f)s(\))p Fj(?)230 b Fw(starting)14 b(in)g(state)g Fi(n)p Fw(0)393 1528 y(\()296 b(f)12 b Fj(_)d Fw(f)s(\))p Fj(?)246 b Fw(read)15 b(\()p Fi(;)20 b Fw(goto)13 b Fi(n)p Fw(4)393 1578 y(\(f)283 b Fj(_)p Fw(f)s(\))p Fj(?)d Fw(read)15 b(f,)27 b(goto)13 b Fi(r)q Fw(3)393 1628 y(\(C)266 b Fj(_)p Fw(f)s(\))p Fj(?)280 b Fw(apply)13 b Fi(r)q Fw(3)705 1677 y(\(C)c Fj(_)g Fw(f)s(\))p Fj(?)216 b Fw(starting)14
b(in)g(state)g Fi(n)p Fw(0)393 1727 y(\()296 b(C)9 b Fj(_)g Fw(f)s(\))p Fj(?)232 b Fw(read)15 b(\()p Fi(;)20 b Fw(goto)13 b Fi(n)p Fw(4)393 1777 y(\(C)266 b Fj(_)p Fw(f)s(\))p Fj(?)280 b Fw(read)15 b(C,)27 b(goto)13 b Fi(r)q Fw(2)393 1827 y(\(D)264 b Fj(_)p Fw(f)s(\))p Fj(?)280 b Fw(apply)13 b Fi(r)q Fw(2)705 1877 y(\(D)c Fj(_)g Fw(f)s(\))p Fj(?)214 b Fw(starting)14 b(in)g(state)g Fi(n)p Fw(0)393 1926 y(\()296 b(D)9 b Fj(_)g Fw(f)s(\))p Fj(?)230 b Fw(read)15 b(\()p Fi(;)20 b Fw(goto)13 b Fi(n)p Fw(4)393 1976 y(\(D)264 b Fj(_)p Fw(f)s(\))p Fj(?)280 b Fw(read)15 b(D,)27 b(goto)13 b Fi(n)p Fw(1)393 2026 y(\(D)p Fj(_)236 b Fw(f)s(\))p Fj(?)308 b Fw(read)15 b Fj(_)p Fi(;)20 b Fw(goto)13 b Fi(n)p Fw(6)393 2076 y(\(D)c Fj(_)g Fw(f)205 b(\))p Fj(?)324 b Fw(read)15 b(f,)27 b(goto)13 b Fi(r)q Fw(3)393 2126 y(\(D)c Fj(_)g Fw(C)188 b(\))p Fj(?)324 b Fw(apply)13 b Fi(r)q Fw(3)705 2176 y(\(D)c Fj(_)g Fw(C\))p Fj(?)200 b Fw(starting)14 b(in)g(state)g Fi(n)p Fw(0)393 2225 y(\()296 b(D)9 b Fj(_)g Fw(C\))p Fj(?)216 b Fw(read)15 b(\()p Fi(;)20 b Fw(goto)13 b Fi(n)p Fw(4)393 2275 y(\(D)264 b Fj(_)p Fw(C\))p Fj(?)i Fw(read)15 b(D,)27 b(goto)13 b Fi(n)p Fw(1)393 2325 y(\(D)p Fj(_)236 b Fw(C\))p Fj(?)294 b Fw(read)15 b Fj(_)p Fi(;)20 b Fw(goto)13 b Fi(n)p Fw(6)393 2375 y(\(D)c Fj(_)g Fw(C)188 b(\))p Fj(?)324 b Fw(read)15 b(C,)27 b(goto)13 b Fi(r)q Fw(1)p eop %%Page: 195 19 195 18 bop 300 50 a Fr(5.5.)31 b(A)o(UTOMA)m(TIC)14 b(P)m(ARSING)814 b Fw(195)393 166 y(\(D)264 b(\))p Fj(?)324 b Fw(apply)13 b Fi(r)q Fw(1)705 216 y(\(D\))p Fj(?)276 b Fw(starting)14 b(in)g(state)g Fi(n)p Fw(0)393 266 y(\()296 b(D\))p Fj(?)c Fw(read)15 b(\()p Fi(;)20 b Fw(goto)13 b Fi(n)p Fw(4)393 316 y(\(D)264 b(\))p Fj(?)324 b Fw(read)15 b(D,)27 b(goto)13 b Fi(n)p Fw(7)393 366 y(\(D\))248 b Fj(?)340 b Fw(read)15 b(\))p Fi(;)20 b Fw(goto)13 b Fi(r)q Fw(4)393 415 y(C)282 b Fj(?)340 b Fw(apply)13 b Fi(r)q Fw(4)705 465 y(C)p Fj(?)310 b Fw(starting)14 b(in)g(state)g Fi(n)p Fw(0)393 515 y(C)282 b Fj(?)340 b Fw(read)15 b(C,)27 b(goto)13 b Fi(r)q Fw(2)393 565 y(D)280 b Fj(?)340 b Fw(apply)13 b Fi(r)q Fw(2)705 615 y(D)p Fj(?)308 b Fw(starting)14 b(in)g(state)g Fi(n)p Fw(0)393 664 y(D)280 b Fj(?)340 b Fw(read)15 b(D,)27 b(goto)13 b Fi(n)p Fw(1)393 714 y(D)p Fj(?)620 b Fw(read)15 b Fj(?)p Fi(;)k Fw(goto)14 b Fi(r)q Fw(0)705 764 y(P)345 b(apply)13 b Fi(r)q Fw(0,)g(quit)362 856 y(The)i(canonical)e(parse)i(is)e Fi(r)q Fw(3)p Fi(;)7 b(r)q Fw(2)p Fi(;)g(r)q Fw(3)p Fi(;)g(r)q Fw(1)p Fi(;)g(r)p Fw(4)p Fi(;)g(r)p Fw(2)p Fi(;)g(r)q Fw(0)o(.)362 906 y(If)15 b Fi(\032)g Fw(is)f(the)h(parse)h(stac)o(k)f(and)g Fi(\016)h Fw(the)f(unread)h(input,)e(the)h(in)o(v)n(arian)o(t)e(G)g Fj(!)1558 891 y Fg(\003)1589 906 y Fi(\032\016)k Fw(holds)300 956 y(throughout)e(the)h(parse.)24 b(Note)15 b(that)h(the)g(actions)f (after)h(restarting)g(the)g Fu(df)l(a)g Fw(are)g(rep)q(e-)300 1006 y(titious.)j(This)14 b(is)g(a)g(consequence)j(of)d(the)h(parse)g (stac)o(k)g(not)f(c)o(hanging)f(to)i(the)g(left)f(of)f(the)300 1055 y(substitution.)300 1164 y Fo(Exercises)330 1241 y Fw(36.)20 b([1,1])13 b(V)m(erify)h(the)i(in)o(v)n(arian)o(t)e(G)f Fj(!)955 1226 y Fg(\003)987 1241 y Fi(\032\016)k Fw(where)g(G)c(=)h(P)h (for)g(the)g(parse)h(of)f(\(f)e Fj(_)d Fw(f)s(\))p Fj(?)404 1291 y Fw(sho)o(wn)k(in)f(the)i(previous)f(example.)330 1374 y(37.)20 b([1,1])9 b(Apply)i(the)h Fu(lr)p Fw(\(0\))f(mac)o(hine)f (to)h(strings)g(f)p Fj(?)p Fw(,)g(\(f)s(\))p Fj(?)p Fw(,)f(f)d
Fj(_)t Fw(f)f Fj(_)t Fw(f)s Fj(?)p Fw(,)k(\(f)d Fj(_)t Fw(f)s(\))p Fj(?)p Fw(.)15 b(What)404 1424 y(is)f(the)g(canonical)f (parse)i(in)e(eac)o(h)i(case?)330 1508 y(38.)20 b([1,1])12 b(Apply)i(the)g Fu(lr)q Fw(\(0\))g(mac)o(hine)e(to)i(string)g(ff)p Fj(?)p Fw(.)k(What)13 b(kind)h(of)f(diagnostic)h(can)404 1558 y(b)q(e)g(generated)i(in)d(this)h(case?)19 b(In)14 b(general?)330 1641 y(39.)20 b([1,1])d(In)o(v)o(en)o(t)h(a)h(hac)o(k)g (to)f(a)o(v)o(oid)g(the)h(rep)q(etitious)g(transitions)g(across)h(the)f (parse)404 1691 y(stac)o(k)13 b Fi(\032)g Fw(after)g(a)f(substitution.) 18 b(Hin)o(t:)f(if)12 b Fi(p)g Fw(is)h(the)g(length)f(of)g(the)i (canonical)e(parse,)404 1741 y(and)i Fi(i)h Fw(is)g(the)g(length)f(of)g (the)i(input)e(excluding)g Fj(?)p Fw(,)g(the)h(n)o(um)o(b)q(er)f Fu(df)l(a)h Fw(steps)h(\()p Fv(shift)404 1791 y Fw(or)e Fv(r)n(e)n(duc)n(e)p Fw(\))g(should)f(b)q(e)i(only)e(2)p Fi(p)c Fw(+)g Fi(i)p Fw(.)300 1899 y Fo(Using)14 b(the)h(LR\(0\))f(DF) -5 b(A)16 b(more)f(e\016cien)o(tly)e(\(example\))300 1976 y Fw(The)g(last)g(exercise)h(ab)q(o)o(v)o(e)f(hin)o(ted)g(at)g(an) f(ine\016ciency)h(in)g(the)g(applications)f(of)g(the)i Fu(lr)p Fw(\(0\))300 2026 y(mac)o(hine.)i(Supp)q(ose)e(w)o(e)f(start)g (this)g(time)f(with)g(the)h(v)n(alid)f(text)h(f)d Fj(_)c Fw(\(f)s(\))p Fj(?)p Fw(.)17 b(The)d(canonical)300 2076 y(parse)19 b(will)e(b)q(e)h(a)g(sequence)i(of)e(rule)g(applications)f (resulting)h(in)g(a)g(sequence)i(of)d(forms)300 2126 y(ev)o(en)o(tually)c(con)o(v)o(erging)h(to)g(the)g(goal)f(sym)o(b)q(ol) f(P)m(.)362 2176 y(It)j(is)f(con)o(v)o(enien)o(t)h(to)g(write)g(the)g (string)g(and)f(the)h(states)h(of)e(the)h Fu(df)l(a)h Fw(together,)f(with)300 2225 y(the)e(states)h(b)q(et)o(w)o(een)h(and)d (b)q(elo)o(w)h(the)g(sym)o(b)q(ols)f(of)g(the)h(string.)18 b(Whenev)o(er)c(the)g Fu(df)l(a)f Fw(gets)300 2275 y(to)k(a)g(\014nal)f (state)i(the)g(rewriting)f(is)g(done,)h(remo)o(ving)d(some)h(sym)o(b)q (ols)g(from)f(the)j(string)300 2325 y(and)g(replacing)f(them)g(b)o(y)h (the)g(phrase)h(name.)29 b(When)18 b(sym)o(b)q(ols)e(are)i(remo)o(v)o (ed,)g(so)g(are)300 2375 y(the)e(in)o(terp)q(olated)g(states)g(whic)o (h)f(b)q(ecome)h(in)o(v)n(alid)d(after)j(the)g(substitution.)22 b(The)16 b(e\013ect)p eop %%Page: 196 20 196 19 bop 300 50 a Fw(196)677 b Fr(CHAPTER)14 b(5.)31 b(AD)o(V)-5 b(ANCED)14 b(TOPICS)300 166 y Fw(is)f(that)g(one)g(do)q(es) h(not)f(need)h(to)f(start)g(from)f(the)h(left)g(end)h(of)e(the)i(parse) g(stac)o(k)f(after)g(eac)o(h)300 216 y(rewrite.)20 b(T)m(o)14 b(reestablish)h(the)g(state,)g(the)g(new)g(phrase)g(name)e(is)i(tac)o (k)o(ed)f(on)o(to)g(the)h(fron)o(t)300 266 y(of)e(the)i(input)e(and)h (a)g(non-terminal)e(transition)h(gets)i(things)e(going)g(again.)543 347 y Fi(inter)q(pol)q(ated)174 b(\016)182 b Fw(the)15 b(stac)o(k)f(and)g(input)543 397 y Fi(stack)543 453 y Fp(n0)942 447 y Fw(f)e Fj(_)d Fw(\(f)s(\))p Fj(?)57 b Fw(start)543 503 y Fp(n0)579 497 y Fw(f)592 503 y Fp(r3)942 497 y Fj(_)p Fw(\(f)s(\))p Fj(?)91 b Fw(shift)14 b(o)o(v)o(er)g(f)543 552 y Fp(n0)942 546 y Fw(C)9 b Fj(_)g Fw(\(f)s(\))p Fj(?)43 b Fw(apply)13 b(rule)h Fi(r)q Fw(3)543 602 y Fp(n0)579 596 y Fw(C)609 602 y Fp(r2)942 596 y Fj(_)p Fw(\(f)s(\))p Fj(?)91 b Fw(shift)14 b(o)o(v)o(er)g(C)543 652 y Fp(n0)942 646 y Fw(D)9 b Fj(_)g Fw(\(f)s(\))p Fj(?)41 b Fw(apply)13 b(rule)h Fi(r)q Fw(2)543 702 y Fp(n0)579 696 y Fw(D)611 702 y Fp(n1)942 696 y Fj(_)p Fw(\(f)s(\))p Fj(?)91 b Fw(shift)14 b(o)o(v)o(er)g(D)543 752 y Fp(n0)579 746 y Fw(D)611 752 y Fp(n1)648 746 y Fj(_)676 752 y Fp(n6)942 746 y Fw(\(f)s(\))p Fj(?)119 b Fw(shift)14 b(o)o(v)o(er)g Fj(_)543 802 y Fp(n0)579 796 y Fw(D)611 802 y Fp(n1)657
796 y Fj(_)685 802 y Fp(n6)731 796 y Fw(\()747 802 y Fp(n4)942 796 y Fw(f)s(\))p Fj(?)135 b Fw(shift)14 b(o)o(v)o(er)g(\() 543 851 y Fp(n0)579 845 y Fw(D)611 851 y Fp(n1)657 845 y Fj(_)685 851 y Fp(n6)731 845 y Fw(\()747 851 y Fp(n4)784 845 y Fw(f)797 851 y Fp(r3)942 845 y Fw(\))p Fj(?)151 b Fw(shift)14 b(o)o(v)o(er)g(f)543 901 y Fp(n0)579 895 y Fw(D)611 901 y Fp(n1)657 895 y Fj(_)685 901 y Fp(n6)731 895 y Fw(\()747 901 y Fp(n4)942 895 y Fw(C\))p Fj(?)121 b Fw(apply)13 b(rule)h Fi(r)q Fw(3)543 951 y Fp(n0)579 945 y Fw(D)611 951 y Fp(n1)657 945 y Fj(_)685 951 y Fp(n6)731 945 y Fw(\()747 951 y Fp(n4)784 945 y Fw(C)814 951 y Fp(r2)942 945 y Fw(\))p Fj(?)151 b Fw(shift)14 b(o)o(v)o(er)g(C)543 1001 y Fp(n0)579 995 y Fw(D)611 1001 y Fp(n1)657 995 y Fj(_)685 1001 y Fp(n6)731 995 y Fw(\()747 1001 y Fp(n4)942 995 y Fw(D\))p Fj(?)119 b Fw(apply)13 b(rule)h Fi(r)q Fw(2)543 1051 y Fp(n0)579 1045 y Fw(D)611 1051 y Fp(n1)657 1045 y Fj(_)685 1051 y Fp(n6)731 1045 y Fw(\()747 1051 y Fp(n4)784 1045 y Fw(D)816 1051 y Fp(n7)942 1045 y Fw(\))p Fj(?)151 b Fw(shift)14 b(o)o(v)o(er)g(D)543 1100 y Fp(n0)579 1094 y Fw(D)611 1100 y Fp(n1)657 1094 y Fj(_)685 1100 y Fp(n6)731 1094 y Fw(\()747 1100 y Fp(n4)784 1094 y Fw(D)816 1100 y Fp(n7)853 1094 y Fw(\))869 1100 y Fp(r4)942 1094 y Fj(?)167 b Fw(shift)14 b(o)o(v)o(er)g(\))543 1150 y Fp(n0)579 1144 y Fw(D)611 1150 y Fp(n1)648 1144 y Fj(_)676 1150 y Fp(n6)942 1144 y Fw(C)p Fj(?)137 b Fw(apply)13 b(rule)h Fi(r)q Fw(4)543 1200 y Fp(n0)579 1194 y Fw(D)611 1200 y Fp(n1)657 1194 y Fj(_)685 1200 y Fp(n6)731 1194 y Fw(C)761 1200 y Fp(r1)942 1194 y Fj(?)167 b Fw(shift)14 b(o)o(v)o(er)g(C)543 1250 y Fp(n0)942 1244 y Fw(D)p Fj(?)135 b Fw(apply)13 b(rule)h Fi(r)q Fw(1)543 1300 y Fp(n0)579 1294 y Fw(D)611 1300 y Fp(n1)942 1294 y Fj(?)167 b Fw(shift)14 b(o)o(v)o(er)g(D)543 1349 y Fp(n0)579 1343 y Fw(D)611 1349 y Fp(n1)648 1343 y Fj(?)681 1349 y Fp(r0)942 1343 y Fi(\025)176 b Fw(shift)14 b(o)o(v)o(er)g Fj(?)543 1399 y Fp(n0)942 1393 y Fw(P)172 b(apply)13 b(rule)h Fi(r)q Fw(0,)f(quit)362 1475 y(The)h(canonical)e(parse)i(is)f Fi(r)q Fw(3)p Fi(;)7 b(r)q Fw(2)p Fi(;)g(r)q Fw(3)p Fi(;)g(r)q Fw(2)p Fi(;)f(r)q Fw(4)p Fi(;)g(r)q Fw(1)p Fi(;)g(r)q Fw(0.)15 b(No)o(w)e(one)g(can)g(observ)o(e)h(that)f(the)300 1525 y(sym)o(b)q(ols)h(in)i(the)g(in)o(terp)q(olated)g(stac)o(k)h(\(as) f(con)o(trasted)h(to)f(the)g(states\))h(are)g(nev)o(er)g(used.)300 1575 y(That)10 b(is,)h(the)g(reduce)h(step)g(m)o(ust)d(discard)i(one)g (state)g(and)f(sym)o(b)q(ol)f(from)g(the)i(in)o(terp)q(olated)300 1625 y(stac)o(k)17 b(for)f(ev)o(ery)h(sym)o(b)q(ol)e(on)h(the)h(righ)o (t)f(side)h(of)f(the)h(applied)f(rule)h(but)g(need)g(examine)300 1675 y(none)f(of)f(them)f(while)h(doing)g(so.)22 b(The)16 b(newly)f(exp)q(osed)i(top-of-stac)o(k)e(is)g(the)h(restarting)300 1724 y(state.)j(A)13 b(new)h(stac)o(k)g(consisting)f(of)g(only)f(the)i (states,)h(can)e(b)q(e)h(used)h(in)d(place)i(of)f Fi(\032)p Fw(.)18 b(This)300 1774 y(is)11 b(the)h(form)d(of)i(parse)h(stac)o(k)g (used)g(in)f(the)g(rest)i(of)d(this)i(section.)17 b(W)m(e)11 b(will)f(call)g(it)h(the)h(parse)300 1824 y(state)i(stac)o(k)f(to)f (distinguish)g(from)f(the)j(parse)f(stac)o(k)g(and)g(use)g(sym)o(b)q (ol)e Fi(\033)j Fw(to)f(represen)o(t)i(it.)300 1931 y Fo(Exercises)330 2007 y Fw(40.)20 b([1,1])13 b(Once)k(again)d(apply)h (the)h Fu(lr)q Fw(\(0\))f(mac)o(hine)f(to)i(strings)g(f)p Fj(?)p Fw(,)e(\(f)s(\))p Fj(?)p Fw(,)h(f)e Fj(_)d Fw(f)j Fj(_)d Fw(f)s Fj(?)p Fw(,)404 2057 y(\(f)j Fj(_)d Fw(f)s(\))p Fj(?)p Fw(,)k(but)h(in)g(this)g(case)h(use)g(the)g(parse)g(state)g
(stac)o(k)f Fi(\033)h Fw(instead)g(of)e(the)i(parse)404 2107 y(stac)o(k)e Fi(\032)p Fw(.)330 2187 y(41.)20 b([1,1])12 b(Do)q(es)j(using)g Fi(\033)g Fw(a\013ect)h(the)f(qualit)o(y)e(of)h (the)h(diagnostics)g(that)f(can)h(b)q(e)g(gener-)404 2237 y(ated?)330 2317 y(42.)20 b([1,1])e(Giv)o(en)h(the)h Fu(lr)q Fw(\(0\))g Fu(df)l(a)g Fw(and)g(some)f(parse)i(state)f(stac)o (k)h Fi(\033)q Fw(,)g(sho)o(w)f(ho)o(w)f(to)404 2367 y(compute)13 b(the)i(corresp)q(onding)f(parse)h(stac)o(k)f Fi(\032)p Fw(.)p eop %%Page: 197 21 197 20 bop 300 50 a Fr(5.5.)31 b(A)o(UTOMA)m(TIC)14 b(P)m(ARSING)814 b Fw(197)300 166 y Fo(The)16 b(F)l(ailures)d(of)i(LR\(0\))p 501 281 70 2 v 501 341 2 60 v 515 320 a Fi(p)p Fw(0)p 569 341 V 501 343 70 2 v 646 291 a(E)p 576 304 175 2 v 709 303 a Fc(-)p 750 281 70 2 v 750 341 2 60 v 764 320 a Fi(p)p Fw(1)p 818 341 V 750 343 70 2 v 895 291 a Fj(?)p 825 304 175 2 v 958 303 a Fc(-)p 999 267 99 2 v 999 355 2 88 v 1013 281 70 2 v 1013 341 2 60 v 1027 320 a Fi(p)p Fw(4)p 1082 341 V 1013 343 70 2 v 1096 355 2 88 v 999 357 99 2 v 262 w(G)c Fj(!)g Fw(E)p Fj(?)1331 403 y Fw(E)h Fj(!)f Fw(T)1331 486 y(T)h Fj(!)f Fw(Tx)1331 569 y(T)h Fj(!)f Fw(x)p 604 636 2 333 v 646 457 a(T)p 605 470 146 2 v 709 469 a Fc(-)p 750 433 99 2 v 750 521 2 88 v 764 447 70 2 v 764 507 2 60 v 778 486 a Fi(p)p Fw(2)p 833 507 V 764 509 70 2 v 847 521 2 88 v 750 523 99 2 v 895 457 a(x)p 846 470 154 2 v 958 469 a Fc(-)p 999 433 99 2 v 999 521 2 88 v 1013 447 70 2 v 1013 507 2 60 v 1027 486 a Fi(p)p Fw(5)p 1082 507 V 1013 509 70 2 v 1096 521 2 88 v 999 523 99 2 v 655 623 a(x)p 605 636 146 2 v 709 635 a Fc(-)p 750 599 99 2 v 750 687 2 88 v 764 613 70 2 v 764 673 2 60 v 778 652 a Fi(p)p Fw(3)p 833 673 V 764 675 70 2 v 847 687 2 88 v 750 689 99 2 v 709 798 a(Figure)j(5.3.)j(Inadequate)e(LR\(0\))e Fu(df)l(a)362 890 y Fw(T)m(aking)i(the)j(gramma)o(r)c(from)h(Exercise)j(28,)f(w)o(e)g (get)g(the)g Fu(df)l(a)g Fw(in)f(Figure)h(5.3.)26 b(The)300 940 y(problem)14 b(arises)i(with)f(state)h Fi(p)p Fw(2,)f(whic)o(h)g (is)g(b)q(oth)h(a)f(reduce)i(state)f(for)f(rule)g(E)f Fj(!)g Fw(T)h(and)300 990 y(also)g(a)g(shift)h(state)g(carrying)g(on)f (b)o(y)g(x)h(to)f(state)i Fi(p)p Fw(5.)22 b(Ha)o(ving)15 b(arriv)o(ed)h(in)f(state)h Fi(p)p Fw(2,)f(the)300 1040 y(sargean)o(t)451 1025 y Fp(8)484 1040 y Fw(w)o(on't)e(kno)o(w)g(what)h (command)d(to)j(giv)o(e.)362 1090 y(The)d(answ)o(er)h(can)f(b)q(e)g (found)f(b)o(y)h(examining)d(the)j Fu(lr)p Fw(\(0\))g Fu(df)l(a)q Fw(.)17 b(The)11 b(only)f(allo)o(w)o(ed)f(shift)300 1139 y(in)16 b(state)h Fi(p)p Fw(2)e(is)h(on)g(sym)o(b)q(ol)e(x.)25 b(One)17 b(could)f(tak)o(e)g(the)h(attitude)f(\\shift)g(when)g(y)o(ou)g (can")300 1189 y(and)e(that)g(w)o(ould)f(w)o(ork)h(in)g(this)g(case.)20 b(One)14 b(can)h(also)e(imagine)f(doing)h(a)h(trial)f(reduction)300 1239 y(and)f(see)h(where)g(that)e(w)o(ould)g(lea)o(v)o(e)h(the)g Fu(lr)q Fw(\(0\))f(mac)o(hine.)16 b(In)c(state)h Fi(p)p Fw(2,)e(E)h(will)f(b)q(e)h(pushed)300 1289 y(on)o(to)g(the)g(head)g(of) g(the)g(input)g(and)g(the)h(top)f(of)f Fi(\033)i Fw(will)e(b)q(e)h Fi(p)p Fw(0.)17 b(Then,)c(shifting)e(the)h(E)h(go)q(es)300 1339 y(to)g(state)i Fi(p)p Fw(1)e(in)g(whic)o(h)g(only)g Fj(?)g Fw(is)g(v)n(alid)f(input.)18 b(Th)o(us,)13 b(if)g(reduce)i(b)o (y)f(rule)f(E)f Fj(!)f Fw(T)j(\(from)300 1389 y(state)h Fi(p)p Fw(2\))f(w)o(as)g(the)h(righ)o(t)f(answ)o(er,)h(the)g(next)f (sym)o(b)q(ol)f(will)g(surely)h(b)q(e)h Fj(?)p Fw(.)k(This)14 b(resolv)o(es)300 1438 y(the)f(sargean)o(t's)f(dilema:)j(when)e(in)f
(state)h Fi(p)p Fw(2,)e(an)h(x)g(gets)h(shifted)g(but)f Fj(?)f Fw(is)h(left)g(alone)g(and)300 1488 y(a)i(reduce)h(b)o(y)f(rule) g(E)e Fj(!)f Fw(T)j(is)f(done)h(instead.)362 1538 y(The)h(next)f(task)g (is)g(to)f(generalize)i(and)f(formalize)e(this)i(insigh)o(t.)300 1647 y Fo(Exercises)330 1724 y Fw(43.)20 b([1,1])e(Sho)o(w)h(that)h (merely)g(letting)f(shift)h(tak)o(e)g(precedence)j(o)o(v)o(er)d(reduce) i(is)d(the)404 1774 y(correct)c(solution)e(for)h(the)g Fu(lr)q Fw(\(0\))g(mac)o(hine)e(in)i(Figure)g(5.3.)330 1858 y(44.)20 b([1,1])c(Eac)o(h)i(of)g(the)g(grammars)d(in)j(the)h(set) g(starting)f(with)f(Exercise)j(27)d(fails)g(to)404 1907 y(b)q(e)h Fu(lr)p Fw(\(0\).)28 b(Iden)o(tify)17 b(the)h(failure\(s\).) 29 b(See)18 b(if)e(the)i Fu(lr)q Fw(\(0\))f(mac)o(hine)f(con)o(tains)h (the)404 1957 y(resolution)d(to)f(the)i(problem\(s\))e(as)h(in)f(the)i (example)d(w)o(ork)o(ed)i(ab)q(o)o(v)o(e.)300 2074 y Fq(Lo)r(ok)m(ahead)300 2151 y Fw(The)g Fu(lr)q Fw(\(0\))f(mac)o(hine)g (m)o(ust)g(b)q(e)h(augmen)o(ted)f(with)g(lo)q(ok)n(ahead)g(for)g (practical)h(languages.)300 2201 y(In)19 b(fact,)g(the)g Fu(df)l(a)h Fw(has)e(b)q(een)i(used)f(as)g(a)f(stepping)h(pro)q (cedure,)j(taking)17 b(the)i(form)e(\(in)300 2251 y(language)c Fu(x)p Fw(\):)362 2301 y Fi(\033)o(;)7 b(\016)13 b Fw(:=)f Fs(")o Fi(step)p Fs(")g Fw(:=)g Fi(\033)o(;)7 b(\016)p 300 2336 576 2 v 346 2363 a Fn(8)364 2375 y Fm(See)j(page)h(186.)p eop %%Page: 198 22 198 21 bop 300 50 a Fw(198)677 b Fr(CHAPTER)14 b(5.)31 b(AD)o(V)-5 b(ANCED)14 b(TOPICS)300 166 y Fw(where)g(at)e(eac)o(h)h (step)h(either)f(a)g(shift)f(tak)o(es)h(a)f(sym)o(b)q(ol)f(from)g Fi(\016)j Fw(and)f(places)g(the)g(new)g(state)300 216 y(on)i(the)h(top)f(of)f Fi(\033)q Fw(,)h(or)g(a)g(reduce)i(p)q(ops)e (some)f(states)j(o\013)e Fi(\033)h Fw(and)f(puts)g(a)g(non)o(terminal)e (on)300 266 y(the)h(fron)o(t)g(of)f Fi(\016)r Fw(.)362 316 y(The)j(argumen)o(ts)f(of)g Fi(step)i Fw(range)f(o)o(v)o(er)f (in\014nite)h(sets)h(therefore)g(they)f(cannot)g(b)q(e)h(di-)300 366 y(rectly)e(tabulated.)20 b(The)15 b(top)g(of)f(the)h(parse)h(state) f(stac)o(k)g Fi(s)g Fw(and)f(the)i(leading)d(sym)o(b)q(ol)g(on)300 415 y(the)j(input)f(D)f(are)i(the)g(k)o(eys.)22 b(One)16 b(can)f(implemen)o(t)d Fi(step)k Fw(with)f(\014nite)g(tables)g (recording)300 465 y(all)c(of)h(the)h(p)q(ossible)g(decisions.)18 b(There)c(are)f(three)h(p)q(ossibilities)e(follo)o(wing)e(the)j(arriv)n (al)e(at)300 515 y(an)o(y)i(particular)h(state)h Fi(s)p Fw(:)362 596 y Fj(\017)21 b Fv(shift)13 b Fw(\(if)g(there)i(is)f(a)g (transition)f(from)f Fi(s)j Fw(de\014ned)g(on)e(D\))362 678 y Fj(\017)21 b Fv(r)n(e)n(duc)n(e)14 b Fw(\(if)f(D)g(is)h(in)f(the) i(lo)q(ok)n(ahead\))362 761 y Fj(\017)21 b Fv(r)n(eje)n(ct)13 b Fw(\(a)g(syn)o(tax)h(error)h(has)f(b)q(een)h(disco)o(v)o(ered\))362 842 y(What)f(is)f(needed)j(are)e(t)o(w)o(o)g(functions:)362 923 y Fj(\017)21 b Fi(s)12 b Fw(:=)f Fv(shift)s Fw(\()p Fi(s;)c Fw(D\);)14 b(whic)o(h)g(lo)q(oks)f(up)h(a)f(new)i(state)f Fi(s)362 1005 y Fj(\017)21 b Fi(r)12 b Fw(:=)g Fv(r)n(e)n(duc)n(e)s Fw(\()p Fi(s;)7 b Fw(D\),)13 b(whic)o(h)h(lo)q(oks)f(up)h(the)g(rule)h Fi(r)f Fw(to)g(b)q(e)g(applied.)300 1087 y(Information)f(ab)q(out)i (rules,)h(suc)o(h)h(as)e(the)h(length)g(of)f(the)h(rule)f(and)h(whic)o (h)f(non)o(terminal)300 1137 y(it)f(de\014nes,)g(m)o(ust)f(also)g(b)q (e)i(a)o(v)n(ailable)d(to)h(the)i(algorithm)c(implemen)o(ting)f Fv(step)p Fw(.)362 1186 y(The)j(functions)g Fv(shift)p Fw(,)e(and)i Fv(r)n(e)n(duc)n(e)i Fw(can)e(eac)o(h)g(b)q(e)g(represen)o (ted)i(b)o(y)d(a)g(\014xed)h(size)h(table.)300 1236 y(F)m(or)h Fu(cf)o(g)f Fj(G)i Fw(=)d Fj(h)p Fi(V)589 1242 y Fh(T)616 1236 y Fi(;)7 b(V)659 1242 y Fh(N)690 1236 y Fi(;)g(G;)g
Fw(\005)p Fj(i)14 b Fw(and)g(deriv)o(ed)i Fu(lr)p Fw(\(0\))f Fu(df)l(a)h Fj(A)1282 1221 y Fg(0)1307 1236 y Fw(=)e Fj(h)p Fi(V)1403 1221 y Fg(0)1393 1248 y Fh(T)1419 1236 y Fi(;)7 b(V)1471 1221 y Fg(0)1462 1248 y Fh(N)1494 1236 y Fi(;)g(G)1546 1221 y Fg(0)1556 1236 y Fi(;)g Fw(\005)1606 1221 y Fg(0)1618 1236 y Fj(i)p Fw(,)14 b(eac)o(h)300 1286 y(table)g(is)f(a)h(matrix)e(of)h(with)g(size)q(\()p Fi(V)871 1271 y Fg(0)861 1297 y Fh(N)893 1286 y Fw(\))c Fj(\002)g Fw(\(size)q(\()p Fi(V)1080 1292 y Fh(T)1116 1286 y Fj([)g Fi(V)1177 1292 y Fh(N)1208 1286 y Fw(\))14 b(elemen)o(ts.)k(That)c(is,)f(for)g(ev)o(ery)300 1336 y(state)i(of)f(the)h Fu(df)l(a)h Fw(and)e(ev)o(ery)i(sym)o(b)q(ol)c(of) i(the)h Fu(cf)o(g)p Fw(,)f(there)i(is)e(one)h(en)o(try)g(in)f(eac)o(h)i (table.)300 1386 y(The)e(en)o(tries)h(for)f Fv(shift)j Fw(can)d(b)q(e)h(pic)o(k)o(ed)f(directly)g(o\013)g(the)g Fu(lr)p Fw(\(0\))g(mac)o(hine:)670 1475 y(A)701 1458 y Fg(0)724 1475 y Fj(!)d Fw(DC)839 1458 y Fg(0)862 1475 y Fj(2)h Fw(\005)933 1458 y Fg(0)986 1475 y Fw(i\013)44 b Fv(shift)t Fw(\(A)1195 1458 y Fg(0)1207 1475 y Fi(;)7 b Fw(D\))j(=)i(C)1358 1458 y Fg(0)1634 1475 y Fw(\(5.22\))362 1564 y(Supp)q(ose)18 b(that)e(A)651 1549 y Fg(0)679 1564 y Fj(2)g Fi(V)756 1549 y Fg(0)747 1575 y Fh(F)775 1564 y Fw(.)25 b(Then)18 b(there)f(is)g(some)e(rule)i Fi(r)g Fj(2)f Fw(\005)g(from)f(the)i Fu(cf)o(g)f Fw(suc)o(h)300 1614 y(that)i([)p Fi(r)q Fw(])f Fj(2)g Fw(A)531 1599 y Fg(0)543 1614 y Fw(.)29 b(There)19 b(are)f(sev)o(eral)h(strategies)g (for)e(recording)h(v)n(alues)g(for)f Fv(r)n(e)n(duc)n(e)p Fw(.)29 b(If)300 1664 y(there)17 b(is)e(nothing)f(else)i(in)f(A)765 1649 y Fg(0)792 1664 y Fw(except)h([)p Fi(r)q Fw(])e(then)i(the)g (state)g(is)f Fu(lr)p Fw(\(0\).)22 b(Only)15 b(a)g(reduction)300 1713 y(is)i(allo)o(w)o(ed.)25 b(In)16 b(this)h(case)h(it)e(will)f(nev)o (er)j(cause)g(the)f(parser)h(to)e(fail)g(\(although)g(it)g(ma)o(y)300 1763 y(dela)o(y)11 b(the)g(detection)i(of)d(a)h(syn)o(tax)g(error\))h (to)f(giv)o(e)g(the)g(v)n(alue)g Fi(r)h Fw(to)f Fv(r)n(e)n(duc)n(e)j Fw(for)d(all)f(p)q(ossible)300 1813 y(lo)q(ok)n(aheads.)542 1902 y(A)573 1885 y Fg(0)597 1902 y Fw(=)h Fj(f)p Fw([)p Fi(r)q Fw(])p Fj(g)d(^)h Fi(r)j Fj(2)g Fw(\005)d Fj(^)g Fw(D)i Fj(2)g Fi(V)21 b Fj(\))11 b Fv(r)n(e)n(duc)n(e)s Fw(\(A)1261 1885 y Fg(0)1273 1902 y Fi(;)c Fw(D\))k(=)h Fi(r)220 b Fw(\(5.23\))300 1991 y(It)12 b(is)g(only)g(when)g Fj(f)p Fw([)p Fi(r)q Fw(])p Fj(g)e(\032)i Fw(A)750 1976 y Fg(0)768 1991 y Fj(^)6 b Fi(r)12 b Fj(2)f Fw(\005)h(that)h(lo)q(ok)n (ahead)e(m)o(ust)g(b)q(e)i(added.)k(There)d(are)e(t)o(w)o(o)300 2041 y(algorithms)f(of)j(in)o(terest:)19 b Fu(slr)14 b Fw(and)f Fu(lalr)q Fw(,)h(the)g(latter)g(b)q(eing)g(the)g(more)f(p)q (o)o(w)o(erful)g(of)g(the)300 2091 y(t)o(w)o(o.)k(As)c(it)g(happ)q (ens,)g(the)h Fu(slr)f Fw(algorithm)d(can)j(b)q(e)g(used)h(to)f (compute)f(the)h Fu(lalr)h Fw(tables,)300 2141 y(so)g(it)f(mak)o(es)g (sense)j(to)e(presen)o(t)h Fu(slr)f Fw(\014rst.)300 2248 y Fo(SLR\(1\))g(Lo)q(ok)m(ahead)300 2325 y Fw(Whenev)o(er)i(a)f (reduction)g(is)g(applied,)f(a)h(phrase)h(is)e(reduced)j(to)e(a)f(non)o (terminal.)20 b(What-)300 2375 y(ev)o(er)d(comes)e(next)h(in)f(the)i (input)e(m)o(ust)g(follo)o(w)f(that)h(non)o(terminal)f(in)h(the)i Fu(cf)o(g)o Fw(.)24 b(In)15 b(the)p eop %%Page: 199 23 199 22 bop 300 50 a Fr(5.5.)31 b(A)o(UTOMA)m(TIC)14 b(P)m(ARSING)814 b Fw(199)300 166 y(little)16 b Fu(cf)o(g)h Fw(b)q(elo)o(w)g(\(from)f (Figure)h(5.3\),)g(E)g(o)q(ccurs)i(once)f(on)f(the)h(righ)o(t-hand)e (side)i(of)e(a)300 216 y(rule.)j(There)d(is)e(a)g(terminal)e(sym)o(b)q
(ol)g(to)i(its)h(imm)o(ediate)d(righ)o(t.)19 b(Therefore,)c(whenev)o (er)g(a)300 266 y(reduction)g(to)e(E)h(is)g(made,)e(the)j(only)e (acceptable)i(follo)o(wing)c(sym)o(b)q(ol)h(is)i Fj(?)p Fw(.)p 501 388 70 2 v 501 448 2 60 v 515 428 a Fi(p)p Fw(0)p 569 448 V 501 450 70 2 v 646 399 a(E)p 576 412 175 2 v 709 411 a Fc(-)p 750 388 70 2 v 750 448 2 60 v 764 428 a Fi(p)p Fw(1)p 818 448 V 750 450 70 2 v 895 399 a Fj(?)p 825 412 175 2 v 958 411 a Fc(-)p 999 374 99 2 v 999 462 2 88 v 1013 388 70 2 v 1013 448 2 60 v 1027 428 a Fi(p)p Fw(4)p 1082 448 V 1013 450 70 2 v 1096 462 2 88 v 999 464 99 2 v 262 w(G)d Fj(!)g Fw(E)p Fj(?)1331 511 y Fw(E)h Fj(!)f Fw(T)1331 594 y(T)h Fj(!)f Fw(Tx)1331 677 y(T)h Fj(!)f Fw(x)p 604 743 2 333 v 646 565 a(T)p 605 578 146 2 v 709 577 a Fc(-)p 750 540 99 2 v 750 628 2 88 v 764 554 70 2 v 764 614 2 60 v 778 594 a Fi(p)p Fw(2)p 833 614 V 764 616 70 2 v 847 628 2 88 v 750 630 99 2 v 895 565 a(x)p 846 578 154 2 v 958 577 a Fc(-)p 999 540 99 2 v 999 628 2 88 v 1013 554 70 2 v 1013 614 2 60 v 1027 594 a Fi(p)p Fw(5)p 1082 614 V 1013 616 70 2 v 1096 628 2 88 v 999 630 99 2 v 655 731 a(x)p 605 744 146 2 v 709 743 a Fc(-)p 750 706 99 2 v 750 794 2 88 v 764 720 70 2 v 764 780 2 60 v 778 760 a Fi(p)p Fw(3)p 833 780 V 764 782 70 2 v 847 794 2 88 v 750 796 99 2 v 362 901 a(There)i(is)f(one)g(other)g(non)o(terminal)e(in)h(the)i (little)e Fu(cf)o(g)o Fw(.)18 b(It)12 b(o)q(ccurs)h(at)e(the)i(righ)o (t)e(end)i(of)300 951 y(a)e(rule)g(de\014ning)g(E,)g(whic)o(h)g(sa)o (ys)g(that)h(whatev)o(er)f(follo)o(ws)f(E)h(ma)o(y)e(also)i(follo)o(w)e (T.)i(T)g(is)g(also)300 1000 y(follo)o(w)o(ed)i(b)o(y)h(x)g(in)g(the)g Fu(cf)o(g)p Fw(.)19 b(W)m(e)14 b(deduce)i(that)e(whenev)o(er)i(a)e(T)g (is)g(made,)f(the)i(follo)o(wing)300 1050 y(sym)o(b)q(ol)e(ma)o(y)g(b)q (e)i(either)g(x)g(or)f Fj(?)p Fw(.)20 b(The)15 b Fu(slr)p Fw(\(1\))g(lo)q(ok)n(ahead)f(for)g(the)h(application)e(of)h(an)o(y)300 1100 y(rule)d(A)h Fj(!)f Fi(\014)j Fw(is)d(the)g(set)h(of)f(sym)o(b)q (ols)f(that)h(can)g(follo)o(w)e(A.)i(In)g(the)h(little)e Fu(cf)o(g)o Fw(,)i(the)f(problem)300 1150 y(arises)k(in)f(state)i Fi(p)p Fw(2)e(where)i Fv(shift)s Fw(\()p Fi(p)p Fw(2)p Fi(;)7 b Fw(x\))12 b(=)h Fi(p)p Fw(5)i(and)f Fv(r)n(e)n(duc)n(e)s Fw(\()p Fi(p)p Fw(2)p Fi(;)7 b Fj(?)p Fw(\))12 b(=)h(E)e Fj(!)g Fw(T.)21 b(Without)300 1200 y(the)14 b(lo)q(ok)n(ahead)f Fj(?)g Fw(it)h(w)o(ould)f(not)h(b)q(e)g(clear)g(what)g(to)g(do)g(in)f (state)i Fi(p)p Fw(2.)362 1250 y(The)g(relation)f Fi(F)o(B)r Fw(,)h(meaning)d(\\follo)o(w)o(ed-b)o(y)m(,")g(is)i(the)h(information)d (needed)k(to)e(com-)300 1299 y(pute)h(the)f Fu(slr)p Fw(\(1\))g(lo)q(ok)n(ahead.)j(Both)d(are)h(de\014ned)g(b)q(elo)o(w:)712 1399 y(A)f Fi(F)o(B)i Fw(D)903 1376 y Fp(def)910 1399 y Fw(=)49 b Fj(9)p Fi(\013;)7 b(\016)o(:)13 b Fw(G)e Fj(!)1188 1382 y Fg(\003)1218 1399 y Fi(\013)p Fw(AD)p Fi(\016)308 b Fw(\(5.24\))432 1524 y Fi(r)12 b Fw(=)g(A)g Fj(!)f Fi(\014)g Fj(^)e Fw([)p Fi(r)q Fw(])i Fj(2)g Fw(A)799 1506 y Fg(0)820 1524 y Fj(^)e Fi(r)j Fj(2)g Fw(\005)d Fj(^)g Fw(A)14 b Fi(F)o(B)i Fw(D)55 b Fj(\))41 b Fv(r)n(e)n(duc)n(e)s Fw(\(A)1455 1506 y Fg(0)1467 1524 y Fi(;)7 b Fw(D)o(\))12 b(=)f Fi(r)27 b Fw(\(5.25\))362 1611 y(The)20 b(computation)d(of)h(the) i Fi(F)o(B)h Fw(relation)d(is)h(complicated)f(b)o(y)h(erasure.)1550 1596 y Fp(9)1603 1611 y Fw(If)g(some)300 1661 y(sym)o(b)q(ol)14 b(in)h(a)h(rule)g(migh)o(t)d(disapp)q(ear,)k(then)f(things)g(to)f(the)i (righ)o(t)e(of)g(what)h(w)o(as)g(erased)300 1710 y(m)o(ust)9 b(also)g(b)q(e)i(recorded)g(as)f(follo)o(wing,)e(and)i(so)f(on.)17 b(All)9 b(follo)o(w)o(ed-b)o(y)f(sym)o(b)q(ols)g(ultimately)300
1760 y(deriv)o(e)15 b(from)e(sym)o(b)q(ols)g(next)i(to)g(eac)o(h)g (other)g(in)f(some)g(gramma)o(r)e(rule.)20 b(Supp)q(osing)15 b(that)300 1810 y(the)f Fu(cf)o(g)g Fw(is)g(not)f(pathological,)806 1795 y Fp(10)853 1810 y Fw(then)945 1887 y(A)h Fi(F)o(B)i Fw(D)1001 1937 y(i\013)881 1987 y Fj(9)p Fi(\015)r(;)7 b Fw(X)p Fi(;)g(\026;)g Fw(B)p Fi(;)g Fw(C)p Fi(;)g(\027)954 2036 y(\015)14 b Fj(!)1031 2021 y Fg(\003)1062 2036 y Fi(\025)866 2086 y Fw(X)d Fj(!)g Fi(\026)p Fw(B)p Fi(\015)r Fw(C)p Fi(\027)k Fj(2)c Fw(\005)831 2136 y(B)g Fj(!)913 2121 y Fg(\003)944 2136 y Fi(\021)q Fw(A)e Fj(^)g Fw(C)i Fj(!)1126 2121 y Fg(\003)1157 2136 y Fw(D)p Fi(\020)300 2214 y Fw(Sym)o(b)q(ols)g(B)j(and)f(C)g(are)h(\\next)f(to")g(eac)o(h)h (other.)k(An)o(y)13 b(non)o(terminal)f(ending)h(B)g(m)o(ust)f(b)q(e)300 2264 y(created)g(when)f(an)o(y)f(sym)o(b)q(ol)f(that)i(can)f(b)q(e)i(a) e(head)h(of)f(C)g(app)q(ears.)18 b(All)9 b(of)h(the)i(p)q(ossibilities) p 300 2297 576 2 v 346 2323 a Fn(9)364 2335 y Fm(See)e(page)h(999.)331 2363 y Fn(10)364 2375 y Fm(See)f(page)h(999.)p eop %%Page: 200 24 200 23 bop 300 50 a Fw(200)677 b Fr(CHAPTER)14 b(5.)31 b(AD)o(V)-5 b(ANCED)14 b(TOPICS)300 166 y Fw(can)f(b)q(e)h(collected)g (in)o(to)e(three)j(situations,)d(erasure)j(o)q(ccurs)g(on)e(the)g(left) g(of)g(a)g(rule,)g(in)f(the)300 216 y(middle)h(of)i(a)g(rule,)h(or)f (on)g(the)h(righ)o(t)f(of)g(a)g(rule.)22 b(Supp)q(ose)16 b(that)g(string)f Fi(\015)i Fj(!)1548 201 y Fg(\003)1580 216 y Fi(\025)f Fw(\(i.e.,)e Fi(\015)300 266 y Fw(can)i(b)q(e)h (erased\).)26 b(Then)16 b(w)o(e)h(ha)o(v)o(e)f(three)h(relations)f (that)g(can)g(b)q(e)h(read)f(directly)h(out)f(of)300 316 y(the)e(grammar:)736 417 y(C)803 387 y Fh(:)792 417 y Fi(<)e Fw(D)909 394 y Fp(def)916 417 y Fw(=)48 b(C)12 b Fj(!)f Fi(\015)r Fw(D)p Fi(\016)j Fj(2)d Fw(\005)386 b(\(5.26\))739 492 y(B)805 469 y Fh(:)794 492 y Fw(=)11 b(C)909 469 y Fp(def)916 492 y Fw(=)48 b(X)12 b Fj(!)f Fi(\013)p Fw(B)p Fi(\015)r Fw(C)p Fi(\016)j Fj(2)d Fw(\005)331 b(\(5.27\))738 567 y(A)805 536 y Fh(:)794 567 y Fi(>)12 b Fw(B)909 544 y Fp(def)916 567 y Fw(=)48 b(B)12 b Fj(!)f Fi(\013)p Fw(A)p Fi(\015)j Fj(2)d Fw(\005)381 b(\(5.28\))300 659 y(They)14 b(ma)o(y)e(b)q(e)j(describ)q(ed)g(b)o(y)f(the)h(phrases) 602 760 y(C)668 730 y Fh(:)657 760 y Fi(<)d Fw(D)774 737 y Fp(def)781 760 y Fw(=)90 b(D)13 b(starts)i(some)e(rule)h (de\014ning)g(C)604 833 y(B)670 809 y Fh(:)659 833 y Fw(=)e(C)774 809 y Fp(def)781 833 y Fw(=)90 b(B)14 b(precedes)i(C)e(in) g(some)f(rule)603 905 y(A)671 875 y Fh(:)659 905 y Fi(>)f Fw(B)774 882 y Fp(def)781 905 y Fw(=)90 b(A)14 b(ends)h(some)e(rule)h (de\014ning)g(B)300 997 y(The)k(p)q(oin)o(t)o(y)f(end)h(of)f(the)h (relation)f(is)h(to)o(w)o(ard)f(the)h(visible)f(sym)o(b)q(ol)f(within)h (whic)o(h)g(the)300 1047 y(other)j(sym)o(b)q(ols)f(in)g(the)h(relation) g(\\hide".)35 b(The)20 b(re\015exiv)o(e)h(transitiv)o(e)f(closure)g(of) f(the)300 1096 y(relations)14 b(\(i.e.,)574 1066 y Fh(:)563 1096 y Fi(<)596 1069 y Fg(\003)628 1096 y Fw(and)720 1066 y Fh(:)709 1096 y Fi(>)741 1069 y Fg(\003)761 1096 y Fw(\))g(exp)q(oses)h(the)f(hidden)g(comp)q(onen)o(ts.)k(One)d(can)f (sho)o(w)811 1194 y Fi(F)o(B)58 b Fw(=)1024 1163 y Fh(:)1013 1194 y Fi(>)1045 1166 y Fg(\003)1069 1194 y Fj(\016)1113 1171 y Fh(:)1101 1194 y Fw(=)12 b Fj(\016)1189 1163 y Fh(:)1178 1194 y Fi(<)1210 1166 y Fg(\003)1634 1194 y Fw(\(5.29\))300 1285 y(One)17 b(can)g(understand)g(the)h(comp)q(ound)d (relation)h(ab)q(o)o(v)o(e)g(b)o(y)g(inserting)h(the)g(sym)o(b)q(ols)e (B)300 1335 y(and)f(C;)869 1385 y(A)923 1355 y Fh(:)912 1385 y Fi(>)944 1357 y Fg(\003)968 1385 y Fj(\016)1011
1362 y Fh(:)1000 1385 y Fw(=)e Fj(\016)1087 1355 y Fh(:)1076 1385 y Fi(<)1108 1357 y Fg(\003)1139 1385 y Fw(D)300 1460 y(means)i(there)i(are)f(sym)o(b)q(ols)e(B)i(and)f(C,)h(where)g(A)g (is)g(a)f(tail)f(of)h(B)h(and)g(D)f(is)h(a)f(head)h(of)f(C)300 1510 y(and)g(B)g(is)g(next)g(to)g(C)g(in)f(some)g(rule.)18 b(That)c(is:)794 1601 y Fj(9)p Fw(B)p Fi(;)7 b Fw(C)p Fi(:)13 b Fw(A)974 1571 y Fh(:)963 1601 y Fi(>)995 1574 y Fg(\003)1025 1601 y Fw(B)1078 1578 y Fh(:)1066 1601 y Fw(=)f(C)1163 1571 y Fh(:)1152 1601 y Fi(<)1184 1574 y Fg(\003)1215 1601 y Fw(D)300 1693 y(The)h(computation)e(of)h(the)h (follo)o(w)o(ed-b)o(y)d(relation)i(is)h(thereb)o(y)h(reduced)g(to)e (pic)o(king)g(infor-)300 1743 y(mation)g(out)h(of)h(the)g Fu(cf)o(g)g Fw(and)f(doing)g(some)g(standard)i(computations)d(on)i (relations.)362 1793 y(If)k Fv(r)n(e)n(duc)n(e)k Fw(is)c(not)h (single-v)n(alued,)f(the)h(language)e(is)i(not)f Fu(slr)p Fw(\(1\))h(\()p Fv(a)g(r)n(e)n(duc)n(e-r)n(e)n(duc)n(e)300 1842 y(c)n(on\015ict)p Fw(\).)32 b(If)18 b(b)q(oth)g Fv(shift)k Fw(and)c Fv(r)n(e)n(duc)n(e)j Fw(are)e(de\014ned)g(for)f (some)g(parameters)g Fi(s)h Fw(and)f(D,)300 1892 y(the)f(language)f(is) g(not)g Fu(slr)q Fw(\(1\))g(\()p Fv(a)i(shift-r)n(e)n(duc)n(e)e(c)n (on\015ict)p Fw(\).)26 b(If)16 b(neither)i Fv(shift)i Fw(nor)c Fv(r)n(e)n(duc)n(e)300 1942 y Fw(is)h(de\014ned)h(for)f(some)g (parameters)g Fi(s)g Fw(and)h(D,)e(there)i(is)g(no)e(de\014ned)j (action.)27 b(If)17 b(suc)o(h)h(a)300 1992 y(situation)g(arises)i (during)f(translation,)g(a)f(syn)o(tax)h(error)h(has)f(b)q(een)i(disco) o(v)o(ered)e(and)g(a)300 2042 y(diagnostic)13 b(ma)o(y)f(b)q(e)j (issued.)763 2027 y Fp(11)817 2042 y Fw(Otherwise)g(the)g(function)e (that)h(is)g(de\014ned)h(is)f(ob)q(ey)o(ed.)362 2092 y(Since)k(in)f(practice)i Fv(shift)h Fw(and)e Fv(r)n(e)n(duc)n(e)i Fw(are)e(nev)o(er)g(b)q(oth)g(de\014ned)h(at)e(an)o(y)g(one)h(\014xed) 300 2141 y(lo)q(cation)13 b([)p Fi(s;)7 b Fw(D],)12 b(an)i(implem)o(en) o(tation)d(ma)o(y)h(use)j(a)e(single)h(matrix)e(to)i(store)g(them)f(b)q (oth.)p 300 2179 576 2 v 331 2205 a Fn(11)364 2217 y Fm(It)d(is)h(m)o(y)g(p)q(ersonal)e(opinion)g(that)h(the)g(translator...