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Showing content from http://reports-archive.adm.cs.cmu.edu/anon/1999/CMU-CS-99-156.ps below:

%!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: tr.dvi %%Pages: 22 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%EndComments %DVIPSCommandLine: dvips tr.dvi -f %DVIPSParameters: dpi=300, compressed, comments removed %DVIPSSource: TeX output 1999.07.11:0910 %%BeginProcSet: texc.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 4 get round 4 exch put dup dup 5 get round 5 exch put 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 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/gp 0 N /cp 0 N{ rc 0 ne{rc 1 sub /rc X rw}{G}ifelse}imagemask restore}B /G{{id gp get /gp gp 1 add N dup 18 mod S 18 idiv pl S get exec}loop}B /adv{cp add /cp X}B /chg{rw cp id gp 4 index getinterval putinterval dup gp add /gp X adv}B /nd{/cp 0 N rw exit}B /lsh{rw cp 2 copy get dup 0 eq{pop 1}{dup 255 eq{pop 254}{dup dup add 255 and S 1 and or}ifelse}ifelse put 1 adv} B /rsh{rw cp 2 copy get dup 0 eq{pop 128}{dup 255 eq{pop 127}{dup 2 idiv S 128 and or}ifelse}ifelse put 1 adv}B /clr{rw cp 2 index string putinterval adv}B /set{rw cp fillstr 0 4 index getinterval putinterval adv}B /fillstr 18 string 0 1 17{2 copy 255 put pop}for N /pl[{adv 1 chg} {adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{ adv rsh nd}{1 add adv}{/rc X nd}{1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]dup{bind pop}forall N /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 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f(b)q(ound)g(to)h(b)q(e)g(100.)k(The)c(third)g(creator)g(uses)h(its)f (in)o(teger)f(argumen)o(t)g(to)g(create)i(a)e(singleton)0 959 y(bag.)18 b(The)c(assertion)g Fk(new)p Fn($)p Fg(x)p Fn($)g(stands)h(for)e(the)i(predicate:)104 1037 y Fg(x)c Ff(2)g Fg(dom)p Fn($)p Fg(post)p Fn($)f Ff(\000)g Fg(dom)p Fn($)p Fg(pr)q(e)p Fn($)0 1116 y(Recall)j(that)h(ob)r(jects)h(are)f (nev)o(er)h(destro)o(y)o(ed)g(so)f(that)g Fg(dom)p Fn($)p Fg(pr)q(e)p Fn($)e Ff(\022)g Fg(dom)p Fn($)p Fg(post)p Fn($.)p 0 1174 1950 1 v 0 1249 a Fh(b)n(ag)p 60 1249 13 2 v 16 w(cr)n(e)n(ate)p Fn(=)h Fk(pro)q(c)h Fn($)p Fh(n:)k Fn(in)o(t$)c Fk(returns)d Fn($bag$)178 1299 y Fk(requires)h Fg(n)f Ff(\025)h Fn(0)178 1348 y Fk(ensures)25 b(new)p Fh($r)n(esult$)13 b Ff(^)g Fg(r)q(esul)q(t)739 1354 y Fe(post)815 1348 y Fn(=)26 b Ff(hfg)p Fg(;)7 b(n)p Ff(i)0 1448 y Fh(b)n(ag)p 60 1448 V 16 w(cr)n(e)n(ate)p 182 1448 V 14 w(smal)r(l)14 b Fn(=)g Fk(pro)q(c)f Fn($)h($)g Fk(returns)e Fn($bag$)178 1498 y Fk(ensures)25 b(new)p Fh($r)n(esult$)13 b Ff(^)g Fg(r)q(esul)q(t)739 1504 y Fe(post)815 1498 y Fn(=)26 b Ff(hfg)p Fg(;)7 b Fn(100)p Ff(i)0 1597 y Fh(b)n(ag)p 60 1597 V 16 w(cr)n(e)n(ate)p 182 1597 V 14 w(single)14 b Fn(=)g Fk(pro)q(c)g Fn($)p Fh(i:)k Fn(in)o(t$)13 b Fk(returns)f Fn($bag$)178 1647 y Fk(ensures)25 b(new)p Fh($r)n(esult$)13 b Ff(^)g Fg(r)q(esul)q(t)739 1653 y Fe(post)815 1647 y Fn(=)26 b Ff(hf)p Fg(i)p Ff(g)p Fg(;)7 b Fn(100)p Ff(i)603 1737 y Fn(Figure)14 b(2:)j(Creator)e(Sp)q(eci\014cations)g(for)e(Bags)p 0 1786 1950 1 v 0 1939 a Fc(4.3)56 b(T)n(yp)r(e)18 b(Sp)r (eci\014cations)f(Need)h(Explicit)e(In)n(v)m(arian)n(ts)0 2015 y Fn(By)11 b(not)g(including)f(creators)i(in)e(t)o(yp)q(e)i(sp)q (eci\014cations)g(and)e(b)o(y)h(allo)o(wing)e(subt)o(yp)q(es)j(to)f (extend)g(sup)q(ert)o(yp)q(es)j(with)c(m)o(utators)0 2065 y(w)o(e)i(lose)g(a)g(p)q(o)o(w)o(erful)f(reasoning)h(to)q(ol:)17 b(data)11 b(t)o(yp)q(e)i(induction.)k(Data)11 b(t)o(yp)q(e)h(induction) g(is)g(used)g(to)g(pro)o(v)o(e)g(t)o(yp)q(e)h(in)o(v)n(arian)o(ts.)0 2115 y(The)j(base)h(case)g(of)e(the)h(rule)g(requires)h(that)f(eac)o(h) h(creator)f(of)g(the)g(t)o(yp)q(e)g(establish)h(the)f(in)o(v)n(arian)o (t;)f(the)h(inductiv)o(e)g(case)0 2165 y(requires)c(that)f(eac)o(h)h (metho)q(d)e($in)h(particular)f(eac)o(h)i(m)o(utator$)e(preserv)o(e)j (the)e(in)o(v)n(arian)o(t.)16 b(Without)10 b(the)h(creators,)i(w)o(e)e (ha)o(v)o(e)0 2215 y(no)k(base)h(case.)24 b(Without)15 b(kno)o(wing)f(all)g(m)o(utators)g(of)h(t)o(yp)q(e)h Fg(\034)k Fn($as)c(added)g(b)o(y)f Fg(\034)5 b Fn('s)15 b(subt)o(yp)q(es$,)i(w)o(e)f(ha)o(v)o(e)f(an)g(incomplete)0 2264 y(inductiv)o(e)f(case.)19 b(With)13 b(no)h(data)f(t)o(yp)q(e)i (induction)e(rule,)h(w)o(e)g(cannot)g(pro)o(v)o(e)g(t)o(yp)q(e)g(in)o (v)n(arian)o(ts!)62 2314 y(T)m(o)i(comp)q(ensate)g(for)g(the)h(lac)o(k) f(of)g(a)g(data)g(t)o(yp)q(e)g(induction)g(rule,)h(w)o(e)g(state)g(the) g(in)o(v)n(arian)o(t)d(explicitly)i(in)g(the)g(t)o(yp)q(e)0 2364 y(sp)q(eci\014cation)e(through)g(an)f Fk(in)o(v)m(arian)o(t)d Fn(clause;)k(if)e(the)i(in)o(v)n(arian)o(t)e(is)h(trivial)f($i.e.,)g (iden)o(tical)h(to)g(\\true"$,)h(the)g(clause)g(can)0 2414 y(b)q(e)h(omitted.)h(The)e(in)o(v)n(arian)o(t)f(de\014nes)i(the)g Fh(le)n(gal)e Fn(v)n(alues)g(of)h(its)g(t)o(yp)q(e)g Fg(\034)5 b Fn(.)18 b(F)m(or)13 b(example,)f(w)o(e)j(include)104 2492 y Fk(in)o(v)m(arian)o(t)c Ff(j)i Fg(b)345 2498 y Fe(\032)364 2492 y Fg(:el)q(ems)i Ff(j)c(\024)h Fg(b)582 2498 y Fe(\032)601 2492 y Fg(:bound)0 2570 y Fn(in)j(the)h(t)o(yp)q(e)g (sp)q(eci\014cation)g(of)e(Figure)i(1)f(to)g(state)h(that)g(the)f(size) i(of)d(a)h(b)q(ounded)h(bag)f(nev)o(er)h(exceeds)i(its)d(b)q(ound.)23 b(The)0 2620 y(predicate)15 b Fg(\036)p Fn($)p Fg(x)245 2626 y Fe(\032)264 2620 y Fn($)e(app)q(earing)g(in)g(an)h Fk(in)o(v)m(arian)o(t)c Fn(clause)k(for)g(t)o(yp)q(e)g Fg(\034)k Fn(stands)c(for)f(the)h(predicate:)19 b(F)m(or)14 b(all)e(computations,)0 2670 y Fg(c)p Fn(,)h(and)h(all)f(states)i Fg(\032)f Fn(in)g Fg(c)p Fn(,)965 2795 y(6)p eop %%Page: 7 9 7 8 bop 104 42 a Ff(8)p Fg(x)11 b Fn(:)g Fg(\034)19 b(:)13 b(x)e Ff(2)h Fg(dom)p Fn($)p Fg(\032)p Fn($)g Ff(\))f Fg(\036)p Fn($)p Fg(x)583 48 y Fe(\032)602 42 y Fn($)0 125 y(An)o(y)16 b(additional)e(in)o(v)n(arian)o(t)h(prop)q(ert)o(y)i(m) o(ust)e(follo)o(w)f(from)h(the)i(conjunction)f(of)f(the)i(t)o(yp)q(e's) g(in)o(v)n(arian)o(t)d(and)i(in)o(v)n(arian)o(ts)0 174 y(that)11 b(hold)g(for)f(the)i(en)o(tire)g(v)n(alue)e(space.)18 b(F)m(or)11 b(example,)f(w)o(e)h(could)g(sho)o(w)g(that)g(the)h(size)g (of)e(a)h(bag)g(is)f(nonnegativ)o(e)h(b)q(ecause)0 224 y(this)j(is)g(true)g(for)g(all)f(mathematical)d(m)o(ultiset)j(v)n (alues.)62 274 y(As)h(part)g(of)f(sp)q(ecifying)g(a)h(t)o(yp)q(e)g(and) f(its)h(creators)h(w)o(e)e(m)o(ust)g(sho)o(w)g(that)h(the)g(in)o(v)n (arian)o(t)e(holds)h(for)g(all)g(ob)r(jects)h(of)f(the)0 324 y(t)o(yp)q(e.)19 b(All)13 b(creators)i(for)e(a)h(t)o(yp)q(e)g Fg(\034)19 b Fn(m)o(ust)13 b Fh(establish)g Fg(\034)5 b Fn('s)14 b Fh(invariant)p Fn(,)f Fg(I)1104 330 y Fe(\034)1125 324 y Fn(:)104 407 y(F)m(or)g(eac)o(h)i(creator)g(for)e(t)o(yp)q(e)i Fg(\034)5 b Fn(,)13 b(sho)o(w)h(for)f(all)g Fg(x)5 b Fn(:)g Fg(\034)17 b Fn(that)d Fg(I)1031 413 y Fe(\034)1052 407 y Fn([)p Fg(r)q(esul)q(t)1174 413 y Fe(post)1238 407 y Fg(=x)1283 413 y Fe(\032)1302 407 y Fn(].)0 490 y(where)g Fg(P)6 b Fn([)p Fg(a=b)p Fn(])k(stands)j(for)f(predicate)i Fg(P)j Fn(with)12 b(ev)o(ery)i(o)q(ccurrence)h(of)d Fg(b)g Fn(replaced)h(b)o(y)f Fg(a)p Fn(.)18 b(Similarly)l(,)9 b(eac)o(h)k(pro)q(ducer)h(m)o(ust)0 540 y(establish)g(the)h(in)o(v)n (arian)o(t)d(on)h(its)h(newly-created)h(ob)r(ject.)k(In)14 b(addition,)e(eac)o(h)i(m)o(utator)f(of)g(the)h(t)o(yp)q(e)h(m)o(ust)e Fh(pr)n(eserve)h(the)0 589 y(invariant)p Fn(.)24 b(T)m(o)15 b(pro)o(v)o(e)g(this,)h(w)o(e)g(assume)g(eac)o(h)g(m)o(utator)e(is)i (called)f(on)h(an)f(ob)r(ject)i(of)e(t)o(yp)q(e)h Fg(\034)21 b Fn(with)15 b(a)h(legal)e(v)n(alue)i($one)0 639 y(that)e(satis\014es) h(the)f(in)o(v)n(arian)o(t$,)e(and)i(sho)o(w)g(that)g(an)o(y)f(v)n (alue)g(of)h(a)f Fg(\034)19 b Fn(ob)r(ject)c(it)e(mo)q(di\014es)g(is)h (legal:)104 722 y(F)m(or)f(eac)o(h)i(m)o(utator)d Fg(m)i Fn(of)g Fg(\034)5 b Fn(,)13 b(for)g(all)g Fg(x)5 b Fn(:)g Fg(\034)17 b Fn(assume)d Fg(I)942 728 y Fe(\034)963 722 y Fn([)p Fg(x)999 728 y Fe(pr)q(e)1049 722 y Fg(=x)1094 728 y Fe(\032)1113 722 y Fn(])f(and)h(sho)o(w)g Fg(I)1340 728 y Fe(\034)1361 722 y Fn([)p Fg(x)1397 728 y Fe(post)1460 722 y Fg(=x)1505 728 y Fe(\032)1524 722 y Fn(].)62 805 y(F)m(or)j(example,)f(w)o(e)h(w)o(ould)f(need)i(to)f(sho)o(w)g(that)g (the)h(three)g(creators)g(for)f(bag)g(establish)g(the)g(in)o(v)n(arian) o(t,)f(and)h(that)0 855 y Fh(put)i Fn(and)g Fh(get)g Fn(preserv)o(e)h(the)g(in)o(v)n(arian)o(t)d(for)i(bag.)32 b($W)m(e)19 b(can)g(ignore)f Fh(c)n(ar)n(d)h Fn(and)f Fh(e)n(qual)h Fn(b)q(ecause)i(they)e(are)h(observ)o(ers.$)0 905 y(Informally)14 b(the)j(in)o(v)n(arian)o(t)f(holds)g(b)q(ecause)j (eac)o(h)e(creator)h(guaran)o(tees)g(that)f(the)g(size)h(is)f(no)f (larger)h(than)g(the)g(b)q(ound;)0 955 y Fh(put)p Fn('s)12 b(pre-condition)g(c)o(hec)o(ks)i(that)e(there)h(is)f(enough)h(ro)q(om)d (in)i(the)h(bag)e(for)h(another)h(elemen)o(t;)e(and)h Fh(get)g Fn(either)h(decreases)0 1005 y(the)h(size)h(of)e(the)i(bag)e (or)h(lea)o(v)o(es)g(it)g(the)g(same.)62 1054 y(The)d(loss)f(of)g(data) g(t)o(yp)q(e)h(induction)f(means)g(that)g(additional)f(in)o(v)n(arian)o (ts)g(cannot)i(b)q(e)g(pro)o(v)o(ed.)17 b(Therefore)11 b(the)g(sp)q(eci\014er)0 1104 y(m)o(ust)i(b)q(e)h(careful)g(to)g (de\014ne)h(an)f(in)o(v)n(arian)o(t)e(that)i(is)g(strong)g(enough)g (that)g(all)e(desired)j(in)o(v)n(arian)o(ts)e(follo)o(w)f(from)g(it.)0 1220 y Fc(4.4)56 b(T)n(yp)r(e)18 b(Sp)r(eci\014cations)f(Need)h (Explicit)e(Constrain)n(ts)0 1297 y Fn(W)m(e)j(are)g(in)o(terested)i (in)d(the)i(history)f(prop)q(erties)h(of)f(ob)r(jects)h(in)f(addition)f (to)g(their)i(in)o(v)n(arian)o(t)d(prop)q(erties.)35 b(W)m(e)19 b(can)0 1347 y(form)o(ulate)12 b(history)i(prop)q(erties)h (as)f(predicates)h(o)o(v)o(er)f(state)h(pairs,)e(and)h(pro)o(v)o(e)g (them)f(using)h(the)g Fh(history)h(rule)p Fn(:)104 1430 y Fh(History)f(R)o(ule)p Fn(:)k(F)m(or)c(eac)o(h)g(of)f(the)i Fg(i)f Fn(m)o(utators)f Fg(m)h Fn(of)f Fg(\034)5 b Fn(,)13 b(for)h(all)f Fg(x)e Fn(:)g Fg(\034)5 b Fn(:)609 1521 y Fg(m)645 1527 y Fe(i)659 1521 y Fg(:pr)q(e)k Ff(^)g Fg(m)813 1527 y Fe(i)827 1521 y Fg(:post)j Ff(\))f Fg(\036)p Fn([)p Fg(x)1040 1527 y Fe(pr)q(e)1090 1521 y Fg(=x)1135 1527 y Fe(\032)1154 1521 y Fg(;)c(x)1197 1527 y Fe(post)1260 1521 y Fg(=x)1305 1527 y Fe( )1329 1521 y Fn(])p 609 1540 732 2 v 891 1577 a Fg(\036)p Fn($)p Fg(x)956 1583 y Fe(\032)975 1577 y Fg(;)g(x)1018 1583 y Fe( )1043 1577 y Fn($)0 1668 y(W)m(e)18 b(cannot)g(use)g(this)g(history)g(rule)g (directly)m(,)h(ho)o(w)o(ev)o(er.)30 b(It)18 b(is)g(incomplete)f(since) i(subt)o(yp)q(es)g(ma)o(y)d(de\014ne)j(additional)0 1718 y(m)o(utators.)d(If)c(w)o(e)h(use)g(it)f(without)f(considering)i(the)g (extra)f(m)o(utators,)f(it)h(is)g(easy)h(to)f(pro)o(v)o(e)g(prop)q (erties)i(that)e(do)g(not)h(hold)0 1768 y(for)h(subt)o(yp)q(e)g(ob)r (jects!)62 1817 y(T)m(o)h(comp)q(ensate)h(for)g(the)g(lac)o(k)g(of)f (the)h(history)g(rule,)g(w)o(e)g(state)h(history)f(prop)q(erties)h (explicitly)e(in)g(the)i(t)o(yp)q(e)f(sp)q(eci-)0 1867 y(\014cation)f(through)g(a)g Fk(constrain)o(t)d Fn(clause)675 1852 y Fi(2)694 1867 y Fn(;)j(if)f(the)i(constrain)o(t)f(is)g(trivial,) f(the)h(clause)h(can)f(b)q(e)h(omitted.)k(F)m(or)15 b(example,)0 1917 y(the)f(constrain)o(t)104 2008 y Fk(constrain)o(t)d Fg(b)345 2014 y Fe(\032)364 2008 y Fg(:bound)f Fn(=)i Fg(b)557 2014 y Fe( )582 2008 y Fg(:bound)0 2100 y Fn(in)g(the)g(sp)q (eci\014cation)h(of)f(bag)f(declares)j(that)e(a)g(bag's)f(b)q(ound)i (nev)o(er)g(c)o(hanges.)18 b(As)12 b(another)h(example,)e(consider)i(a) f(fat)p 1887 2100 13 2 v 14 w(set)0 2149 y(ob)r(ject)j(that)g(has)f(an) h Fh(insert)e Fn(but)i(no)f Fh(delete)g Fn(metho)q(d;)g(fat)p 935 2149 V 14 w(sets)i(only)e(gro)o(w)g(in)g(size.)20 b(The)15 b(constrain)o(t)g(for)f(fat)p 1765 2149 V 14 w(set)i(w)o(ould)0 2199 y(b)q(e:)104 2282 y Fk(constrain)o(t)11 b Ff(8)i Fg(i)f Fn(:)f Fg(int)j(:)g(i)e Ff(2)f Fg(s)590 2288 y Fe(\032)621 2282 y Ff(\))g Fg(i)h Ff(2)f Fg(s)758 2288 y Fe( )62 2365 y Fn(The)17 b(predicate)h Fg(\036)p Fn($)p Fg(x)398 2371 y Fe(\032)417 2365 y Fg(;)7 b(x)460 2371 y Fe( )484 2365 y Fn($)17 b(app)q(earing)g(in)f(a)g Fk(constrain)o(t)e Fn(clause)j(for)f(t)o(yp)q(e)i Fg(\034)j Fn(stands)d(for)e(the)h(predicate:)25 b(F)m(or)16 b(all)0 2415 y(computations,)c Fg(c)p Fn(,)h(and)h(all)f(states)i Fg(\032)f Fn(and)g Fg( )h Fn(in)f Fg(c)f Fn(suc)o(h)i(that)f Fg(\032)g Fn(precedes)j Fg( )q Fn(,)104 2498 y Ff(8)p Fg(x)11 b Fn(:)g Fg(\034)19 b(:)13 b(x)e Ff(2)h Fg(dom)p Fn($)p Fg(\032)p Fn($)g Ff(\))f Fg(\036)p Fn($)p Fg(x)583 2504 y Fe(\032)602 2498 y Fg(;)c(x)645 2504 y Fe( )669 2498 y Fn($)p 0 2537 780 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Fn($in)o(t$)225 1602 y Fk(ensures)25 b Fg(r)q(esul)q(t)12 b Fn(=)g Fg(l)q(eng)q(th)p Fn($)p Fg(s)724 1608 y Fe(pr)q(e)776 1602 y Fg(:items)p Fn($)104 1702 y Fh(e)n(qual)i Fn(=)g Fk(pro)q(c)f Fn($)p Fh(t:)18 b Fn(bag$)c Fk(returns)d Fn($b)q(o)q(ol$)225 1752 y Fk(ensures)25 b Fg(r)q(esul)q(t)12 b Fn(=)g($)p Fg(s)g Fn(=)g Fg(t)p Fn($)166 1851 y Fk(subt)o(yp)q(e)g (of)h Fn(bag)h($)p Fh(push)g Fk(for)f Fh(put)p Fn(,)h Fh(p)n(op)g Fk(for)f Fh(get)p Fn(,)h Fh(height)g Fk(for)e Fh(c)n(ar)n(d)p Fn($)344 1901 y Ff(8)p Fg(st)g Fn(:)f Fg(S)17 b(:)c(A)p Fn($)p Fg(st)p Fn($)f(=)g Ff(h)p Fg(mk)p 734 1901 V 17 w(el)q(ems)p Fn($)p Fg(st:items)p Fn($)p Fg(;)7 b(st:l)q(im)q(it)p Ff(i)396 1951 y Fk(where)13 b Fg(mk)p 596 1951 V 17 w(el)q(ems)g Fn(:)e Fg(S)r(eq)j Ff(!)d Fg(M)475 2001 y Ff(8)p Fg(i)h Fn(:)f Fg(I)s(nt;)c(sq)12 b Fn(:)g Fg(S)r(eq)586 2051 y(mk)p 648 2051 V 17 w(el)q(ems)p Fn($[)j(]$)c(=)h Ff(f)h(g)586 2100 y Fg(mk)p 648 2100 V 17 w(el)q(ems)p Fn($)p Fg(sq)k Ff(jj)c Fn([)g Fg(i)h Fn(]$)d(=)h Fg(mk)p 1073 2100 V 17 w(el)q(ems)p Fn($)p Fg(sq)q Fn($)f Ff([)e(f)p Fg(i)p Ff(g)166 2200 y Fk(end)k Fn(stac)o(k)781 2274 y(Figure)h(3:)k(Stac)o(k)c(T)o(yp)q(e)p 0 2324 1950 1 v 965 2795 a(9)p eop %%Page: 10 12 10 11 bop 0 1 1950 1 v 0 72 a Fq(Definition)18 b(of)g(the)g(subtype)g (rela)m(tion,)i Ff(\026)p Fn(:)i Fg(\033)31 b Fn(=)g Ff(h)p Fg(O)1021 78 y Fe(\033)1043 72 y Fg(;)7 b(S;)g(M)e Ff(i)15 b Fn(is)h(a)f Fh(subtyp)n(e)h Fn(of)f Fg(\034)35 b Fn(=)c Ff(h)p Fg(O)1621 78 y Fe(\034)1641 72 y Fg(;)7 b(T)s(;)g(N)e Ff(i)16 b Fn(if)0 121 y(there)f(exists)g(an)e (abstraction)i(function,)e Fg(A)e Fn(:)g Fg(S)k Ff(!)c Fg(T)6 b Fn(,)13 b(and)h(a)g(renaming)e(map,)g Fg(R)f Fn(:)g Fg(M)16 b Ff(!)11 b Fg(N)5 b Fn(,)14 b(suc)o(h)g(that:)51 213 y(1.)20 b(Subt)o(yp)q(e)15 b(metho)q(ds)g(preserv)o(e)h(the)g(sup)q (ert)o(yp)q(e)g(metho)q(ds')e(b)q(eha)o(vior.)21 b(If)14 b Fg(m)1310 219 y Fe(\034)1346 213 y Fn(of)g Fg(\034)20 b Fn(is)15 b(the)g(corresp)q(onding)104 262 y(renamed)e(metho)q(d)g Fg(m)459 268 y Fe(\033)496 262 y Fn(of)g Fg(\033)q Fn(,)h(the)g(follo)o (wing)e(rules)i(m)o(ust)f(hold:)154 346 y Ff(\017)20 b Fh(Signatur)n(e)15 b(rule.)228 412 y Fk({)21 b Fh(Contr)n(avarianc)n (e)16 b(of)h(ar)n(guments.)24 b Fg(m)874 418 y Fe(\034)911 412 y Fn(and)16 b Fg(m)1030 418 y Fe(\033)1068 412 y Fn(ha)o(v)o(e)g(the)h(same)e(n)o(um)o(b)q(er)g(of)g(argumen)o(ts.)23 b(If)273 462 y(the)14 b(list)g(of)f(argumen)o(t)g(t)o(yp)q(es)h(of)g Fg(m)840 468 y Fe(\034)875 462 y Fn(is)f Fg(\013)943 468 y Fe(i)971 462 y Fn(and)g(that)h(of)g Fg(m)1225 468 y Fe(\033)1261 462 y Fn(is)g Fg(\014)1326 468 y Fe(i)1340 462 y Fn(,)g(then)g Ff(8)p Fg(i)g(:)g(\013)1564 468 y Fe(i)1589 462 y Ff(\026)e Fg(\014)1656 468 y Fe(i)1670 462 y Fn(.)228 520 y Fk({)21 b Fh(Covarianc)n(e)16 b(of)f(r)n(esult.)21 b Fn(Either)16 b(b)q(oth)f Fg(m)937 526 y Fe(\034)973 520 y Fn(and)g Fg(m)1091 526 y Fe(\033)1128 520 y Fn(ha)o(v)o(e)g(a)g (result)h(or)f(neither)h(has.)21 b(If)15 b(there)273 570 y(is)e(a)h(result,)g(let)g Fg(m)572 576 y Fe(\034)593 570 y Fn('s)g(result)h(t)o(yp)q(e)f(b)q(e)h Fg(\013)e Fn(and)h Fg(m)1058 576 y Fe(\033)1081 570 y Fn('s)f(b)q(e)i Fg(\014)r Fn(.)k(Then)14 b Fg(\014)g Ff(\026)e Fg(\013)p Fn(.)228 628 y Fk({)21 b Fh(Exc)n(eption)14 b(rule.)j Fn(The)c(exceptions)g(signaled)f(b)o(y)g Fg(m)1096 634 y Fe(\033)1132 628 y Fn(are)g(con)o(tained)h(in)f(the)h(set)g(of)f (exceptions)273 678 y(signaled)h(b)o(y)h Fg(m)526 684 y Fe(\034)547 678 y Fn(.)154 744 y Ff(\017)20 b Fh(Metho)n(ds)c(rule.)h Fn(F)m(or)d(all)e Fg(x)g Fn(:)f Fg(\033)q Fn(:)228 810 y Fk({)21 b Fh(Pr)n(e-c)n(ondition)15 b(rule.)i Fg(m)668 816 y Fe(\034)689 810 y Fg(:pr)q(e)p Fn([)p Fg(A)p Fn($)p Fg(x)844 816 y Fe(pr)q(e)895 810 y Fn($)p Fg(=x)956 816 y Fe(pr)q(e)1007 810 y Fn(])11 b Ff(\))g Fg(m)1119 816 y Fe(\033)1142 810 y Fg(:pr)q(e:)228 869 y Fk({)21 b Fh(Post-c)n(ondition)15 b(rule.)j Fg(m)686 875 y Fe(\033)709 869 y Fg(:post)11 b Ff(\))g Fg(m)896 875 y Fe(\034)917 869 y Fg(:post)p Fn([)p Fg(A)p Fn($)p Fg(x)1087 875 y Fe(pr)q(e)1138 869 y Fn($)p Fg(=x)1199 875 y Fe(pr)q(e)1250 869 y Fg(;)c(A)p Fn($)p Fg(x)1340 875 y Fe(post)1403 869 y Fn($)p Fg(=x)1464 875 y Fe(post)1528 869 y Fn(])51 952 y(2.)20 b(Subt)o(yp)q(es)c(preserv)o(e)i(sup)q(ert)o(yp)q(e)f(prop) q(erties.)24 b(F)m(or)15 b(all)f(computations,)f Fg(c)p Fn(,)j(and)f(all)f(states)i Fg(\032)g Fn(and)f Fg( )i Fn(in)e Fg(c)104 1001 y Fn(suc)o(h)f(that)g Fg(\032)h Fn(precedes)h Fg( )q Fn(,)e(for)f(all)g Fg(x)e Fn(:)g Fg(\033)q Fn(:)154 1084 y Ff(\017)20 b Fh(Invariant)15 b(R)o(ule)p Fn(.)j(Subt)o(yp)q(e)d(in)o(v)n(arian)o(ts)e(ensure)i(sup)q (ert)o(yp)q(e)h(in)o(v)n(arian)o(ts.)273 1151 y Fg(I)291 1157 y Fe(\033)325 1151 y Ff(\))11 b Fg(I)396 1157 y Fe(\034)417 1151 y Fn([)p Fg(A)p Fn($)p Fg(x)500 1157 y Fe(\032)519 1151 y Fn($)p Fg(=x)580 1157 y Fe(\032)599 1151 y Fn(])154 1217 y Ff(\017)20 b Fh(Constr)n(aint)14 b(R)o(ule)p Fn(.)19 b(Subt)o(yp)q(e)14 b(constrain)o(ts)h(ensure)g(sup) q(ert)o(yp)q(e)h(constrain)o(ts.)273 1284 y Fg(C)303 1290 y Fe(\033)336 1284 y Ff(\))11 b Fg(C)419 1290 y Fe(\034)440 1284 y Fn([)p Fg(A)p Fn($)p Fg(x)523 1290 y Fe(\032)542 1284 y Fn($)p Fg(=x)603 1290 y Fe(\032)621 1284 y Fg(;)c(A)p Fn($)p Fg(x)711 1290 y Fe( )736 1284 y Fn($)p Fg(=x)797 1290 y Fe( )822 1284 y Fn(])570 1370 y(Figure)14 b(4:)k(De\014nition)13 b(of)h(the)g(Subt)o(yp)q(e)h (Relation)p 0 1420 V 0 1535 a(is)d(completely)f(analogous,)g(using)i (the)g(constrain)o(t)f(rule)h(and)f(the)h(assumption)e(that)i(the)g(t)o (yp)q(e)f(sp)q(eci\014cation)i(satis\014es)f(its)0 1585 y(constrain)o(t.)62 1635 y(W)m(e)19 b(do)f(not)h(include)g(the)h(in)o (v)n(arian)o(t)d(in)i(the)g(metho)q(ds)f($or)h(constrain)o(t$)h(rule) f(directly)m(.)33 b(F)m(or)18 b(example,)h(the)g(pre-)0 1685 y(condition)13 b(rule)h(could)g(ha)o(v)o(e)g(b)q(een)104 1759 y($)p Fg(m)156 1765 y Fe(\034)177 1759 y Fg(:pr)q(e)p Fn([)p Fg(A)p Fn(\()p Fg(x)332 1765 y Fe(pr)q(e)383 1759 y Fn($)p Fg(=x)444 1765 y Fe(pr)q(e)494 1759 y Fn(])9 b Ff(^)g Fg(I)570 1765 y Fe(\034)591 1759 y Fn([)p Fg(A)p Fn($)p Fg(x)674 1765 y Fe(pr)q(e)725 1759 y Fn($)p Fg(=x)786 1765 y Fe(pr)q(e)837 1759 y Fn(]\))i Ff(\))g Fg(m)965 1765 y Fe(\033)988 1759 y Fg(:pr)q(e)0 1833 y Fn(W)m(e)k(omit)e(adding)h(the)i(in)o(v)n(arian)o(t)d(b)q(ecause)k (if)d(it)h(is)g(needed)i(in)d(doing)g(a)h(pro)q(of)g(it)g(can)g(alw)o (a)o(ys)f(b)q(e)i(assumed,)e(since)i(it)f(is)0 1883 y(kno)o(wn)e(to)h (b)q(e)h(true)f(for)g(all)f(ob)r(jects)i(of)e(its)h(t)o(yp)q(e.)62 1933 y(Note)h(that)f(in)f(the)i(v)n(arious)e(rules)i(w)o(e)f(require)h Fg(x)d Fn(:)f Fg(\033)q Fn(,)j(y)o(et)g(x)g(app)q(ears)h(in)f (predicates)h(concerning)g Fg(\034)k Fn(ob)r(jects)c(as)f(w)o(ell.)0 1982 y(This)g(mak)o(es)f(sense)i(b)q(ecause)h Fg(\033)c Ff(\026)g Fg(\034)5 b Fn(.)0 2097 y Fc(5.3)56 b(Applying)18 b(the)g(De\014nition)f(of)i(Subt)n(yping)g(as)g(a)g(Chec)n(klist)0 2173 y Fn(Pro)q(ofs)11 b(of)e(the)j(subt)o(yp)q(e)f(relation)f(are)h (usually)e(ob)o(vious)h(and)g(can)h(b)q(e)g(done)g(b)o(y)f(insp)q (ection.)18 b(T)o(ypically)m(,)8 b(the)j(only)f(in)o(terest-)0 2223 y(ing)h(part)h(is)g(the)h(de\014nition)e(of)h(the)g(abstraction)g (function;)g(the)h(other)f(parts)h(of)e(the)i(pro)q(of)e(are)i(usually) e(straigh)o(tforw)o(ard.)0 2273 y(Ho)o(w)o(ev)o(er,)18 b(this)f(section)g(go)q(es)h(through)f(the)g(steps)i(of)d(an)h (informal)d(pro)q(of)j(just)g(to)g(sho)o(w)f(what)h(kind)g(of)f (reasoning)h(is)0 2323 y(in)o(v)o(olv)o(ed.)g(F)m(ormal)11 b(v)o(ersions)j(of)g(these)h(informal)c(pro)q(ofs)j(are)g(giv)o(en)g (in)f([L)-5 b(W92)o(].)62 2372 y(Let's)15 b(revisit)f(the)g(stac)o(k)g (and)g(bag)g(example)e(using)i(our)g(de\014nition)f(as)h(a)g(c)o(hec)o (klist.)k(Here)104 2447 y Fg(\033)13 b Fn(=)25 b Ff(h)p Fg(O)246 2453 y Fe(stack)327 2447 y Fg(;)7 b(S;)g Ff(f)p Fg(push;)g(pop;)g(sw)q(ap)p 694 2447 13 2 v 14 w(top;)g(heig)q(ht;)g (eq)q(ual)q Ff(gi)104 2496 y Fg(\034)16 b Fn(=)26 b Ff(h)p Fg(O)244 2502 y Fe(bag)296 2496 y Fg(;)7 b(B)r(;)13 b Ff(f)p Fg(put;)g(g)q(et;)h(car)q(d;)7 b(eq)q(ual)q Ff(gi)0 2570 y Fn(Recall)k(that)g(w)o(e)h(represen)o(t)i(a)d(b)q(ounded)i (bag's)e(v)n(alue)g(as)g(a)h(pair,)f Ff(h)p Fg(el)q(ems;)c(bound)p Ff(i)p Fn(,)12 b(of)f(a)g(m)o(ultiset)f(of)h(in)o(tegers)i(and)e(a)h (\014xed)0 2620 y(b)q(ound,)h(and)f(a)h(b)q(ounded)g(stac)o(k's)h(v)n (alue)e(as)h(a)f(pair,)h Ff(h)p Fg(items;)7 b(l)q(imit)p Ff(i)p Fn(,)15 b(of)d(a)h(sequence)i(of)d(in)o(tegers)h(and)g(a)g (\014xed)g(b)q(ound.)18 b(It)0 2670 y(can)c(easily)f(b)q(e)i(sho)o(wn)f (that)g(eac)o(h)g(sp)q(eci\014cation)h(preserv)o(es)h(its)e(in)o(v)n (arian)o(t)e(and)i(satis\014es)h(its)f(constrain)o(t.)954 2795 y(10)p eop %%Page: 11 13 11 12 bop 62 42 a Fn(W)m(e)15 b(use)h(the)g(abstraction)f(function)g (and)f(the)i(renaming)e(map)f(giv)o(en)i(in)f(the)i(sp)q(eci\014cation) g(for)e(stac)o(k)i(in)f(Figure)g(3.)0 91 y(The)f(abstraction)g (function)g(states)h(that)f(for)g(all)e Fg(st)g Fn(:)f Fg(S)104 174 y(A)p Fn($)p Fg(st)p Fn($)h(=)26 b Ff(h)p Fg(mk)p 349 174 13 2 v 16 w(el)q(ems)p Fn($)p Fg(st:items)q 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2670 y(age,)i(and)g(p)q(ossibly)h(its)f(relationship)g(to)g(other)i(p)q (ersons)f($e.g.,)f(its)h(paren)o(ts)g(or)f(c)o(hildren$.)21 b(Studen)o(t)15 b(and)f(emplo)o(y)o(ee)g(are)954 2795 y(12)p eop %%Page: 13 15 13 14 bop 0 1 1950 1 v 608 245 a @beginspecial 23 @llx 652 @lly 199 @urx 701 @ury 1760 @rwi @setspecial %%BeginDocument: fig7.ps 50 dict begin /arrowHeight 8 def /arrowWidth 4 def /none null def /numGraphicParameters 17 def /stringLimit 65535 def /Begin { save numGraphicParameters dict begin } def /End { end restore } def /SetB { dup type /nulltype eq { pop false /brushRightArrow idef false /brushLeftArrow idef true /brushNone idef } { /brushDashOffset idef /brushDashArray idef 0 ne /brushRightArrow idef 0 ne /brushLeftArrow idef /brushWidth idef false /brushNone idef } ifelse } def /SetCFg { /fgblue idef /fggreen idef /fgred idef } def /SetCBg { /bgblue idef /bggreen idef /bgred idef } def /SetF { /printSize idef /printFont idef } def /SetP { dup type /nulltype eq { pop true /patternNone idef } { /patternGrayLevel idef patternGrayLevel -1 eq { /patternString idef } if false /patternNone idef } ifelse } def /BSpl { 0 begin storexyn newpath n 1 gt { 0 0 0 0 0 0 1 1 true subspline n 2 gt { 0 0 0 0 1 1 2 2 false subspline 1 1 n 3 sub { /i exch def i 1 sub dup i dup i 1 add dup i 2 add dup false subspline } for n 3 sub dup n 2 sub dup n 1 sub dup 2 copy false subspline } if n 2 sub dup n 1 sub dup 2 copy 2 copy false subspline patternNone not brushLeftArrow not brushRightArrow not and and { ifill } if brushNone not { istroke } if 0 0 1 1 leftarrow n 2 sub dup n 1 sub dup rightarrow } if end } dup 0 4 dict put def /Circ { newpath 0 360 arc patternNone not { ifill } if brushNone not { istroke } if } def /CBSpl { 0 begin dup 2 gt { storexyn newpath n 1 sub dup 0 0 1 1 2 2 true subspline 1 1 n 3 sub { /i exch def i 1 sub dup i dup i 1 add dup i 2 add dup false subspline } for n 3 sub dup n 2 sub dup n 1 sub dup 0 0 false subspline n 2 sub dup n 1 sub dup 0 0 1 1 false subspline patternNone not { ifill } if brushNone not { istroke } if } { Poly } ifelse end } dup 0 4 dict put def /Elli { 0 begin newpath 4 2 roll translate scale 0 0 1 0 360 arc patternNone not { ifill } if brushNone not { istroke } if end } dup 0 1 dict put def /Line { 0 begin 2 storexyn newpath x 0 get y 0 get moveto x 1 get y 1 get lineto brushNone not { istroke } if 0 0 1 1 leftarrow 0 0 1 1 rightarrow end } dup 0 4 dict put def /MLine { 0 begin storexyn newpath n 1 gt { x 0 get y 0 get moveto 1 1 n 1 sub { /i exch def x i get y i get lineto } for patternNone not brushLeftArrow not brushRightArrow not and and { ifill } if brushNone not { istroke } if 0 0 1 1 leftarrow n 2 sub dup n 1 sub dup rightarrow } if end } dup 0 4 dict put def /Poly { 3 1 roll newpath moveto -1 add { lineto } repeat closepath patternNone not { ifill } if brushNone not { istroke } if } def /Rect { 0 begin /t exch def /r exch def /b exch def /l exch def newpath l b moveto l t lineto r t lineto r b lineto closepath patternNone not { ifill } if brushNone not { istroke } if end } dup 0 4 dict put def /Text { ishow } def /idef { dup where { pop pop pop } { exch def } ifelse } def /ifill { 0 begin gsave patternGrayLevel -1 ne { fgred bgred fgred sub patternGrayLevel mul add fggreen bggreen fggreen sub patternGrayLevel mul add fgblue bgblue fgblue sub patternGrayLevel mul add setrgbcolor eofill } { eoclip originalCTM setmatrix pathbbox /t exch def /r exch def /b exch def /l exch def /w r l sub ceiling cvi def /h t b sub ceiling cvi def /imageByteWidth w 8 div ceiling cvi def /imageHeight h def bgred bggreen bgblue setrgbcolor eofill fgred fggreen fgblue setrgbcolor w 0 gt h 0 gt and { l b translate w h scale w h true [w 0 0 h neg 0 h] { patternproc } imagemask } if } ifelse grestore end } dup 0 8 dict put def /istroke { gsave brushDashOffset -1 eq { [] 0 setdash 1 setgray } { brushDashArray brushDashOffset setdash fgred fggreen fgblue setrgbcolor } ifelse brushWidth setlinewidth originalCTM setmatrix stroke grestore } def /ishow { 0 begin gsave fgred fggreen fgblue setrgbcolor /fontDict printFont findfont printSize scalefont dup setfont def /descender fontDict begin 0 [FontBBox] 1 get FontMatrix end transform exch pop def /vertoffset 0 descender sub printSize sub printFont /Courier ne printFont /Courier-Bold ne and { 1 add } if def { 0 vertoffset moveto show /vertoffset vertoffset printSize sub def } forall grestore end } dup 0 3 dict put def /patternproc { 0 begin /patternByteLength patternString length def /patternHeight patternByteLength 8 mul sqrt cvi def /patternWidth patternHeight def /patternByteWidth patternWidth 8 idiv def /imageByteMaxLength imageByteWidth imageHeight mul stringLimit patternByteWidth sub min def /imageMaxHeight imageByteMaxLength imageByteWidth idiv patternHeight idiv patternHeight mul patternHeight max def /imageHeight imageHeight imageMaxHeight sub store /imageString imageByteWidth imageMaxHeight mul patternByteWidth add string def 0 1 imageMaxHeight 1 sub { /y exch def /patternRow y patternByteWidth mul patternByteLength mod def /patternRowString patternString patternRow patternByteWidth getinterval def /imageRow y imageByteWidth mul def 0 patternByteWidth imageByteWidth 1 sub { /x exch def imageString imageRow x add patternRowString putinterval } for } for imageString end } dup 0 12 dict put def /min { dup 3 2 roll dup 4 3 roll lt { exch } if pop } def /max { dup 3 2 roll dup 4 3 roll gt { exch } if pop } def /arrowhead { 0 begin transform originalCTM itransform /taily exch def /tailx exch def transform originalCTM itransform /tipy exch def /tipx exch def /dy tipy taily sub def /dx tipx tailx sub def /angle dx 0 ne dy 0 ne or { dy dx atan } { 90 } ifelse def gsave originalCTM setmatrix tipx tipy translate angle rotate newpath 0 0 moveto arrowHeight neg arrowWidth 2 div lineto arrowHeight neg arrowWidth 2 div neg lineto closepath patternNone not { originalCTM setmatrix /padtip arrowHeight 2 exp 0.25 arrowWidth 2 exp mul add sqrt brushWidth mul arrowWidth div def /padtail brushWidth 2 div def tipx tipy translate angle rotate padtip 0 translate arrowHeight padtip add padtail add arrowHeight div dup scale arrowheadpath ifill } if brushNone not { originalCTM setmatrix tipx tipy translate angle rotate arrowheadpath istroke } if grestore end } dup 0 9 dict put def /arrowheadpath { newpath 0 0 moveto arrowHeight neg arrowWidth 2 div lineto arrowHeight neg arrowWidth 2 div neg lineto closepath } def /leftarrow { 0 begin y exch get /taily exch def x exch get /tailx exch def y exch get /tipy exch def x exch get /tipx exch def brushLeftArrow { tipx tipy tailx taily arrowhead } if end } dup 0 4 dict put def /rightarrow { 0 begin y exch get /tipy exch def x exch get /tipx exch def y exch get /taily exch def x exch get /tailx exch def brushRightArrow { tipx tipy tailx taily arrowhead } if end } dup 0 4 dict put def /midpoint { 0 begin /y1 exch def /x1 exch def /y0 exch def /x0 exch def x0 x1 add 2 div y0 y1 add 2 div end } dup 0 4 dict put def /thirdpoint { 0 begin /y1 exch def /x1 exch def /y0 exch def /x0 exch def x0 2 mul x1 add 3 div y0 2 mul y1 add 3 div end } dup 0 4 dict put def /subspline { 0 begin /movetoNeeded exch def y exch get /y3 exch def x exch get /x3 exch def y exch get /y2 exch def x exch get /x2 exch def y exch get /y1 exch def x exch get /x1 exch def y exch get /y0 exch def x exch get /x0 exch def x1 y1 x2 y2 thirdpoint /p1y exch def /p1x exch def x2 y2 x1 y1 thirdpoint /p2y exch def /p2x exch def x1 y1 x0 y0 thirdpoint p1x p1y midpoint /p0y exch def /p0x exch def x2 y2 x3 y3 thirdpoint p2x p2y midpoint /p3y exch def /p3x exch def movetoNeeded { p0x p0y moveto } if p1x p1y p2x p2y p3x p3y curveto end } dup 0 17 dict put def /storexyn { /n exch def /y n array def /x n array def n 1 sub -1 0 { /i exch def y i 3 2 roll put x i 3 2 roll put } for } def %I Idraw 7 Grid 8 Begin %I b u %I cfg u %I cbg u %I f u %I p u %I t [ 0.923077 0 0 0.923077 0 0 ] concat /originalCTM matrix currentmatrix def Begin %I Text %I cfg Black 0 0 0 SetCFg %I f -adobe-times-medium-r-normal--10-100-75-75-p-54-iso8859-1 /Times-Roman 10 SetF %I t [ 1 0 0 1 86 756 ] concat %I [ (immutable pair) ] Text End Begin %I Text %I cfg Black 0 0 0 SetCFg %I f -adobe-times-medium-r-normal--10-100-75-75-p-54-iso8859-1 /Times-Roman 10 SetF %I t [ 1 0 0 1 26 717 ] concat %I [ (immutable triple) ] Text End Begin %I Text %I cfg Black 0 0 0 SetCFg %I f -adobe-times-medium-r-normal--10-100-75-75-p-54-iso8859-1 /Times-Roman 10 SetF %I t [ 1 0 0 1 129 717 ] concat %I [ (semi-mutable triple) ] Text End Begin %I Line %I b 65535 1 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0 SetP %I t [ 1 0 0 1 -87 52 ] concat %I 199 691 165 673 Line End Begin %I Line %I b 65535 1 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0 SetP %I t [ 1 0 0 1 -87 52 ] concat %I 199 691 234 673 Line End End %I eop showpage end %%EndDocument @endspecial 729 336 a Fn(Figure)14 b(5:)k(P)o(airs)c(and)f(T)m(riples) p 0 386 V 0 560 a(subt)o(yp)q(es)19 b(of)d(p)q(erson;)k(in)d(eac)o(h)h (case)g(they)g(ha)o(v)o(e)f(additional)f(prop)q(erties,)j(e.g.,)e(a)g (studen)o(t)h(id)f(n)o(um)o(b)q(er,)g(an)g(emplo)o(y)o(ee)0 610 y(emplo)o(y)o(er)e(and)h(salary)m(.)24 b(In)16 b(addition,)f(t)o (yp)q(e)i(studen)o(t)p 867 610 13 2 v 16 w(emplo)o(y)o(ee)e(is)h(a)g (subt)o(yp)q(e)i(of)d(b)q(oth)i(studen)o(t)g(and)f(emplo)o(y)o(ee)f ($and)0 660 y(also)f(p)q(erson,)i(since)g(the)f(subt)o(yp)q(e)h (relation)e(is)h(transitiv)o(e$.)21 b(In)15 b(this)g(example,)e(the)j (subt)o(yp)q(e)g(ob)r(jects)g(ha)o(v)o(e)e(more)g(state)0 710 y(than)g(those)g(of)g(the)g(sup)q(ert)o(yp)q(e)i(as)e(w)o(ell)f(as) h(more)f(metho)q(ds.)p 0 768 1950 1 v 679 1224 a @beginspecial 35 @llx 468 @lly 177 @urx 568 @ury 1420 @rwi @setspecial %%BeginDocument: fig8.ps 50 dict begin /arrowHeight 8 def /arrowWidth 4 def /none null def /numGraphicParameters 17 def /stringLimit 65535 def /Begin { save numGraphicParameters dict begin } def /End { end restore } def /SetB { dup type /nulltype eq { pop false /brushRightArrow idef false /brushLeftArrow idef true /brushNone idef } { /brushDashOffset idef /brushDashArray idef 0 ne /brushRightArrow idef 0 ne /brushLeftArrow idef /brushWidth idef false /brushNone idef } ifelse } def /SetCFg { /fgblue idef /fggreen idef /fgred idef } def /SetCBg { /bgblue idef /bggreen idef /bgred idef } def /SetF { /printSize idef /printFont idef } def /SetP { dup type /nulltype eq { pop true /patternNone idef } { /patternGrayLevel idef patternGrayLevel -1 eq { /patternString idef } if false /patternNone idef } ifelse } def /BSpl { 0 begin storexyn newpath n 1 gt { 0 0 0 0 0 0 1 1 true subspline n 2 gt { 0 0 0 0 1 1 2 2 false subspline 1 1 n 3 sub { /i exch def i 1 sub dup i dup i 1 add dup i 2 add dup false subspline } for n 3 sub dup n 2 sub dup n 1 sub dup 2 copy false subspline } if n 2 sub dup n 1 sub dup 2 copy 2 copy false subspline patternNone not brushLeftArrow not brushRightArrow not and and { ifill } if brushNone not { istroke } if 0 0 1 1 leftarrow n 2 sub dup n 1 sub dup rightarrow } if end } dup 0 4 dict put def /Circ { newpath 0 360 arc patternNone not { ifill } if brushNone not { istroke } if } def /CBSpl { 0 begin dup 2 gt { storexyn newpath n 1 sub dup 0 0 1 1 2 2 true subspline 1 1 n 3 sub { /i exch def i 1 sub dup i dup i 1 add dup i 2 add dup false subspline } for n 3 sub dup n 2 sub dup n 1 sub dup 0 0 false subspline n 2 sub dup n 1 sub dup 0 0 1 1 false subspline patternNone not { ifill } if brushNone not { istroke } if } { Poly } ifelse end } dup 0 4 dict put def /Elli { 0 begin newpath 4 2 roll translate scale 0 0 1 0 360 arc patternNone not { ifill } if brushNone not { istroke } if end } dup 0 1 dict put def /Line { 0 begin 2 storexyn newpath x 0 get y 0 get moveto x 1 get y 1 get lineto brushNone not { istroke } if 0 0 1 1 leftarrow 0 0 1 1 rightarrow end } dup 0 4 dict put def /MLine { 0 begin storexyn newpath n 1 gt { x 0 get y 0 get moveto 1 1 n 1 sub { /i exch def x i get y i get lineto } for patternNone not brushLeftArrow not brushRightArrow not and and { ifill } if brushNone not { istroke } if 0 0 1 1 leftarrow n 2 sub dup n 1 sub dup rightarrow } if end } dup 0 4 dict put def /Poly { 3 1 roll newpath moveto -1 add { lineto } repeat closepath patternNone not { ifill } if brushNone not { istroke } if } def /Rect { 0 begin /t exch def /r exch def /b exch def /l exch def newpath l b moveto l t lineto r t lineto r b lineto closepath patternNone not { ifill } if brushNone not { istroke } if end } dup 0 4 dict put def /Text { ishow } def /idef { dup where { pop pop pop } { exch def } ifelse } def /ifill { 0 begin gsave patternGrayLevel -1 ne { fgred bgred fgred sub patternGrayLevel mul add fggreen bggreen fggreen sub patternGrayLevel mul add fgblue bgblue fgblue sub patternGrayLevel mul add setrgbcolor eofill } { eoclip originalCTM setmatrix pathbbox /t exch def /r exch def /b exch def /l exch def /w r l sub ceiling cvi def /h t b sub ceiling cvi def /imageByteWidth w 8 div ceiling cvi def /imageHeight h def bgred bggreen bgblue setrgbcolor eofill fgred fggreen fgblue setrgbcolor w 0 gt h 0 gt and { l b translate w h scale w h true [w 0 0 h neg 0 h] { patternproc } imagemask } if } ifelse grestore end } dup 0 8 dict put def /istroke { gsave brushDashOffset -1 eq { [] 0 setdash 1 setgray } { brushDashArray brushDashOffset setdash fgred fggreen fgblue setrgbcolor } ifelse brushWidth setlinewidth originalCTM setmatrix stroke grestore } def /ishow { 0 begin gsave fgred fggreen fgblue setrgbcolor /fontDict printFont findfont printSize scalefont dup setfont def /descender fontDict begin 0 [FontBBox] 1 get FontMatrix end transform exch pop def /vertoffset 0 descender sub printSize sub printFont /Courier ne printFont /Courier-Bold ne and { 1 add } if def { 0 vertoffset moveto show /vertoffset vertoffset printSize sub def } forall grestore end } dup 0 3 dict put def /patternproc { 0 begin /patternByteLength patternString length def /patternHeight patternByteLength 8 mul sqrt cvi def /patternWidth patternHeight def /patternByteWidth patternWidth 8 idiv def /imageByteMaxLength imageByteWidth imageHeight mul stringLimit patternByteWidth sub min def /imageMaxHeight imageByteMaxLength imageByteWidth idiv patternHeight idiv patternHeight mul patternHeight max def /imageHeight imageHeight imageMaxHeight sub store /imageString imageByteWidth imageMaxHeight mul patternByteWidth add string def 0 1 imageMaxHeight 1 sub { /y exch def /patternRow y patternByteWidth mul patternByteLength mod def /patternRowString patternString patternRow patternByteWidth getinterval def /imageRow y imageByteWidth mul def 0 patternByteWidth imageByteWidth 1 sub { /x exch def imageString imageRow x add patternRowString putinterval } for } for imageString end } dup 0 12 dict put def /min { dup 3 2 roll dup 4 3 roll lt { exch } if pop } def /max { dup 3 2 roll dup 4 3 roll gt { exch } if pop } def /arrowhead { 0 begin transform originalCTM itransform /taily exch def /tailx exch def transform originalCTM itransform /tipy exch def /tipx exch def /dy tipy taily sub def /dx tipx tailx sub def /angle dx 0 ne dy 0 ne or { dy dx atan } { 90 } ifelse def gsave originalCTM setmatrix tipx tipy translate angle rotate newpath 0 0 moveto arrowHeight neg arrowWidth 2 div lineto arrowHeight neg arrowWidth 2 div neg lineto closepath patternNone not { originalCTM setmatrix /padtip arrowHeight 2 exp 0.25 arrowWidth 2 exp mul add sqrt brushWidth mul arrowWidth div def /padtail brushWidth 2 div def tipx tipy translate angle rotate padtip 0 translate arrowHeight padtip add padtail add arrowHeight div dup scale arrowheadpath ifill } if brushNone not { originalCTM setmatrix tipx tipy translate angle rotate arrowheadpath istroke } if grestore end } dup 0 9 dict put def /arrowheadpath { newpath 0 0 moveto arrowHeight neg arrowWidth 2 div lineto arrowHeight neg arrowWidth 2 div neg lineto closepath } def /leftarrow { 0 begin y exch get /taily exch def x exch get /tailx exch def y exch get /tipy exch def x exch get /tipx exch def brushLeftArrow { tipx tipy tailx taily arrowhead } if end } dup 0 4 dict put def /rightarrow { 0 begin y exch get /tipy exch def x exch get /tipx exch def y exch get /taily exch def x exch get /tailx exch def brushRightArrow { tipx tipy tailx taily arrowhead } if end } dup 0 4 dict put def /midpoint { 0 begin /y1 exch def /x1 exch def /y0 exch def /x0 exch def x0 x1 add 2 div y0 y1 add 2 div end } dup 0 4 dict put def /thirdpoint { 0 begin /y1 exch def /x1 exch def /y0 exch def /x0 exch def x0 2 mul x1 add 3 div y0 2 mul y1 add 3 div end } dup 0 4 dict put def /subspline { 0 begin /movetoNeeded exch def y exch get /y3 exch def x exch get /x3 exch def y exch get /y2 exch def x exch get /x2 exch def y exch get /y1 exch def x exch get /x1 exch def y exch get /y0 exch def x exch get /x0 exch def x1 y1 x2 y2 thirdpoint /p1y exch def /p1x exch def x2 y2 x1 y1 thirdpoint /p2y exch def /p2x exch def x1 y1 x0 y0 thirdpoint p1x p1y midpoint /p0y exch def /p0x exch def x2 y2 x3 y3 thirdpoint p2x p2y midpoint /p3y exch def /p3x exch def movetoNeeded { p0x p0y moveto } if p1x p1y p2x p2y p3x p3y curveto end } dup 0 17 dict put def /storexyn { /n exch def /y n array def /x n array def n 1 sub -1 0 { /i exch def y i 3 2 roll put x i 3 2 roll put } for } def %I Idraw 7 Grid 8 Begin %I b u %I cfg u %I cbg u %I f u %I p u %I t [ 0.923077 0 0 0.923077 0 0 ] concat /originalCTM matrix currentmatrix def Begin %I Text %I cfg Black 0 0 0 SetCFg %I f -adobe-times-medium-r-normal--10-100-75-75-p-54-iso8859-1 /Times-Roman 10 SetF %I t [ 1 0 0 1 100 612 ] concat %I [ (person) ] Text End Begin %I Text %I cfg Black 0 0 0 SetCFg %I f -adobe-times-medium-r-normal--10-100-75-75-p-54-iso8859-1 /Times-Roman 10 SetF %I t [ 1 0 0 1 40 564 ] concat %I [ (student) ] Text End Begin %I Text %I cfg Black 0 0 0 SetCFg %I f -adobe-times-medium-r-normal--10-100-75-75-p-54-iso8859-1 /Times-Roman 10 SetF %I t [ 1 0 0 1 152 565 ] concat %I [ (employee) ] Text End Begin %I Text %I cfg Black 0 0 0 SetCFg %I f -adobe-times-medium-r-normal--10-100-75-75-p-54-iso8859-1 /Times-Roman 10 SetF %I t [ 1 0 0 1 75 518 ] concat %I [ (student_employee) ] Text End Begin %I Line %I b 65535 1 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0 SetP %I t [ 1 0 0 1 -87 52 ] concat %I 199 544 165 509 Line End Begin %I Line %I b 65535 1 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0 SetP %I t [ 1 0 0 1 -87 52 ] concat %I 199 544 234 509 Line End Begin %I Line %I b 65535 1 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0 SetP %I t [ 1 0 0 1 -87 52 ] concat %I 165 509 199 475 Line End Begin %I Line %I b 65535 1 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0 SetP %I t [ 1 0 0 1 -87 52 ] concat %I 234 509 199 475 Line End End %I eop 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g(de\014ned;)i(it)e(can)g(b)q(e)h(an)o(y)e(natural)h(n)o(um)o(b)q(er,)g (th)o(us)0 2670 y(allo)o(wing)d(subt)o(yp)q(es)j(to)f(ha)o(v)o(e)g (di\013eren)o(t)g(b)q(ounds.)22 b(This)15 b(v)n(ariabilit)o(y)d(sho)o (ws)k(up)f(in)f(the)i(sp)q(eci\014cation)f(of)g Fh(put)p Fn(,)g(where)h(w)o(e)954 2795 y(14)p eop %%Page: 15 17 15 16 bop 0 1 1950 1 v 750 215 a @beginspecial 52 @llx 373 @lly 160 @urx 415 @ury 1080 @rwi @setspecial %%BeginDocument: fig9.ps 50 dict begin /arrowHeight 8 def /arrowWidth 4 def /none null def /numGraphicParameters 17 def /stringLimit 65535 def /Begin { save numGraphicParameters dict begin } def /End { end restore } def /SetB { dup type /nulltype eq { pop false /brushRightArrow idef false /brushLeftArrow idef true /brushNone idef } { /brushDashOffset idef /brushDashArray idef 0 ne /brushRightArrow idef 0 ne /brushLeftArrow idef /brushWidth idef false /brushNone idef } ifelse } def /SetCFg { /fgblue idef /fggreen idef /fgred idef } def /SetCBg { /bgblue idef /bggreen idef /bgred idef } def /SetF { /printSize idef /printFont idef } def /SetP { dup type /nulltype eq { pop true /patternNone idef } { /patternGrayLevel idef patternGrayLevel -1 eq { /patternString idef } if false /patternNone idef } ifelse } def /BSpl { 0 begin storexyn newpath n 1 gt { 0 0 0 0 0 0 1 1 true subspline n 2 gt { 0 0 0 0 1 1 2 2 false subspline 1 1 n 3 sub { /i exch def i 1 sub dup i dup i 1 add dup i 2 add dup false subspline } for n 3 sub dup n 2 sub dup n 1 sub dup 2 copy false subspline } if n 2 sub dup n 1 sub dup 2 copy 2 copy false subspline patternNone not brushLeftArrow not brushRightArrow not and and { ifill } if brushNone not { istroke } if 0 0 1 1 leftarrow n 2 sub dup n 1 sub dup rightarrow } if end } dup 0 4 dict put def /Circ { newpath 0 360 arc patternNone not { ifill } if brushNone not { istroke } if } def /CBSpl { 0 begin dup 2 gt { storexyn newpath n 1 sub dup 0 0 1 1 2 2 true subspline 1 1 n 3 sub { /i exch def i 1 sub dup i dup i 1 add dup i 2 add dup false subspline } for n 3 sub dup n 2 sub dup n 1 sub dup 0 0 false subspline n 2 sub dup n 1 sub dup 0 0 1 1 false subspline patternNone not { ifill } if brushNone not { istroke } if } { Poly } ifelse end } dup 0 4 dict put def /Elli { 0 begin newpath 4 2 roll translate scale 0 0 1 0 360 arc patternNone not { ifill } if brushNone not { istroke } if end } dup 0 1 dict put def /Line { 0 begin 2 storexyn newpath x 0 get y 0 get moveto x 1 get y 1 get lineto brushNone not { istroke } if 0 0 1 1 leftarrow 0 0 1 1 rightarrow end } dup 0 4 dict put def /MLine { 0 begin storexyn newpath n 1 gt { x 0 get y 0 get moveto 1 1 n 1 sub { /i exch def x i get y i get lineto } for patternNone not brushLeftArrow not brushRightArrow not and and { ifill } if brushNone not { istroke } if 0 0 1 1 leftarrow n 2 sub dup n 1 sub dup rightarrow } if end } dup 0 4 dict put def /Poly { 3 1 roll newpath moveto -1 add { lineto } repeat closepath patternNone not { ifill } if brushNone not { istroke } if } def /Rect { 0 begin /t exch def /r exch def /b exch def /l exch def newpath l b moveto l t lineto r t lineto r b lineto closepath patternNone not { ifill } if brushNone not { istroke } if end } dup 0 4 dict put def /Text { ishow } def /idef { dup where { pop pop pop } { exch def } ifelse } def /ifill { 0 begin gsave patternGrayLevel -1 ne { fgred bgred fgred sub patternGrayLevel mul add fggreen bggreen fggreen sub patternGrayLevel mul add fgblue bgblue fgblue sub patternGrayLevel mul add setrgbcolor eofill } { eoclip originalCTM setmatrix pathbbox /t exch def /r exch def /b exch def /l exch def /w r l sub ceiling cvi def /h t b sub ceiling cvi def /imageByteWidth w 8 div ceiling cvi def /imageHeight h def bgred bggreen bgblue setrgbcolor eofill fgred fggreen fgblue setrgbcolor w 0 gt h 0 gt and { l b translate w h scale w h true [w 0 0 h neg 0 h] { patternproc } imagemask } if } ifelse grestore end } dup 0 8 dict put def /istroke { gsave brushDashOffset -1 eq { [] 0 setdash 1 setgray } { brushDashArray brushDashOffset setdash fgred fggreen fgblue setrgbcolor } ifelse brushWidth setlinewidth originalCTM setmatrix stroke grestore } def /ishow { 0 begin gsave fgred fggreen fgblue setrgbcolor /fontDict printFont findfont printSize scalefont dup setfont def /descender fontDict begin 0 [FontBBox] 1 get FontMatrix end transform exch pop def /vertoffset 0 descender sub printSize sub printFont /Courier ne printFont /Courier-Bold ne and { 1 add } if def { 0 vertoffset moveto show /vertoffset vertoffset printSize sub def } forall grestore end } dup 0 3 dict put def /patternproc { 0 begin /patternByteLength patternString length def /patternHeight patternByteLength 8 mul sqrt cvi def /patternWidth patternHeight def /patternByteWidth patternWidth 8 idiv def /imageByteMaxLength imageByteWidth imageHeight mul stringLimit patternByteWidth sub min def /imageMaxHeight imageByteMaxLength imageByteWidth idiv patternHeight idiv patternHeight mul patternHeight max def /imageHeight imageHeight imageMaxHeight sub store /imageString imageByteWidth imageMaxHeight mul patternByteWidth add string def 0 1 imageMaxHeight 1 sub { /y exch def /patternRow y patternByteWidth mul patternByteLength mod def /patternRowString patternString patternRow patternByteWidth getinterval def /imageRow y imageByteWidth mul def 0 patternByteWidth imageByteWidth 1 sub { /x exch def imageString imageRow x add patternRowString putinterval } for } for imageString end } dup 0 12 dict put def /min { dup 3 2 roll dup 4 3 roll lt { exch } if pop } def /max { dup 3 2 roll dup 4 3 roll gt { exch } if pop } def /arrowhead { 0 begin transform originalCTM itransform /taily exch def /tailx exch def transform originalCTM itransform /tipy exch def /tipx exch def /dy tipy taily sub def /dx tipx tailx sub def /angle dx 0 ne dy 0 ne or { dy dx atan } { 90 } ifelse def gsave originalCTM setmatrix tipx tipy translate angle rotate newpath 0 0 moveto arrowHeight neg arrowWidth 2 div lineto arrowHeight neg arrowWidth 2 div neg lineto closepath patternNone not { originalCTM setmatrix /padtip arrowHeight 2 exp 0.25 arrowWidth 2 exp mul add sqrt brushWidth mul arrowWidth div def /padtail brushWidth 2 div def tipx tipy translate angle rotate padtip 0 translate arrowHeight padtip add padtail add arrowHeight div dup scale arrowheadpath ifill } if brushNone not { originalCTM setmatrix tipx tipy translate angle rotate arrowheadpath istroke } if grestore end } dup 0 9 dict put def /arrowheadpath { newpath 0 0 moveto arrowHeight neg arrowWidth 2 div lineto arrowHeight neg arrowWidth 2 div neg lineto closepath } def /leftarrow { 0 begin y exch get /taily exch def x exch get /tailx exch def y exch get /tipy exch def x exch get /tipx exch def brushLeftArrow { tipx tipy tailx taily arrowhead } if end } dup 0 4 dict put def /rightarrow { 0 begin y exch get /tipy exch def x exch get /tipx exch def y exch get /taily exch def x exch get /tailx exch def brushRightArrow { tipx tipy tailx taily arrowhead } if end } dup 0 4 dict put def /midpoint { 0 begin /y1 exch def /x1 exch def /y0 exch def /x0 exch def x0 x1 add 2 div y0 y1 add 2 div end } dup 0 4 dict put def /thirdpoint { 0 begin /y1 exch def /x1 exch def /y0 exch def /x0 exch def x0 2 mul x1 add 3 div y0 2 mul y1 add 3 div end } dup 0 4 dict put def /subspline { 0 begin /movetoNeeded exch def y exch get /y3 exch def x exch get /x3 exch def y exch get /y2 exch def x exch get /x2 exch def y exch get /y1 exch def x exch get /x1 exch def y exch get /y0 exch def x exch get /x0 exch def x1 y1 x2 y2 thirdpoint /p1y exch def /p1x exch def x2 y2 x1 y1 thirdpoint /p2y exch def /p2x exch def x1 y1 x0 y0 thirdpoint p1x p1y midpoint /p0y exch def /p0x exch def x2 y2 x3 y3 thirdpoint p2x p2y midpoint /p3y exch def /p3x exch def movetoNeeded { p0x p0y moveto } if p1x p1y p2x p2y p3x p3y curveto end } dup 0 17 dict put def /storexyn { /n exch def /y n array def /x n array def n 1 sub -1 0 { /i exch def y i 3 2 roll put x i 3 2 roll put } for } def %I Idraw 7 Grid 8 Begin %I b u %I cfg u %I cbg u %I f u %I p u %I t [ 0.923077 0 0 0.923077 0 0 ] concat /originalCTM matrix currentmatrix def Begin %I Text %I cfg Black 0 0 0 SetCFg %I f -adobe-times-medium-r-normal--10-100-75-75-p-54-iso8859-1 /Times-Roman 10 SetF %I t [ 1 0 0 1 97 446 ] concat %I [ (elephant) ] Text End Begin %I Text %I cfg Black 0 0 0 SetCFg %I f -adobe-times-medium-r-normal--10-100-75-75-p-54-iso8859-1 /Times-Roman 10 SetF %I t [ 1 0 0 1 58 415 ] concat %I [ (royal) ] Text End Begin %I Text %I cfg Black 0 0 0 SetCFg %I f -adobe-times-medium-r-normal--10-100-75-75-p-54-iso8859-1 /Times-Roman 10 SetF %I t [ 1 0 0 1 147 416 ] concat %I [ (albino) ] Text End Begin %I Line %I b 65535 1 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0 SetP %I t [ 1 0 0 1 -87 52 ] concat %I 199 380 173 362 Line End Begin %I Line %I b 65535 1 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0 SetP %I t [ 1 0 0 1 -87 52 ] concat %I 199 380 225 362 Line End End %I eop showpage end %%EndDocument @endspecial 708 307 a Fn(Figure)14 b(7:)k(Elephan)o(t)c(Hierarc)o(h)o (y)p 0 357 V 0 531 a(do)f(not)h(sa)o(y)f(what)h(sp)q(eci\014c)h(b)q (ound)e(v)n(alue)g(causes)i(the)g(call)d(to)i(fail.)i(Therefore,)f(a)e (user)i(of)e Fh(put)g Fn(m)o(ust)g(b)q(e)h(prepared)h(for)e(a)0 581 y(failure.)k($Of)c(course)h(the)g(user)g(could)f(deduce)h(that)f (a)g(particular)g(call)f(will)f(succeed,)k(based)f(on)e(a)h(previous)g (sequence)j(of)0 631 y(metho)q(d)c(calls)f(and)i(the)g(constrain)o(t)f (that)h(the)g(b)q(ound)f(of)g(a)g(bag)g(do)q(es)h(not)f(c)o(hange.$)18 b(A)13 b(subt)o(yp)q(e)g(of)f(bag)f(migh)o(t)g(limit)e(the)0 680 y(b)q(ound)15 b(to)g(a)f(\014xed)i(v)n(alue,)e(or)h(to)f(a)h (smaller)e(range.)22 b(Sev)o(eral)15 b(subt)o(yp)q(es)h(of)e(bag)h(are) g(sho)o(wn)g(in)f(Figure)h(8;)g(medium)o(bags)0 730 y(ha)o(v)o(e)f(v)n (arious)f(b)q(ounds,)h(so)g(that)g(this)g(t)o(yp)q(e)g(migh)o(t)e(ha)o (v)o(e)i(its)f(o)o(wn)h(subt)o(yp)q(es,)h(e.g.,)d(bag)p 1406 730 13 2 v 15 w(150.)p 0 789 1950 1 v 344 1412 a @beginspecial 256 @llx 493 @lly 559 @urx 633 @ury 3030 @rwi @setspecial %%BeginDocument: fig10.ps /arrowHeight 10 def /arrowWidth 5 def /IdrawDict 52 dict def IdrawDict begin /reencodeISO { dup dup findfont dup length dict begin { 1 index /FID ne { def }{ pop pop } ifelse } forall /Encoding ISOLatin1Encoding def currentdict end definefont } def /ISOLatin1Encoding [ /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /space/exclam/quotedbl/numbersign/dollar/percent/ampersand/quoteright /parenleft/parenright/asterisk/plus/comma/minus/period/slash /zero/one/two/three/four/five/six/seven/eight/nine/colon/semicolon /less/equal/greater/question/at/A/B/C/D/E/F/G/H/I/J/K/L/M/N /O/P/Q/R/S/T/U/V/W/X/Y/Z/bracketleft/backslash/bracketright /asciicircum/underscore/quoteleft/a/b/c/d/e/f/g/h/i/j/k/l/m /n/o/p/q/r/s/t/u/v/w/x/y/z/braceleft/bar/braceright/asciitilde /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/dotlessi/grave/acute/circumflex/tilde/macron/breve /dotaccent/dieresis/.notdef/ring/cedilla/.notdef/hungarumlaut /ogonek/caron/space/exclamdown/cent/sterling/currency/yen/brokenbar /section/dieresis/copyright/ordfeminine/guillemotleft/logicalnot /hyphen/registered/macron/degree/plusminus/twosuperior/threesuperior /acute/mu/paragraph/periodcentered/cedilla/onesuperior/ordmasculine /guillemotright/onequarter/onehalf/threequarters/questiondown /Agrave/Aacute/Acircumflex/Atilde/Adieresis/Aring/AE/Ccedilla /Egrave/Eacute/Ecircumflex/Edieresis/Igrave/Iacute/Icircumflex /Idieresis/Eth/Ntilde/Ograve/Oacute/Ocircumflex/Otilde/Odieresis /multiply/Oslash/Ugrave/Uacute/Ucircumflex/Udieresis/Yacute /Thorn/germandbls/agrave/aacute/acircumflex/atilde/adieresis /aring/ae/ccedilla/egrave/eacute/ecircumflex/edieresis/igrave /iacute/icircumflex/idieresis/eth/ntilde/ograve/oacute/ocircumflex /otilde/odieresis/divide/oslash/ugrave/uacute/ucircumflex/udieresis /yacute/thorn/ydieresis ] def /Times-Roman reencodeISO def /Times-Italic reencodeISO def /none null def /numGraphicParameters 17 def /stringLimit 65535 def /Begin { save numGraphicParameters dict begin } def /End { end restore } def /SetB { dup type /nulltype eq { pop false /brushRightArrow idef false /brushLeftArrow idef true /brushNone idef } { /brushDashOffset idef /brushDashArray idef 0 ne /brushRightArrow idef 0 ne /brushLeftArrow idef /brushWidth idef false /brushNone idef } ifelse } def /SetCFg { /fgblue idef /fggreen idef /fgred idef } def /SetCBg { /bgblue idef /bggreen idef /bgred idef } def /SetF { /printSize idef /printFont idef } def /SetP { dup type /nulltype eq { pop true /patternNone idef } { dup -1 eq { /patternGrayLevel idef /patternString idef } { /patternGrayLevel idef } ifelse false /patternNone idef } ifelse } def /BSpl { 0 begin storexyn newpath n 1 gt { 0 0 0 0 0 0 1 1 true subspline n 2 gt { 0 0 0 0 1 1 2 2 false subspline 1 1 n 3 sub { /i exch def i 1 sub dup i dup i 1 add dup i 2 add dup false subspline } for n 3 sub dup n 2 sub dup n 1 sub dup 2 copy false subspline } if n 2 sub dup n 1 sub dup 2 copy 2 copy false subspline patternNone not brushLeftArrow not brushRightArrow not and and { ifill } if brushNone not { istroke } if 0 0 1 1 leftarrow n 2 sub dup n 1 sub dup rightarrow } if end } dup 0 4 dict put def /Circ { newpath 0 360 arc patternNone not { ifill } if brushNone not { istroke } if } def /CBSpl { 0 begin dup 2 gt { storexyn newpath n 1 sub dup 0 0 1 1 2 2 true subspline 1 1 n 3 sub { /i exch def i 1 sub dup i dup i 1 add dup i 2 add dup false subspline } for n 3 sub dup n 2 sub dup n 1 sub dup 0 0 false subspline n 2 sub dup n 1 sub dup 0 0 1 1 false subspline patternNone not { ifill } if brushNone not { istroke } if } { Poly } ifelse end } dup 0 4 dict put def /Elli { 0 begin newpath 4 2 roll translate scale 0 0 1 0 360 arc patternNone not { ifill } if brushNone not { istroke } if end } dup 0 1 dict put def /Line { 0 begin 2 storexyn newpath x 0 get y 0 get moveto x 1 get y 1 get lineto brushNone not { istroke } if 0 0 1 1 leftarrow 0 0 1 1 rightarrow end } dup 0 4 dict put def /MLine { 0 begin storexyn newpath n 1 gt { x 0 get y 0 get moveto 1 1 n 1 sub { /i exch def x i get y i get lineto } for patternNone not brushLeftArrow not brushRightArrow not and and { ifill } if brushNone not { istroke } if 0 0 1 1 leftarrow n 2 sub dup n 1 sub dup rightarrow } if end } dup 0 4 dict put def /Poly { 3 1 roll newpath moveto -1 add { lineto } repeat closepath patternNone not { ifill } if brushNone not { istroke } if } def /Rect { 0 begin /t exch def /r exch def /b exch def /l exch def newpath l b moveto l t lineto r t lineto r b lineto closepath patternNone not { ifill } if brushNone not { istroke } if end } dup 0 4 dict put def /Text { ishow } def /idef { dup where { pop pop pop } { exch def } ifelse } def /ifill { 0 begin gsave patternGrayLevel -1 ne { fgred bgred fgred sub patternGrayLevel mul add fggreen bggreen fggreen sub patternGrayLevel mul add fgblue bgblue fgblue sub patternGrayLevel mul add setrgbcolor eofill } { eoclip originalCTM setmatrix pathbbox /t exch def /r exch def /b exch def /l exch def /w r l sub ceiling cvi def /h t b sub ceiling cvi def /imageByteWidth w 8 div ceiling cvi def /imageHeight h def bgred bggreen bgblue setrgbcolor eofill fgred fggreen fgblue setrgbcolor w 0 gt h 0 gt and { l b translate w h scale w h true [w 0 0 h neg 0 h] { patternproc } imagemask } if } ifelse grestore end } dup 0 8 dict put def /istroke { gsave brushDashOffset -1 eq { [] 0 setdash 1 setgray } { brushDashArray brushDashOffset setdash fgred fggreen fgblue setrgbcolor } ifelse brushWidth setlinewidth originalCTM setmatrix stroke grestore } def /ishow { 0 begin gsave fgred fggreen fgblue setrgbcolor /fontDict printFont printSize scalefont dup setfont def /descender fontDict begin 0 [FontBBox] 1 get FontMatrix end transform exch pop def /vertoffset 1 printSize sub descender sub def { 0 vertoffset moveto show /vertoffset vertoffset printSize sub def } forall grestore end } dup 0 3 dict put def /patternproc { 0 begin /patternByteLength patternString length def /patternHeight patternByteLength 8 mul sqrt cvi def /patternWidth patternHeight def /patternByteWidth patternWidth 8 idiv def /imageByteMaxLength imageByteWidth imageHeight mul stringLimit patternByteWidth sub min def /imageMaxHeight imageByteMaxLength imageByteWidth idiv patternHeight idiv patternHeight mul patternHeight max def /imageHeight imageHeight imageMaxHeight sub store /imageString imageByteWidth imageMaxHeight mul patternByteWidth add string def 0 1 imageMaxHeight 1 sub { /y exch def /patternRow y patternByteWidth mul patternByteLength mod def /patternRowString patternString patternRow patternByteWidth getinterval def /imageRow y imageByteWidth mul def 0 patternByteWidth imageByteWidth 1 sub { /x exch def imageString imageRow x add patternRowString putinterval } for } for imageString end } dup 0 12 dict put def /min { dup 3 2 roll dup 4 3 roll lt { exch } if pop } def /max { dup 3 2 roll dup 4 3 roll gt { exch } if pop } def /midpoint { 0 begin /y1 exch def /x1 exch def /y0 exch def /x0 exch def x0 x1 add 2 div y0 y1 add 2 div end } dup 0 4 dict put def /thirdpoint { 0 begin /y1 exch def /x1 exch def /y0 exch def /x0 exch def x0 2 mul x1 add 3 div y0 2 mul y1 add 3 div end } dup 0 4 dict put def /subspline { 0 begin /movetoNeeded exch def y exch get /y3 exch def x exch get /x3 exch def y exch get /y2 exch def x exch get /x2 exch def y exch get /y1 exch def x exch get /x1 exch def y exch get /y0 exch def x exch get /x0 exch def x1 y1 x2 y2 thirdpoint /p1y exch def /p1x exch def x2 y2 x1 y1 thirdpoint /p2y exch def /p2x exch def x1 y1 x0 y0 thirdpoint p1x p1y midpoint /p0y exch def /p0x exch def x2 y2 x3 y3 thirdpoint p2x p2y midpoint /p3y exch def /p3x exch def movetoNeeded { p0x p0y moveto } if p1x p1y p2x p2y p3x p3y curveto end } dup 0 17 dict put def /storexyn { /n exch def /y n array def /x n array def n 1 sub -1 0 { /i exch def y i 3 2 roll put x i 3 2 roll put } for } def /SSten { fgred fggreen fgblue setrgbcolor dup true exch 1 0 0 -1 0 6 -1 roll matrix astore } def /FSten { dup 3 -1 roll dup 4 1 roll exch newpath 0 0 moveto dup 0 exch lineto exch dup 3 1 roll exch lineto 0 lineto closepath bgred bggreen bgblue setrgbcolor eofill SSten } def /Rast { exch dup 3 1 roll 1 0 0 -1 0 6 -1 roll matrix astore } def /arrowhead { 0 begin transform originalCTM itransform /taily exch def /tailx exch def transform originalCTM itransform /tipy exch def /tipx exch def /dy tipy taily sub def /dx tipx tailx sub def /angle dx 0 ne dy 0 ne or { dy dx atan } { 90 } ifelse def gsave originalCTM setmatrix tipx tipy translate angle rotate newpath arrowHeight neg arrowWidth 2 div moveto 0 0 lineto arrowHeight neg arrowWidth 2 div neg lineto patternNone not { originalCTM setmatrix /padtip arrowHeight 2 exp 0.25 arrowWidth 2 exp mul add sqrt brushWidth mul arrowWidth div def /padtail brushWidth 2 div def tipx tipy translate angle rotate padtip 0 translate arrowHeight padtip add padtail add arrowHeight div dup scale arrowheadpath ifill } if brushNone not { originalCTM setmatrix tipx tipy translate angle rotate arrowheadpath istroke } if grestore end } dup 0 9 dict put def /arrowheadpath { newpath arrowHeight neg arrowWidth 2 div moveto 0 0 lineto arrowHeight neg arrowWidth 2 div neg lineto } def /leftarrow { 0 begin y exch get /taily exch def x exch get /tailx exch def y exch get /tipy exch def x exch get /tipx exch def brushLeftArrow { tipx tipy tailx taily arrowhead } if end } dup 0 4 dict put def /rightarrow { 0 begin y exch get /tipy exch def x exch get /tipx exch def y exch get /taily exch def x exch get /tailx exch def brushRightArrow { tipx tipy tailx taily arrowhead } if end } dup 0 4 dict put def %I Idraw 10 Grid 8 8 Begin %I b u %I cfg u %I cbg u %I f u %I p u %I t [ 0.923077 0 0 0.923077 0 0 ] concat /originalCTM matrix currentmatrix def Begin %I Text %I cfg Black 0 0 0 SetCFg %I f -adobe-times-medium-r-normal--10-100-75-75-p-54-iso8859-1 Times-Roman 10 SetF %I t [ 1 0 0 1 444 682 ] concat %I [ (bag) ] Text End Begin %I Text %I cfg Black 0 0 0 SetCFg %I f -adobe-times-medium-r-normal--10-100-75-75-p-54-iso8859-1 Times-Roman 10 SetF %I t [ 1 0 0 1 293 630 ] concat %I [ (largebag) ] Text End Begin %I Text %I cfg Black 0 0 0 SetCFg %I f -adobe-times-medium-r-normal--10-100-75-75-p-54-iso8859-1 Times-Roman 10 SetF %I t [ 1 0 0 1 423 630 ] concat %I [ (mediumbag) ] Text End Begin %I Text %I cfg Black 0 0 0 SetCFg %I f -adobe-times-medium-r-normal--10-100-75-75-p-54-iso8859-1 Times-Roman 10 SetF %I t [ 1 0 0 1 561 630 ] concat %I [ (smallbag) ] Text End Begin %I Text %I cfg Black 0 0 0 SetCFg %I f -adobe-times-medium-i-normal--10-100-75-75-p-52-iso8859-1 Times-Italic 10 SetF %I t [ 1 0 0 1 431 563 ] concat %I [ (bag_150) ] Text End Begin %I Line %I b 65535 1 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0 SetP %I t [ 1 0 0 1 -87 52 ] concat %I 536 613 536 587 Line %I 1 End Begin %I Line %I b 65535 1 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0 SetP %I t [ 1 0 0 1 -87 52 ] concat %I 536 613 665 587 Line %I 1 End Begin %I Line %I b 65535 1 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0 SetP %I t [ 1 0 0 1 -87 52 ] concat %I 536 613 406 587 Line %I 1 End Begin %I Line %I b 65535 1 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0 SetP %I t [ 1 0 0 1 -87 52 ] concat %I 536 544 536 518 Line %I 1 End Begin %I Text %I cfg Black 0 0 0 SetCFg %I f -adobe-times-medium-i-normal--10-100-75-75-p-52-iso8859-1 Times-Italic 10 SetF %I t [ 1 0 0 1 327.737 619.733 ] concat %I [ (32) ] Text End Begin %I Text %I cfg Black 0 0 0 SetCFg %I f -adobe-times-medium-i-normal--10-100-75-75-p-52-iso8859-1 Times-Italic 10 SetF %I t [ 1 0 0 1 403.335 613 ] concat %I [ ($100 <= bound <= 1000$) ] Text End Begin %I Text %I cfg Black 0 0 0 SetCFg %I f -adobe-times-medium-i-normal--10-100-75-75-p-52-iso8859-1 Times-Italic 10 SetF %I t [ 1 0 0 1 421.064 545.134 ] concat %I [ ($bound = 150$) ] Text End Begin %I Text %I cfg Black 0 0 0 SetCFg %I f -adobe-times-medium-i-normal--10-100-75-75-p-52-iso8859-1 Times-Italic 10 SetF %I t [ 1 0 0 1 552.624 613.134 ] concat %I [ ($bound = 20$) ] Text End Begin %I Text %I cfg Black 0 0 0 SetCFg %I f -adobe-times-medium-i-normal--10-100-75-75-p-52-iso8859-1 Times-Italic 10 SetF %I t [ 1 0 0 1 279.268 613 ] concat %I [ ($bound = 2 $) ] Text End End %I eop showpage end %%EndDocument @endspecial 664 1503 a(Figure)i(8:)j(A)d(T)o(yp)q(e)h(F)m(amily)10 b(for)k(Bags)p 0 1553 V 62 1694 a(The)20 b(bag)e(hierarc)o(h)o(y)i(ma)o (y)d(seem)i(coun)o(terin)o(tuitiv)o(e,)h(since)f(w)o(e)h(migh)o(t)d (exp)q(ect)j(that)f(bags)g(with)g(smaller)e(b)q(ounds)0 1744 y(should)g(b)q(e)g(subt)o(yp)q(es)h(of)f(bags)f(with)h(larger)g(b) q(ounds.)27 b(F)m(or)17 b(example,)f(w)o(e)h(migh)o(t)e(exp)q(ect)j (smallbag)c(to)j(b)q(e)g(a)g(subt)o(yp)q(e)0 1794 y(of)d(largebag.)19 b(Ho)o(w)o(ev)o(er,)14 b(the)i(sp)q(eci\014cations)f(for)f(the)h(t)o(w) o(o)f(t)o(yp)q(es)i(are)f(incompatible:)i(the)e(b)q(ound)f(of)g(ev)o (ery)h(largebag)f(is)0 1844 y(2)21 1829 y Fi(32)56 1844 y Fn(,)f(whic)o(h)f(is)h(clearly)g(not)g(true)h(for)e(smallbags.)k(F)m (urthermore,)c(this)h(di\013erence)i(is)e(observ)n(able)g(via)f(the)i (metho)q(ds:)j(It)c(is)0 1894 y(legal)g(to)g(call)g(the)h Fh(put)g Fn(metho)q(d)f(on)g(a)g(largebag)g(whose)h(size)h(is)e (greater)i(than)e(or)h(equal)f(to)h(20,)e(but)i(the)g(call)f(is)g(not)h (legal)0 1943 y(for)g(a)f(smallbag.)i(Therefore)h(the)e(pre-condition)g 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w(bags)0 2242 y(ha)o(v)o(e)14 b(an)f(additional)f(m)o(utator,)g Fh(change)p 646 2242 V 17 w(b)n(ound)p Fn(:)104 2325 y Fh(change)p 229 2325 V 16 w(b)n(ound)j Fn(=)f Fk(pro)q(c)g Fn($)p Fh(n:)k Fn(in)o(t$)227 2375 y Fk(requires)12 b Fg(n)f Ff(\025)h(j)p Fg(b)518 2381 y Fe(pr)q(e)568 2375 y Fg(:el)q(ems)p Ff(j)227 2425 y Fk(mo)q(di\014es)7 b Fg(b)227 2475 y Fk(ensures)25 b Fg(b)426 2481 y Fe(post)489 2475 y Fg(:el)q(ems)13 b Fn(=)f Fg(b)682 2481 y Fe(pr)q(e)733 2475 y Fg(:el)q(ems)e Ff(^)f Fg(b)916 2481 y Fe(post)980 2475 y Fg(:bound)h Fn(=)i Fg(n)0 2558 y Fn(Notice)i(that)g(other)h(t)o (yp)q(es)g(in)e(the)h(family)d(need)k(not)f(ha)o(v)o(e)g(a)f Fh(change)p 1111 2558 V 17 w(b)n(ound)h Fn(metho)q(d.)62 2608 y(This)j(example)f(illustrates)h(the)h(di\013eren)o(t)g(w)o(a)o (ys)f(that)g(subt)o(yp)q(es)h(reduce)h(v)n(ariabilit)o(y)m(.)25 b(All)16 b(v)n(arying)p 1698 2608 V 14 w(bag)h(subt)o(yp)q(es)0 2657 y(reduce)i(v)n(ariabilit)o(y)c(in)i(the)h(sp)q(eci\014cation)h (for)e(the)h Fh(put)g Fn(metho)q(d;)g(v)n(arying)p 1226 2657 V 13 w(bag's)f Fh(put)h Fn(metho)q(d)f(is)g(non-deterministic,)954 2795 y(15)p eop %%Page: 16 18 16 17 bop 0 1 1950 1 v 442 686 a @beginspecial 93 @llx 516 @lly 349 @urx 671 @ury 2560 @rwi @setspecial %%BeginDocument: fig11.ps /arrowHeight 10 def /arrowWidth 5 def /IdrawDict 52 dict def IdrawDict begin /reencodeISO { dup dup findfont dup length dict begin { 1 index /FID ne { def }{ pop pop } ifelse } forall /Encoding ISOLatin1Encoding def currentdict end definefont } def /ISOLatin1Encoding [ /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /space/exclam/quotedbl/numbersign/dollar/percent/ampersand/quoteright /parenleft/parenright/asterisk/plus/comma/minus/period/slash /zero/one/two/three/four/five/six/seven/eight/nine/colon/semicolon /less/equal/greater/question/at/A/B/C/D/E/F/G/H/I/J/K/L/M/N /O/P/Q/R/S/T/U/V/W/X/Y/Z/bracketleft/backslash/bracketright /asciicircum/underscore/quoteleft/a/b/c/d/e/f/g/h/i/j/k/l/m /n/o/p/q/r/s/t/u/v/w/x/y/z/braceleft/bar/braceright/asciitilde /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef/.notdef /.notdef/dotlessi/grave/acute/circumflex/tilde/macron/breve /dotaccent/dieresis/.notdef/ring/cedilla/.notdef/hungarumlaut /ogonek/caron/space/exclamdown/cent/sterling/currency/yen/brokenbar /section/dieresis/copyright/ordfeminine/guillemotleft/logicalnot /hyphen/registered/macron/degree/plusminus/twosuperior/threesuperior /acute/mu/paragraph/periodcentered/cedilla/onesuperior/ordmasculine /guillemotright/onequarter/onehalf/threequarters/questiondown /Agrave/Aacute/Acircumflex/Atilde/Adieresis/Aring/AE/Ccedilla /Egrave/Eacute/Ecircumflex/Edieresis/Igrave/Iacute/Icircumflex /Idieresis/Eth/Ntilde/Ograve/Oacute/Ocircumflex/Otilde/Odieresis /multiply/Oslash/Ugrave/Uacute/Ucircumflex/Udieresis/Yacute /Thorn/germandbls/agrave/aacute/acircumflex/atilde/adieresis /aring/ae/ccedilla/egrave/eacute/ecircumflex/edieresis/igrave /iacute/icircumflex/idieresis/eth/ntilde/ograve/oacute/ocircumflex /otilde/odieresis/divide/oslash/ugrave/uacute/ucircumflex/udieresis /yacute/thorn/ydieresis ] def /Times-Roman reencodeISO def /Times-Italic reencodeISO def /none null def /numGraphicParameters 17 def /stringLimit 65535 def /Begin { save numGraphicParameters dict begin } def /End { end restore } def /SetB { dup type /nulltype eq { pop false /brushRightArrow idef false /brushLeftArrow idef true /brushNone idef } { /brushDashOffset idef /brushDashArray idef 0 ne /brushRightArrow idef 0 ne /brushLeftArrow idef /brushWidth idef false /brushNone idef } ifelse } def /SetCFg { /fgblue idef /fggreen idef /fgred idef } def /SetCBg { /bgblue idef /bggreen idef /bgred idef } def /SetF { /printSize idef /printFont idef } def /SetP { dup type /nulltype eq { pop true /patternNone idef } { dup -1 eq { /patternGrayLevel idef /patternString idef } { /patternGrayLevel idef } ifelse false /patternNone idef } ifelse } def /BSpl { 0 begin storexyn newpath n 1 gt { 0 0 0 0 0 0 1 1 true subspline n 2 gt { 0 0 0 0 1 1 2 2 false subspline 1 1 n 3 sub { /i exch def i 1 sub dup i dup i 1 add dup i 2 add dup false subspline } for n 3 sub dup n 2 sub dup n 1 sub dup 2 copy false subspline } if n 2 sub dup n 1 sub dup 2 copy 2 copy false subspline patternNone not brushLeftArrow not brushRightArrow not and and { ifill } if brushNone not { istroke } if 0 0 1 1 leftarrow n 2 sub dup n 1 sub dup rightarrow } if end } dup 0 4 dict put def /Circ { newpath 0 360 arc patternNone not { ifill } if brushNone not { istroke } if } def /CBSpl { 0 begin dup 2 gt { storexyn newpath n 1 sub dup 0 0 1 1 2 2 true subspline 1 1 n 3 sub { /i exch def i 1 sub dup i dup i 1 add dup i 2 add dup false subspline } for n 3 sub dup n 2 sub dup n 1 sub dup 0 0 false subspline n 2 sub dup n 1 sub dup 0 0 1 1 false subspline patternNone not { ifill } if brushNone not { istroke } if } { Poly } ifelse end } dup 0 4 dict put def /Elli { 0 begin newpath 4 2 roll translate scale 0 0 1 0 360 arc patternNone not { ifill } if brushNone not { istroke } if end } dup 0 1 dict put def /Line { 0 begin 2 storexyn newpath x 0 get y 0 get moveto x 1 get y 1 get lineto brushNone not { istroke } if 0 0 1 1 leftarrow 0 0 1 1 rightarrow end } dup 0 4 dict put def /MLine { 0 begin storexyn newpath n 1 gt { x 0 get y 0 get moveto 1 1 n 1 sub { /i exch def x i get y i get lineto } for patternNone not brushLeftArrow not brushRightArrow not and and { ifill } if brushNone not { istroke } if 0 0 1 1 leftarrow n 2 sub dup n 1 sub dup rightarrow } if end } dup 0 4 dict put def /Poly { 3 1 roll newpath moveto -1 add { lineto } repeat closepath patternNone not { ifill } if brushNone not { istroke } if } def /Rect { 0 begin /t exch def /r exch def /b exch def /l exch def newpath l b moveto l t lineto r t lineto r b lineto closepath patternNone not { ifill } if brushNone not { istroke } if end } dup 0 4 dict put def /Text { ishow } def /idef { dup where { pop pop pop } { exch def } ifelse } def /ifill { 0 begin gsave patternGrayLevel -1 ne { fgred bgred fgred sub patternGrayLevel mul add fggreen bggreen fggreen sub patternGrayLevel mul add fgblue bgblue fgblue sub patternGrayLevel mul add setrgbcolor eofill } { eoclip originalCTM setmatrix pathbbox /t exch def /r exch def /b exch def /l exch def /w r l sub ceiling cvi def /h t b sub ceiling cvi def /imageByteWidth w 8 div ceiling cvi def /imageHeight h def bgred bggreen bgblue setrgbcolor eofill fgred fggreen fgblue setrgbcolor w 0 gt h 0 gt and { l b translate w h scale w h true [w 0 0 h neg 0 h] { patternproc } imagemask } if } ifelse grestore end } dup 0 8 dict put def /istroke { gsave brushDashOffset -1 eq { [] 0 setdash 1 setgray } { brushDashArray brushDashOffset setdash fgred fggreen fgblue setrgbcolor } ifelse brushWidth setlinewidth originalCTM setmatrix stroke grestore } def /ishow { 0 begin gsave fgred fggreen fgblue setrgbcolor /fontDict printFont printSize scalefont dup setfont def /descender fontDict begin 0 [FontBBox] 1 get FontMatrix end transform exch pop def /vertoffset 1 printSize sub descender sub def { 0 vertoffset moveto show /vertoffset vertoffset printSize sub def } forall grestore end } dup 0 3 dict put def /patternproc { 0 begin /patternByteLength patternString length def /patternHeight patternByteLength 8 mul sqrt cvi def /patternWidth patternHeight def /patternByteWidth patternWidth 8 idiv def /imageByteMaxLength imageByteWidth imageHeight mul stringLimit patternByteWidth sub min def /imageMaxHeight imageByteMaxLength imageByteWidth idiv patternHeight idiv patternHeight mul patternHeight max def /imageHeight imageHeight imageMaxHeight sub store /imageString imageByteWidth imageMaxHeight mul patternByteWidth add string def 0 1 imageMaxHeight 1 sub { /y exch def /patternRow y patternByteWidth mul patternByteLength mod def /patternRowString patternString patternRow patternByteWidth getinterval def /imageRow y imageByteWidth mul def 0 patternByteWidth imageByteWidth 1 sub { /x exch def imageString imageRow x add patternRowString putinterval } for } for imageString end } dup 0 12 dict put def /min { dup 3 2 roll dup 4 3 roll lt { exch } if pop } def /max { dup 3 2 roll dup 4 3 roll gt { exch } if pop } def /midpoint { 0 begin /y1 exch def /x1 exch def /y0 exch def /x0 exch def x0 x1 add 2 div y0 y1 add 2 div end } dup 0 4 dict put def /thirdpoint { 0 begin /y1 exch def /x1 exch def /y0 exch def /x0 exch def x0 2 mul x1 add 3 div y0 2 mul y1 add 3 div end } dup 0 4 dict put def /subspline { 0 begin /movetoNeeded exch def y exch get /y3 exch def x exch get /x3 exch def y exch get /y2 exch def x exch get /x2 exch def y exch get /y1 exch def x exch get /x1 exch def y exch get /y0 exch def x exch get /x0 exch def x1 y1 x2 y2 thirdpoint /p1y exch def /p1x exch def x2 y2 x1 y1 thirdpoint /p2y exch def /p2x exch def x1 y1 x0 y0 thirdpoint p1x p1y midpoint /p0y exch def /p0x exch def x2 y2 x3 y3 thirdpoint p2x p2y midpoint /p3y exch def /p3x exch def movetoNeeded { p0x p0y moveto } if p1x p1y p2x p2y p3x p3y curveto end } dup 0 17 dict put def /storexyn { /n exch def /y n array def /x n array def n 1 sub -1 0 { /i exch def y i 3 2 roll put x i 3 2 roll put } for } def /SSten { fgred fggreen fgblue setrgbcolor dup true exch 1 0 0 -1 0 6 -1 roll matrix astore } def /FSten { dup 3 -1 roll dup 4 1 roll exch newpath 0 0 moveto dup 0 exch lineto exch dup 3 1 roll exch lineto 0 lineto closepath bgred bggreen bgblue setrgbcolor eofill SSten } def /Rast { exch dup 3 1 roll 1 0 0 -1 0 6 -1 roll matrix astore } def /arrowhead { 0 begin transform originalCTM itransform /taily exch def /tailx exch def transform originalCTM itransform /tipy exch def /tipx exch def /dy tipy taily sub def /dx tipx tailx sub def /angle dx 0 ne dy 0 ne or { dy dx atan } { 90 } ifelse def gsave originalCTM setmatrix tipx tipy translate angle rotate newpath arrowHeight neg arrowWidth 2 div moveto 0 0 lineto arrowHeight neg arrowWidth 2 div neg lineto patternNone not { originalCTM setmatrix /padtip arrowHeight 2 exp 0.25 arrowWidth 2 exp mul add sqrt brushWidth mul arrowWidth div def /padtail brushWidth 2 div def tipx tipy translate angle rotate padtip 0 translate arrowHeight padtip add padtail add arrowHeight div dup scale arrowheadpath ifill } if brushNone not { originalCTM setmatrix tipx tipy translate angle rotate arrowheadpath istroke } if grestore end } dup 0 9 dict put def /arrowheadpath { newpath arrowHeight neg arrowWidth 2 div moveto 0 0 lineto arrowHeight neg arrowWidth 2 div neg lineto } def /leftarrow { 0 begin y exch get /taily exch def x exch get /tailx exch def y exch get /tipy exch def x exch get /tipx exch def brushLeftArrow { tipx tipy tailx taily arrowhead } if end } dup 0 4 dict put def /rightarrow { 0 begin y exch get /tipy exch def x exch get /tipx exch def y exch get /taily exch def x exch get /tailx exch def brushRightArrow { tipx tipy tailx taily arrowhead } if end } dup 0 4 dict put def %I Idraw 10 Grid 8 8 Begin %I b u %I cfg u %I cbg u %I f u %I p u %I t [ 0.923077 0 0 0.923077 0 0 ] concat /originalCTM matrix currentmatrix def Begin %I Text %I cfg Black 0 0 0 SetCFg %I f -adobe-times-medium-r-normal--10-100-75-75-p-54-iso8859-1 Times-Roman 10 SetF %I t [ 1 0 0 1 201.284 723.701 ] concat %I [ (varying_bag) ] Text End Begin %I Text %I cfg Black 0 0 0 SetCFg %I f -adobe-times-medium-r-normal--10-100-75-75-p-54-iso8859-1 Times-Roman 10 SetF %I t [ 1 0 0 1 199.784 647.641 ] concat %I [ (dynamic_bag) ] Text End Begin %I Text %I cfg Black 0 0 0 SetCFg %I f -adobe-times-medium-r-normal--10-100-75-75-p-54-iso8859-1 Times-Roman 10 SetF %I t [ 1 0 0 1 306.017 647.641 ] concat %I [ (bag) ] Text End Begin %I Line %I b 65535 1 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0 SetP %I t [ 1 0 0 1 -303.587 292.508 ] concat %I 622 311 622 285 Line %I 1 End Begin %I Line %I b 65535 1 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0 SetP %I t [ 1 0 0 1 -303.587 292.508 ] concat %I 622 311 579 285 Line %I 1 End Begin %I Line %I b 65535 1 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0 SetP %I t [ 1 0 0 1 -303.587 292.508 ] concat %I 622 311 665 285 Line %I 1 End Begin %I Text %I cfg Black 0 0 0 SetCFg %I f -adobe-times-medium-r-normal--10-100-75-75-p-54-iso8859-1 Times-Roman 10 SetF %I t [ 1 0 0 1 109.647 649.641 ] concat %I [ (flexible_bag) ] Text End Begin %I Text %I cfg Black 0 0 0 SetCFg %I f -*-times-medium-i-normal-*-10-*-*-*-*-*-*-* Times-Italic 10 SetF %I t [ 1 0 0 1 193.784 705.949 ] concat %I [ (I: size <= bound) (C: true) ] Text End Begin %I Text %I cfg Black 0 0 0 SetCFg %I f -*-times-medium-i-normal-*-10-*-*-*-*-*-*-* Times-Italic 10 SetF %I t [ 1 0 0 1 102.654 632.337 ] concat %I [ (I: size <= bound) (C: true) ] Text End Begin %I Text %I cfg Black 0 0 0 SetCFg %I f -*-times-medium-i-normal-*-10-*-*-*-*-*-*-* Times-Italic 10 SetF %I t [ 1 0 0 1 280.478 632.337 ] concat %I [ (I: size <= bound) (C: bound stays the same) ] Text End Begin %I Text %I cfg Black 0 0 0 SetCFg %I f -*-times-medium-i-normal-*-10-*-*-*-*-*-*-* Times-Italic 10 SetF %I t [ 1 0 0 1 196.284 632.337 ] concat %I [ (I: size = bound) (C: true) ] Text End Begin %I Line %I b 65535 1 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0 SetP %I t [ 0.960396 0 0 0.90625 -299.686 342.813 ] concat %I 536 371 622 345 Line %I 1 End Begin %I Line %I b 65535 1 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0 SetP %I t [ -0.724846 -0.515783 0.516428 -0.723941 412.008 1224.19 ] concat %I 536 371 622 345 Line %I 1 End Begin %I Line %I b 65535 1 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0 SetP %I t [ 0.263055 -0.270926 0.88107 0.136294 -252.752 773.59 ] concat %I 536 371 622 345 Line %I 1 End Begin %I Text %I cfg Black 0 0 0 SetCFg %I f -adobe-times-medium-r-normal--10-100-75-75-p-54-iso8859-1 Times-Roman 10 SetF %I t [ 1 0 0 1 284.28 570.107 ] concat %I [ ([...as in Fig. 8...]) ] Text End End %I eop showpage end %%EndDocument @endspecial 605 778 a Fn(Figure)14 b(9:)k(Another)c(T)o(yp)q(e)h(F)m (amily)10 b(for)k(Bags)p 0 827 V 0 990 a(since)k(it)f(migh)o(t)e(add)i (the)g(elemen)o(t)g($and)g(c)o(hange)g(the)h(b)q(ound$)f(if)g(the)g (size)h(is)f(the)h(same)e(as)h(the)h(b)q(ound,)f(or)g(it)g(migh)o(t)0 1040 y(not.)32 b(Bag)18 b(and)h(\015exible)p 409 1040 13 2 v 15 w(bag)f(reduce)i(this)f(v)n(ariabilit)o(y)d(b)o(y)i(not)g (adding)g(the)h(elemen)o(t,)g(whereas)h(dynamic)p 1776 1040 V 13 w(bag)e(do)q(es)0 1090 y(add)f(the)h(elemen)o(t.)29 b(In)17 b(addition,)g(bag)g(reduces)i(v)n(ariabilit)o(y)c(b)o(y)i (restricting)i(the)f(constrain)o(t:)25 b(the)19 b(trivial)d(constrain)o (t)0 1139 y(for)f(v)n(arying)p 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b(Ho)o(w)o(ev)o(er,)19 b(w)o(e)g(clearly)f(w)o(ould)g(lik)o(e)g(in)o(tegers)h(of)f(di\013eren) o(t)h(sizes)h(to)e(b)q(e)0 2421 y(related.)33 b(This)18 b(is)g(accomplished)g(b)o(y)g(designing)g(a)g(virtual)g(sup)q(ert)o(yp) q(e)i(that)f(includes)g(them.)31 b(Suc)o(h)19 b(a)f(hierarc)o(h)o(y)h (is)0 2471 y(sho)o(wn)13 b(in)g(Figure)g(11,)g(where)h(in)o(teger)g(is) f(a)g(virtual)g(t)o(yp)q(e)g(whose)h(in)o(v)n(arian)o(t)e(simply)f(sa)o (ys)i(that)h(the)g(size)g(of)e(an)h(in)o(teger)h(is)0 2521 y(greater)g(than)g(zero.)19 b(In)o(teger)14 b(t)o(yp)q(es)g(with)f (di\013eren)o(t)i(sizes)f(are)g(subt)o(yp)q(es)h(of)d(in)o(teger.)19 b(In)13 b(addition,)f(small)f(in)o(teger)j(t)o(yp)q(es)0 2570 y(are)f(subt)o(yp)q(es)i(of)d(regular)p 415 2570 V 15 w(in)o(t,)g(another)i(virtual)e(t)o(yp)q(e;)h(the)h(in)o(v)n (arian)o(t)d(in)h(the)i(sp)q(eci\014cation)g(for)e(regular)p 1681 2570 V 15 w(in)o(t)h(states)h(that)0 2620 y(the)j(size)f(of)g(an)g (in)o(teger)g(is)g(either)h(16)e(bits)i(or)f(32)f(bits.)25 b(An)16 b(in)o(teger)g(family)d(migh)o(t)h(ha)o(v)o(e)i(a)g(structure)i (lik)o(e)d(this,)h(or)g(it)0 2670 y(migh)o(t)c(b)q(e)i(\015atter)h(b)o (y)f(ha)o(ving)e(all)h(in)o(teger)h(t)o(yp)q(es)h(b)q(e)g(direct)f (subt)o(yp)q(es)i(of)d(in)o(teger.)954 2795 y(16)p eop %%Page: 17 19 17 18 bop 0 1 1950 1 v 469 390 a @beginspecial 49 @llx 615 @lly 292 @urx 699 @ury 2430 @rwi @setspecial %%BeginDocument: fig12.ps 50 dict begin /arrowHeight 8 def /arrowWidth 4 def /none null def /numGraphicParameters 17 def /stringLimit 65535 def /Begin { save numGraphicParameters dict begin } def /End { end restore } def /SetB { dup type /nulltype eq { pop false /brushRightArrow idef false /brushLeftArrow idef true /brushNone idef } { /brushDashOffset idef /brushDashArray idef 0 ne /brushRightArrow idef 0 ne /brushLeftArrow idef /brushWidth idef false /brushNone idef } ifelse } def /SetCFg { /fgblue idef /fggreen idef /fgred idef } def /SetCBg { /bgblue idef /bggreen idef /bgred idef } def /SetF { /printSize idef /printFont idef } def /SetP { dup type /nulltype eq { pop true /patternNone idef } { /patternGrayLevel idef patternGrayLevel -1 eq { /patternString idef } if false /patternNone idef } ifelse } def /BSpl { 0 begin storexyn newpath n 1 gt { 0 0 0 0 0 0 1 1 true subspline n 2 gt { 0 0 0 0 1 1 2 2 false subspline 1 1 n 3 sub { /i exch def i 1 sub dup i dup i 1 add dup i 2 add dup false subspline } for n 3 sub dup n 2 sub dup n 1 sub dup 2 copy false subspline } if n 2 sub dup n 1 sub dup 2 copy 2 copy false subspline patternNone not brushLeftArrow not brushRightArrow not and and { ifill } if brushNone not { istroke } if 0 0 1 1 leftarrow n 2 sub dup n 1 sub dup rightarrow } if end } dup 0 4 dict put def /Circ { newpath 0 360 arc patternNone not { ifill } if brushNone not { istroke } if } def /CBSpl { 0 begin dup 2 gt { storexyn newpath n 1 sub dup 0 0 1 1 2 2 true subspline 1 1 n 3 sub { /i exch def i 1 sub dup i dup i 1 add dup i 2 add dup false subspline } for n 3 sub dup n 2 sub dup n 1 sub dup 0 0 false subspline n 2 sub dup n 1 sub dup 0 0 1 1 false subspline patternNone not { ifill } if brushNone not { istroke } if } { Poly } ifelse end } dup 0 4 dict put def /Elli { 0 begin newpath 4 2 roll translate scale 0 0 1 0 360 arc patternNone not { ifill } if brushNone not { istroke } if end } dup 0 1 dict put def /Line { 0 begin 2 storexyn newpath x 0 get y 0 get moveto x 1 get y 1 get lineto brushNone not { istroke } if 0 0 1 1 leftarrow 0 0 1 1 rightarrow end } dup 0 4 dict put def /MLine { 0 begin storexyn newpath n 1 gt { x 0 get y 0 get moveto 1 1 n 1 sub { /i exch def x i get y i get lineto } for patternNone not brushLeftArrow not brushRightArrow not and and { ifill } if brushNone not { istroke } if 0 0 1 1 leftarrow n 2 sub dup n 1 sub dup rightarrow } if end } dup 0 4 dict put def /Poly { 3 1 roll newpath moveto -1 add { lineto } repeat closepath patternNone not { ifill } if brushNone not { istroke } if } def /Rect { 0 begin /t exch def /r exch def /b exch def /l exch def newpath l b moveto l t lineto r t lineto r b lineto closepath patternNone not { ifill } if brushNone not { istroke } if end } dup 0 4 dict put def /Text { ishow } def /idef { dup where { pop pop pop } { exch def } ifelse } def /ifill { 0 begin gsave patternGrayLevel -1 ne { fgred bgred fgred sub patternGrayLevel mul add fggreen bggreen fggreen sub patternGrayLevel mul add fgblue bgblue fgblue sub patternGrayLevel mul add setrgbcolor eofill } { eoclip originalCTM setmatrix pathbbox /t exch def /r exch def /b exch def /l exch def /w r l sub ceiling cvi def /h t b sub ceiling cvi def /imageByteWidth w 8 div ceiling cvi def /imageHeight h def bgred bggreen bgblue setrgbcolor eofill fgred fggreen fgblue setrgbcolor w 0 gt h 0 gt and { l b translate w h scale w h true [w 0 0 h neg 0 h] { patternproc } imagemask } if } ifelse grestore end } dup 0 8 dict put def /istroke { gsave brushDashOffset -1 eq { [] 0 setdash 1 setgray } { brushDashArray brushDashOffset setdash fgred fggreen fgblue setrgbcolor } ifelse brushWidth setlinewidth originalCTM setmatrix stroke grestore } def /ishow { 0 begin gsave fgred fggreen fgblue setrgbcolor /fontDict printFont findfont printSize scalefont dup setfont def /descender fontDict begin 0 [FontBBox] 1 get FontMatrix end transform exch pop def /vertoffset 0 descender sub printSize sub printFont /Courier ne printFont /Courier-Bold ne and { 1 add } if def { 0 vertoffset moveto show /vertoffset vertoffset printSize sub def } forall grestore end } dup 0 3 dict put def /patternproc { 0 begin /patternByteLength patternString length def /patternHeight patternByteLength 8 mul sqrt cvi def /patternWidth patternHeight def /patternByteWidth patternWidth 8 idiv def /imageByteMaxLength imageByteWidth imageHeight mul stringLimit patternByteWidth sub min def /imageMaxHeight imageByteMaxLength imageByteWidth idiv patternHeight idiv patternHeight mul patternHeight max def /imageHeight imageHeight imageMaxHeight sub store /imageString imageByteWidth imageMaxHeight mul patternByteWidth add string def 0 1 imageMaxHeight 1 sub { /y exch def /patternRow y patternByteWidth mul patternByteLength mod def /patternRowString patternString patternRow patternByteWidth getinterval def /imageRow y imageByteWidth mul def 0 patternByteWidth imageByteWidth 1 sub { /x exch def imageString imageRow x add patternRowString putinterval } for } for imageString end } dup 0 12 dict put def /min { dup 3 2 roll dup 4 3 roll lt { exch } if pop } def /max { dup 3 2 roll dup 4 3 roll gt { exch } if pop } def /arrowhead { 0 begin transform originalCTM itransform /taily exch def /tailx exch def transform originalCTM itransform /tipy exch def /tipx exch def /dy tipy taily sub def /dx tipx tailx sub def /angle dx 0 ne dy 0 ne or { dy dx atan } { 90 } ifelse def gsave originalCTM setmatrix tipx tipy translate angle rotate newpath 0 0 moveto arrowHeight neg arrowWidth 2 div lineto arrowHeight neg arrowWidth 2 div neg lineto closepath patternNone not { originalCTM setmatrix /padtip arrowHeight 2 exp 0.25 arrowWidth 2 exp mul add sqrt brushWidth mul arrowWidth div def /padtail brushWidth 2 div def tipx tipy translate angle rotate padtip 0 translate arrowHeight padtip add padtail add arrowHeight div dup scale arrowheadpath ifill } if brushNone not { originalCTM setmatrix tipx tipy translate angle rotate arrowheadpath istroke } if grestore end } dup 0 9 dict put def /arrowheadpath { newpath 0 0 moveto arrowHeight neg arrowWidth 2 div lineto arrowHeight neg arrowWidth 2 div neg lineto closepath } def /leftarrow { 0 begin y exch get /taily exch def x exch get /tailx exch def y exch get /tipy exch def x exch get /tipx exch def brushLeftArrow { tipx tipy tailx taily arrowhead } if end } dup 0 4 dict put def /rightarrow { 0 begin y exch get /tipy exch def x exch get /tipx exch def y exch get /taily exch def x exch get /tailx exch def brushRightArrow { tipx tipy tailx taily arrowhead } if end } dup 0 4 dict put def /midpoint { 0 begin /y1 exch def /x1 exch def /y0 exch def /x0 exch def x0 x1 add 2 div y0 y1 add 2 div end } dup 0 4 dict put def /thirdpoint { 0 begin /y1 exch def /x1 exch def /y0 exch def /x0 exch def x0 2 mul x1 add 3 div y0 2 mul y1 add 3 div end } dup 0 4 dict put def /subspline { 0 begin /movetoNeeded exch def y exch get /y3 exch def x exch get /x3 exch def y exch get /y2 exch def x exch get /x2 exch def y exch get /y1 exch def x exch get /x1 exch def y exch get /y0 exch def x exch get /x0 exch def x1 y1 x2 y2 thirdpoint /p1y exch def /p1x exch def x2 y2 x1 y1 thirdpoint /p2y exch def /p2x exch def x1 y1 x0 y0 thirdpoint p1x p1y midpoint /p0y exch def /p0x exch def x2 y2 x3 y3 thirdpoint p2x p2y midpoint /p3y exch def /p3x exch def movetoNeeded { p0x p0y moveto } if p1x p1y p2x p2y p3x p3y curveto end } dup 0 17 dict put def /storexyn { /n exch def /y n array def /x n array def n 1 sub -1 0 { /i exch def y i 3 2 roll put x i 3 2 roll put } for } def %I Idraw 7 Grid 8 Begin %I b u %I cfg u %I cbg u %I f u %I p u %I t [ 0.923077 0 0 0.923077 0 0 ] concat /originalCTM matrix currentmatrix def Begin %I Text %I cfg Black 0 0 0 SetCFg %I f -adobe-times-medium-r-normal--10-100-75-75-p-54-iso8859-1 /Times-Roman 10 SetF %I t [ 1 0 0 1 184.5 754 ] concat %I [ (counter) ] Text End Begin %I Text %I cfg Black 0 0 0 SetCFg %I f -adobe-times-medium-i-normal--10-100-75-75-p-52-iso8859-1 /Times-Italic 10 SetF %I t [ 1 0 0 1 151.5 739 ] concat %I [ ($value never decreases$) ] Text End Begin %I Text %I cfg Black 0 0 0 SetCFg %I f -adobe-times-medium-r-normal--10-100-75-75-p-54-iso8859-1 /Times-Roman 10 SetF %I t [ 1 0 0 1 105.5 695 ] concat %I [ (incrementer) ] Text End Begin %I Text %I cfg Black 0 0 0 SetCFg %I f -adobe-times-medium-i-normal--10-100-75-75-p-52-iso8859-1 /Times-Italic 10 SetF %I t [ 1 0 0 1 54.5 677 ] concat %I [ ($value never decreases$) ] Text End Begin %I Text %I cfg Black 0 0 0 SetCFg %I f -adobe-times-medium-r-normal--10-100-75-75-p-54-iso8859-1 /Times-Roman 10 SetF %I t [ 1 0 0 1 182.5 694 ] concat %I [ (doubler) ] Text End Begin %I Text %I cfg Black 0 0 0 SetCFg %I f -adobe-times-medium-i-normal--10-100-75-75-p-52-iso8859-1 /Times-Italic 10 SetF %I t [ 1 0 0 1 167.5 677 ] concat %I [ ($value doubles$) ] Text End Begin %I Text %I cfg Black 0 0 0 SetCFg %I f -adobe-times-medium-r-normal--10-100-75-75-p-54-iso8859-1 /Times-Roman 10 SetF %I t [ 1 0 0 1 244 694 ] concat %I [ (multiplier) ] Text End Begin %I Text %I cfg Black 0 0 0 SetCFg %I f -adobe-times-medium-i-normal--10-100-75-75-p-52-iso8859-1 /Times-Italic 10 SetF %I t [ 1 0 0 1 243 678 ] concat %I [ ($value multiplies$) ] Text End Begin %I Line %I b 65535 1 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0 SetP %I t [ 1 0 0 1 -31 51 ] concat %I 230 673 230 649 Line End Begin %I Line %I b 65535 1 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0 SetP %I t [ 1 0 0 1 -29 51 ] concat %I 229 674 168 648 Line End Begin %I Line %I b 65535 1 0 0 [] 0 SetB %I cfg Black 0 0 0 SetCFg %I cbg White 1 1 1 SetCBg %I p 0 SetP %I t [ 1 0 0 1 -31 52 ] concat %I 231 674 280 648 Line End End %I eop showpage end %%EndDocument @endspecial 638 482 a Fn(Figure)14 b(10:)j(T)o(yp)q(e)e(F)m(amily)c (for)i(Coun)o(ters)p 0 532 V 0 632 V 608 997 a @beginspecial 330 @llx 161 @lly 506 @urx 239 @ury 1760 @rwi @setspecial %%BeginDocument: fig13.ps 50 dict begin /arrowHeight 8 def /arrowWidth 4 def /none null def /numGraphicParameters 17 def /stringLimit 65535 def /Begin { save numGraphicParameters dict begin } def /End { end restore } def /SetB { dup type /nulltype eq { pop false /brushRightArrow idef false /brushLeftArrow idef true /brushNone idef } { /brushDashOffset idef /brushDashArray idef 0 ne /brushRightArrow idef 0 ne /brushLeftArrow idef /brushWidth idef false /brushNone idef } ifelse } def /SetCFg { /fgblue idef /fggreen idef /fgred idef } def /SetCBg { /bgblue idef /bggreen idef /bgred idef } def /SetF { /printSize idef /printFont idef } def /SetP { dup type /nulltype eq { pop true /patternNone idef } { /patternGrayLevel idef patternGrayLevel -1 eq { /patternString idef } if false /patternNone idef } ifelse } def /BSpl { 0 begin storexyn newpath n 1 gt { 0 0 0 0 0 0 1 1 true subspline n 2 gt { 0 0 0 0 1 1 2 2 false subspline 1 1 n 3 sub { /i exch def i 1 sub dup i dup i 1 add dup i 2 add dup false subspline } for n 3 sub dup n 2 sub dup n 1 sub dup 2 copy false subspline } if n 2 sub dup n 1 sub dup 2 copy 2 copy false subspline patternNone not brushLeftArrow not brushRightArrow not and and { ifill } if brushNone not { istroke } if 0 0 1 1 leftarrow n 2 sub dup n 1 sub dup rightarrow } if end } dup 0 4 dict put def /Circ { newpath 0 360 arc patternNone not { ifill } if brushNone not { istroke } if } def /CBSpl { 0 begin dup 2 gt { storexyn newpath n 1 sub dup 0 0 1 1 2 2 true subspline 1 1 n 3 sub { /i exch def i 1 sub dup i dup i 1 add dup i 2 add dup false subspline } for n 3 sub dup n 2 sub dup n 1 sub dup 0 0 false subspline n 2 sub 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lineto r t lineto r b lineto closepath patternNone not { ifill } if brushNone not { istroke } if end } dup 0 4 dict put def /Text { ishow } def /idef { dup where { pop pop pop } { exch def } ifelse } def /ifill { 0 begin gsave patternGrayLevel -1 ne { fgred bgred fgred sub patternGrayLevel mul add fggreen bggreen fggreen sub patternGrayLevel mul add fgblue bgblue fgblue sub patternGrayLevel mul add setrgbcolor eofill } { eoclip originalCTM setmatrix pathbbox /t exch def /r exch def /b exch def /l exch def /w r l sub ceiling cvi def /h t b sub ceiling cvi def /imageByteWidth w 8 div ceiling cvi def /imageHeight h def bgred bggreen bgblue setrgbcolor eofill fgred fggreen fgblue setrgbcolor w 0 gt h 0 gt and { l b translate w h scale w h true [w 0 0 h neg 0 h] { patternproc } imagemask } if } ifelse grestore end } dup 0 8 dict put def /istroke { gsave brushDashOffset -1 eq { [] 0 setdash 1 setgray } { brushDashArray brushDashOffset setdash fgred fggreen fgblue setrgbcolor } ifelse brushWidth setlinewidth originalCTM setmatrix stroke grestore } def /ishow { 0 begin gsave fgred fggreen fgblue setrgbcolor /fontDict printFont findfont printSize scalefont dup setfont def /descender fontDict begin 0 [FontBBox] 1 get FontMatrix end transform exch pop def /vertoffset 0 descender sub printSize sub printFont /Courier ne printFont /Courier-Bold ne and { 1 add } if def { 0 vertoffset moveto show /vertoffset vertoffset printSize sub def } forall grestore end } dup 0 3 dict put def /patternproc { 0 begin /patternByteLength patternString length def /patternHeight patternByteLength 8 mul sqrt cvi def /patternWidth patternHeight def /patternByteWidth patternWidth 8 idiv def /imageByteMaxLength imageByteWidth imageHeight mul stringLimit patternByteWidth sub min def /imageMaxHeight imageByteMaxLength imageByteWidth idiv patternHeight idiv patternHeight mul patternHeight max def /imageHeight imageHeight imageMaxHeight sub store /imageString imageByteWidth imageMaxHeight mul patternByteWidth add string def 0 1 imageMaxHeight 1 sub { /y exch def /patternRow y patternByteWidth mul patternByteLength mod def /patternRowString patternString patternRow patternByteWidth getinterval def /imageRow y imageByteWidth mul def 0 patternByteWidth imageByteWidth 1 sub { /x exch def imageString imageRow x add patternRowString putinterval } for } for imageString end } dup 0 12 dict put def /min { dup 3 2 roll dup 4 3 roll lt { exch } if pop } def /max { dup 3 2 roll dup 4 3 roll gt { exch } if pop } def /arrowhead { 0 begin transform originalCTM itransform /taily exch def /tailx exch def transform originalCTM itransform /tipy exch def /tipx exch def /dy tipy taily sub def /dx tipx tailx sub def /angle dx 0 ne dy 0 ne or { dy dx atan } { 90 } ifelse def gsave originalCTM setmatrix tipx tipy translate angle rotate newpath 0 0 moveto arrowHeight neg arrowWidth 2 div lineto arrowHeight neg arrowWidth 2 div neg lineto closepath patternNone not { originalCTM setmatrix /padtip arrowHeight 2 exp 0.25 arrowWidth 2 exp mul add sqrt brushWidth mul arrowWidth div def /padtail brushWidth 2 div def tipx tipy translate angle rotate padtip 0 translate arrowHeight padtip add padtail add arrowHeight div dup scale arrowheadpath ifill } if brushNone not { originalCTM setmatrix tipx tipy translate angle rotate arrowheadpath istroke } if grestore end } dup 0 9 dict put def /arrowheadpath { newpath 0 0 moveto arrowHeight neg arrowWidth 2 div lineto arrowHeight neg arrowWidth 2 div neg lineto closepath } def /leftarrow { 0 begin y exch get /taily exch def x exch get /tailx exch def y exch get /tipy exch def x exch get /tipx exch def brushLeftArrow { tipx tipy tailx taily arrowhead } if end } dup 0 4 dict put def /rightarrow { 0 begin y exch get /tipy exch def x exch get /tipx exch def y exch get /taily exch def x exch get /tailx exch def brushRightArrow { tipx tipy tailx taily arrowhead } if end } dup 0 4 dict put def /midpoint { 0 begin /y1 exch def /x1 exch def /y0 exch 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b(G.)f(T.)22 b(Mo)q(dular)16 b(sp)q(eci\014cation)g(and)f(v)o(eri\014cation)g(of)g(ob)r(ject-orien)o (ted)i(programs.)k Fh(IEEE)c(Soft-)206 2588 y(war)n(e)p Fn(,)c(8$4$:72{80,)e(July)j(1991.)0 2670 y([LG85])81 b(Lisk)o(o)o(v,)13 b(B.)h(and)f(Guttag,)g(J.)18 b Fh(A)o(bstr)n(action) d(and)g(Sp)n(e)n(ci\014c)n(ation)h(in)f(Pr)n(o)n(gr)n(am)f(Design)p Fn(.)k(MIT)c(Press,)h(1985.)954 2795 y(19)p eop %%Page: 20 22 20 21 bop 0 42 a Fn([Lip92])79 b(Lip)q(ec)o(k,)20 b(U.)32 b(Seman)o(tics)17 b(and)i(usage)g(of)f(defaults)g(in)g(sp)q (eci\014cations.)33 b(In)19 b Fh(F)m(oundations)h(of)f(Information)206 91 y(Systems)c(Sp)n(e)n(ci\014c)n(ation)h(and)g(Design)p Fn(,)e(Marc)o(h)g(1992.)j(Dagstuhl)c(Seminar)g(9212)g(Rep)q(ort)h(35.)0 174 y([Lis92])86 b(Lisk)o(o)o(v,)19 b(B.)34 b(Preliminary)17 b(design)i(of)g(the)g(Thor)g(ob)r(ject-orien)o(ted)i(database)e (system.)33 b(In)19 b Fh(Pr)n(o)n(c.)g(of)h(the)206 224 y(Softwar)n(e)c(T)m(e)n(chnolo)n(gy)h(Confer)n(enc)n(e)p 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Fn(,)c(pages)g(1811{1841,)d(No)o(v)o(em)o(b)q(er)i (1994.)0 756 y([Mey88])62 b(Mey)o(er,)15 b(B.)j Fh(Obje)n(ct-oriente)n (d)c(Softwar)n(e)g(Construction)p Fn(.)k(Pren)o(tice)d(Hall,)e(New)h(Y) m(ork,)f(1988.)0 839 y([MS90])79 b(Maier,)11 b(D.)e(and)h(Stein,)h(J.)h (Dev)o(elopmen)o(t)c(and)i(implemen)o(tation)d(of)i(an)h(ob)r (ject-orien)o(ted)i(DBMS.)g(In)e(Zdonik,)206 888 y(S.)h(and)g(Maier,)h (D.,)e(editors,)i Fh(R)n(e)n(adings)i(in)e(Obje)n(ct-Oriente)n(d)g (Datab)n(ase)h(Systems)p Fn(,)f(pages)f(167{185.)e(Morgan)206 938 y(Kaufmann,)j(1990.)0 1021 y([Nel91])79 b(Nelson,)14 b(G.)j Fh(Systems)f(Pr)n(o)n(gr)n(amming)e(with)g(Mo)n(dula-3)p Fn(.)19 b(Pren)o(tice)c(Hall,)d(1991.)0 1104 y([SCB)94 1089 y Fi(+)122 1104 y Fn(86])30 b(Sc)o(ha\013ert,)14 b(C.,)d(Co)q(op)q(er,)i(T.,)f(Bullis,)f(B.,)h(Kilian,)f(M.,)h(and)g (Wilp)q(olt,)e(C.)16 b(An)c(in)o(tro)q(duction)g(to)g(Trellis/Owl.)206 1154 y(In)i Fh(Pr)n(o)n(c)n(e)n(e)n(dings)h(of)g(OOPSLA)g('86)p Fn(,)e(pages)h(9{16,)f(Septem)o(b)q(er)h(1986.)0 1237 y([SH92])86 b(Sc)o(heid,)17 b(J.)f(and)f(Holtsb)q(erg,)i(S.)24 b(Ina)16 b(Jo)g(sp)q(eci\014cation)h(language)e(reference)j(man)o(ual.) k(T)m(ec)o(hnical)16 b(Rep)q(ort)206 1287 y(TM-6021/001/06,)11 b(P)o(aramax)h(Systems)i(Corp)q(oration,)e(A)i(Unisys)h(Compan)o(y)m(,) 10 b(June)15 b(1992.)0 1370 y([Str86])85 b(Stroustrup,)15 b(B.)j Fh(The)d(C++)f(Pr)n(o)n(gr)n(amming)g(L)n(anguage)p Fn(.)20 b(Addison-W)m(esley)m(,)12 b(1986.)0 1453 y([ZW97])72 b(Zaremski,)19 b(A.)f(and)g(Wing,)g(J.)32 b(Sp)q(eci\014cation)19 b(matc)o(hing)e(of)h(soft)o(w)o(are)g(comp)q(onen)o(ts.)31 b Fh(A)o(CM)19 b(T)m(r)n(ans.)f(on)206 1503 y(Softwar)n(e)c(Engine)n (ering)i(and)f(Metho)n(dolo)n(gy)p Fn(,)g(6$4$:333{369,)c(Octob)q(er) k(1997.)954 2795 y(20)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF

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