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| 1919 | % Copyright (C) 1997 American Mathematical Society. All Rights Reserved.
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| 2093 |
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| 2094 | %%EndFont
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| 2095 | TeXDict begin 39158280 55380996 1000 600 600 (prolog-exercise.dvi)
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| 2096 | @start /Fa 145[43 4[24 105[{}2 66.4176 /CMMI8 rf /Fb
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| 2097 | 205[35 35 5[55 43[{}3 66.4176 /CMR8 rf /Fc 134[47 45
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| 2098 | 1[45 1[32 40 41 1[50 50 55 80 25 2[30 50 45 30 45 50
|
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| 2099 | 45 1[50 45[50 50 50 3[35 30 2[40 40 27[55 12[{}27 99.6264
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| 2100 | /CMTI12 rf /Fd 134[71 71 3[52 53 55 1[75 1[75 4[37 75
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|---|
| 2101 | 2[61 75 60 1[65 12[94 75 11[106 85 88 103 1[96 7[37 7[67
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| 2102 | 67 49[{}23 119.552 /CMBX12 rf /Fe 139[44 44 3[56 62 93
|
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| 2103 | 3[31 3[51 1[50 62 25[88 13[31 58[{}11 99.6264 /CMBX12
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| 2104 | rf /Ff 134[48 55 70 47 2[46 44 2[47 58 1[29 2[33 56 47
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| 2105 | 1[45 51 1[41 51 8[81 9[78 6[77 5[73 2[76 2[27 27 58[{}23
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| 2106 | 99.6264 /CMMI12 rf /Fg 152[50 50 7[66 43[66 17[100 13[77
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| 2107 | 2[50 14[77{}8 99.6264 /CMSY10 rf /Fh 134[51 51 51 51
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| 2108 | 51 51 51 51 1[51 51 51 51 51 2[51 51 51 51 51 51 51 51
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| 2109 | 51 3[51 1[51 51 51 51 3[51 3[51 2[51 5[51 5[51 1[51 1[51
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| 2110 | 2[51 2[51 51 51 51 51 51 51 1[51 51 51 51 2[51 51 51
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| 2111 | 39[{}49 99.6264 /CMTT12 rf /Fi 134[51 51 70 51 54 38
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| 2112 | 38 38 51 54 49 54 81 27 51 30 27 54 49 30 43 54 43 54
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| 2113 | 49 3[27 49 27 1[73 73 100 73 73 70 54 72 1[66 76 73 1[61
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| 2114 | 2[35 73 77 64 66 75 70 69 73 76 2[76 1[27 27 1[49 1[49
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| 2116 | 99.6264 /CMR12 rf /Fj 135[85 117 1[90 63 64 66 1[90 81
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| 2117 | 90 134 45 2[45 1[81 49 74 1[72 90 78 14[120 1[110 7[126
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| 2118 | 2[106 17[81 81 81 49[{}25 143.462 /CMBX12 rf /Fk 139[46
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| 2119 | 1[46 2[59 9[52 1[52 65 18[91 22[59 59 4[59 59 59 3[33
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| 2120 | 44[{}13 119.552 /CMR12 rf /Fl 135[83 2[88 61 62 61 1[88
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| 2121 | 79 88 133 43 2[43 1[79 47 70 1[70 1[79 16[108 12[115
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| 2122 | 8[43 58[{}19 172.188 /CMR17 rf end
|
|---|
| 2123 | %%EndProlog
|
|---|
| 2124 | %%BeginSetup
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|---|
| 2125 | %%Feature: *Resolution 600dpi
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| 2126 | TeXDict begin
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|---|
| 2127 | %%PaperSize: A4
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| 2128 |
|
|---|
| 2129 | %%EndSetup
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|---|
| 2130 | %%Page: 1 1
|
|---|
| 2131 | 1 0 bop 57 253 a Fl(Concepts)51 b(of)i(programming)e(languages:)70
|
|---|
| 2132 | b(Prolog)52 b(exercise)1443 783 y Fk(Octob)s(er)38 b(18,)g(2007)-104
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|---|
| 2133 | 1185 y Fj(1)161 b(Prolog)53 b(implemen)l(tation)-104
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|---|
| 2134 | 1404 y Fi(F)-8 b(or)32 b(this)h(exercise,)i(w)m(e)f(use)f(the)g
|
|---|
| 2135 | (SWI-Prolog)g([1)o(])g(implemen)m(tation)h(of)f(the)g(Prolog)f
|
|---|
| 2136 | (language.)42 1524 y(The)52 b(Prolog)e(in)m(terpreter)i(has)f(b)s(een)g
|
|---|
| 2137 | (compiled)h(for)e(Lin)m(ux/x86)i(systems,)58 b(and)50
|
|---|
| 2138 | b(is)h(a)m(v)-5 b(ailable)51 b(as)-104 1644 y Fh
|
|---|
| 2139 | (/home/csalp/bin.linux/pro)q(log)q(/pl)p Fi(.)81 b(V)-8
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|---|
| 2140 | b(ersions)44 b(for)e(other)h(systems)j(are)d(a)m(v)-5
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|---|
| 2141 | b(aible;)48 b(see)c(the)g(w)m(eb-)-104 1765 y(page.)42
|
|---|
| 2142 | 1885 y(Y)-8 b(ou)33 b(can)g(run)g(the)g(in)m(terpreter)h(using)f(the)g
|
|---|
| 2143 | (follo)m(wing)g(commands:)-54 2038 y Fh(/home/csalp/bin.linux/pro)q
|
|---|
| 2144 | (log/)q(pl)772 b Fi(run)33 b(the)g(Prolog)f(in)m(terpreter)-54
|
|---|
| 2145 | 2159 y Fh(/home/csalp/bin.linux/pro)q(log/)q(pl)57 b(-f)52
|
|---|
| 2146 | b(myprog.pl)102 b Fi(.)16 b(.)g(.)g(with)34 b Fh(myprog.pl)h
|
|---|
| 2147 | Fi(b)s(eing)e(the)2301 2279 y(name)g(of)f(a)h(Prolog)f(program)-104
|
|---|
| 2148 | 2392 y(NOTE:)42 2512 y(When)38 b(using)f(the)g(\014rst)g(v)m(ersion,)i
|
|---|
| 2149 | (b)s(e)d(sure)i(to)e(t)m(yp)s(e)h Fh(consult\('myprog.pl'\).)61
|
|---|
| 2150 | b Fi(at)36 b(the)h(b)s(eginning)-104 2632 y(and)c(ev)m(ery)h(time)f(y)m
|
|---|
| 2151 | (ou)h(c)m(hange)f(the)g(program.)-104 2794 y(Once)g(y)m(ou)h(ha)m(v)m
|
|---|
| 2152 | (e)g(started)f(the)g(in)m(terpreter,)h(y)m(ou)f(can)g(test)g(y)m(our)g
|
|---|
| 2153 | (program)g(or)f(edit)h(it.)-104 2914 y(Y)-8 b(ou)33 b(can)f(c)m(hec)m
|
|---|
| 2154 | (k)j(Prolog)d(out)h(b)m(y)g(t)m(yping)h(a)e(goal:)-104
|
|---|
| 2155 | 3076 y Fh(?-)52 b(voegsamen\([1,2,3,4,5],)57 b([6,7],)c(X\).)-104
|
|---|
| 2156 | 3238 y Fi(Presuming)34 b Fh(voegsamen)h Fi(is)e(de\014ned,)h(the)f(in)m
|
|---|
| 2157 | (terpreter)h(will)g(pro)s(duce:)-104 3399 y Fh(X)52 b(=)f
|
|---|
| 2158 | ([1,2,3,4,5,6,7])-104 3561 y Fi(No)m(w)38 b(there)g(are)f(t)m(w)m(o)h
|
|---|
| 2159 | (p)s(ossibilities:)54 b(y)m(ou)38 b(can)g(en)m(ter)g(a)f(';')i(to)e
|
|---|
| 2160 | (let)h(the)f(in)m(terpreter)i(explore)f(alternativ)m(e)-104
|
|---|
| 2161 | 3681 y(v)-5 b(alues)37 b(for)e(X.)h(If)g(none)g(are)g(found,)h(it)f
|
|---|
| 2162 | (will)h(pro)s(duce)f('No'.)54 b(By)36 b(just)g(pressing)i('en)m(ter')f
|
|---|
| 2163 | (the)f(in)m(terpreter)-104 3802 y(will)g(stop)g(searc)m(hing)g(for)f
|
|---|
| 2164 | (alternativ)m(e)h(v)-5 b(alues)36 b(for)f(X.)h(The)g(in)m(terpreter)g
|
|---|
| 2165 | (will)g(pro)s(duce)g(a)f('Y)-8 b(es')36 b(and)g(y)m(ou)-104
|
|---|
| 2166 | 3922 y(can)d(en)m(ter)g(a)g(new)g(command.)-104 4084
|
|---|
| 2167 | y(If)g(y)m(ou)g(w)m(an)m(t)g(to)g(edit)g(in)f Fh(vi)h
|
|---|
| 2168 | Fi(y)m(ou)h(can)f(en)m(ter:)-104 4245 y Fh(?-)52 b(edit)p
|
|---|
| 2169 | Fi(.)-104 4407 y(This)38 b(option)g(will)f(only)h(w)m(ork)g(if)f(y)m
|
|---|
| 2170 | (ou)h(run)g(Prolog)f(using)h(the)f('-f)7 b(')37 b(option.)57
|
|---|
| 2171 | b(Otherwise)39 b(y)m(ou'll)f(ha)m(v)m(e)h(to)-104 4527
|
|---|
| 2172 | y(en)m(ter:)-104 4689 y Fh(?-)52 b(edit\('myprog.pl'\).)-104
|
|---|
| 2173 | 4851 y Fi(NOTE:)41 5045 y Fg(\017)d Fi(Be)33 b(sure)g(to)g(use)g
|
|---|
| 2174 | (capital)g(letters)g(when)h(using)f(v)-5 b(ariables:)45
|
|---|
| 2175 | b(use)33 b('Name')g(instead)h(of)e('name'.)41 5240 y
|
|---|
| 2176 | Fg(\017)49 b Fi(When)38 b(using)g(functions,)i(DO)d(NOT)g(put)h(a)f
|
|---|
| 2177 | (space)i(b)s(et)m(w)m(een)g(the)f(function)g(name)g(and)f(the)h
|
|---|
| 2178 | (\014rst)140 5360 y(brac)m(k)m(et:)45 b Fh(voegsamen\(...\))i
|
|---|
| 2179 | Fi(instead)34 b(of)e Fh(voegsamen)53 b(\(...\))p Fi(.)41
|
|---|
| 2180 | 5555 y Fg(\017)c Fi(Be)33 b(sure)g(to)g(put)f(a)h('.')43
|
|---|
| 2181 | b(at)33 b(the)g(end)g(of)f(ev)m(ery)i(command,)g(otherwise)g(they)f(w)m
|
|---|
| 2182 | (on't)h(b)s(e)e(executed.)41 5750 y Fg(\017)49 b Fi(T)-8
|
|---|
| 2183 | b(o)33 b(use)g(the)g(implication)h(sym)m(b)s(ol)f(\()p
|
|---|
| 2184 | Fg( )p Fi(\))f(in)h(SWI-Prolog,)g(y)m(ou)g(ha)m(v)m(e)h(to)e(use)i(')p
|
|---|
| 2185 | Fh(:-)p Fi(':)140 5907 y Ff(bir)s(d)p Fi(\()p Ff(X)8
|
|---|
| 2186 | b Fi(\))60 b Fg( )g Ff(l)r(ay)t(s)p 883 5907 30 4 v 34
|
|---|
| 2187 | w(eg)t(g)t(s)p Fi(\()p Ff(X)8 b Fi(\))21 b Fg(^)h Ff(has)p
|
|---|
| 2188 | 1537 5907 V 36 w(w)s(ing)t(s)p Fi(\()p Ff(X)8 b Fi(\))31
|
|---|
| 2189 | b Fe(b)s(ecomes)140 6065 y Fh(bird\(X\))53 b(:-)f(lays)p
|
|---|
| 2190 | 914 6065 31 4 v 38 w(eggs\(X\),)h(has)p 1566 6065 V 38
|
|---|
| 2191 | w(wings\(X\).)1856 6129 y Fi(1)p eop
|
|---|
| 2192 | %%Page: 2 2
|
|---|
| 2193 | 2 1 bop -104 -150 a Fi(Y)-8 b(ou)33 b(can)f(exit)i(SWI-Prolog)e(bij)h
|
|---|
| 2194 | (en)m(tering)g(CTRL-C)h(follo)m(w)m(ed)f(b)m(y)h(an)e('e'.)-104
|
|---|
| 2195 | 180 y Fj(2)161 b(Exercises)-104 429 y Fd(Exercise)46
|
|---|
| 2196 | b(1:)60 b(Finite)46 b(Directed)g(Graphs)-104 613 y Fi(W)-8
|
|---|
| 2197 | b(e)28 b(shall)g(consider)h(three)f(problems)h(on)e(\014nite)h
|
|---|
| 2198 | (directed)h(graphs.)42 b(T)-8 b(o)28 b(this)g(end,)h(\014rst)f(some)g
|
|---|
| 2199 | (preliminary)-104 734 y(notions)k(on)g(graphs)g(are)g(giv)m(en.)45
|
|---|
| 2200 | b(Recall)32 b(that)f(a)h Fc(dir)-5 b(e)g(cte)g(d)34 b(gr)-5
|
|---|
| 2201 | b(aph)31 b Ff(G)h Fi(is)g(a)g(pair)g(\()p Ff(N)5 b(;)17
|
|---|
| 2202 | b(A)p Fi(\),)31 b(where)j Ff(N)42 b Fi(is)32 b(a)g(set)-104
|
|---|
| 2203 | 854 y(and)37 b Ff(A)g Fi(is)g(a)g(binary)h(relation)f(suc)m(h)h(that)f
|
|---|
| 2204 | Ff(A)e Fg(\022)g Ff(N)1809 818 y Fb(2)1849 854 y Fi(.)57
|
|---|
| 2205 | b(The)37 b(elemen)m(ts)j(of)c Ff(N)47 b Fi(are)37 b(called)h(the)f
|
|---|
| 2206 | Fc(no)-5 b(des)36 b Fi(of)h Ff(G)-104 974 y Fi(and)f(the)g(pairs)f
|
|---|
| 2207 | (whic)m(h)j(b)s(elong)d(to)g Ff(A)h Fi(are)f(called)i(the)e
|
|---|
| 2208 | Fc(ar)-5 b(cs)36 b Fi(of)f Ff(G)p Fi(.)52 b(If)35 b(the)h(set)g
|
|---|
| 2209 | Ff(N)46 b Fi(is)36 b(\014nite,)h(the)f(directed)-104
|
|---|
| 2210 | 1095 y(graph)h(is)h Fc(\014nite)p Fi(.)58 b(A)37 b(graph)g(is)h(called)
|
|---|
| 2211 | g(acyclic)h(if)e(it)g(con)m(tains)i(no)e(cycles.)60 b(D)m(A)m(G)37
|
|---|
| 2212 | b(stands)h(for)f(\\Directed)-104 1215 y(Acyclic)d(Graph".)42
|
|---|
| 2213 | 1336 y(Prolog)41 b(do)s(es)g(not)g(ha)m(v)m(e)h(an)m(y)f(built)g(in)g
|
|---|
| 2214 | (facilities)h(that)f(deal)g(with)g(graphs.)68 b(W)-8
|
|---|
| 2215 | b(e)42 b(represen)m(t)g(here)g(a)-104 1456 y(\014nite)32
|
|---|
| 2216 | b(directed)g(graph)f(\(in)g(short:)43 b(a)31 b Fc(gr)-5
|
|---|
| 2217 | b(aph)p Fi(\))30 b(b)m(y)i(a)e(\(ground\))h(list)h(of)e(its)i(arcs,)g
|
|---|
| 2218 | (where)g(an)f(arc)g(from)f(no)s(de)-104 1576 y Ff(a)39
|
|---|
| 2219 | b Fi(to)f(no)s(de)h Ff(b)g Fi(is)g(represen)m(ted)i(b)m(y)f(the)f(list)
|
|---|
| 2220 | g([)p Ff(a;)17 b(b)p Fi(].)63 b(In)39 b(this)g(represen)m(tation)h(the)
|
|---|
| 2221 | f(isolated)h(no)s(des)f(of)f(the)-104 1697 y(graph)h(are)f(omitted.)63
|
|---|
| 2222 | b(Ho)m(w)m(ev)m(er,)43 b(w)m(e)c(consider)i(here)e(only)g(problems)h
|
|---|
| 2223 | (dealing)f(with)h(paths)f(in)g(graphs,)-104 1817 y(and)33
|
|---|
| 2224 | b(consequen)m(tly)i(suc)m(h)f(a)f(\(mis\)represen)m(tation)i(is)e
|
|---|
| 2225 | (adequate)h(for)e(our)g(purp)s(oses.)42 1937 y(A)h Fc(p)-5
|
|---|
| 2226 | b(ath)35 b(in)f(a)h(gr)-5 b(aph)34 b Ff(g)k Fc(fr)-5
|
|---|
| 2227 | b(om)35 b Ff(a)g Fc(to)g Ff(b)e Fi(is)g(the)g(sequence)i
|
|---|
| 2228 | Ff(a)2143 1952 y Fb(1)2183 1937 y Ff(;)17 b(:)g(:)g(:)f(;)h(a)2453
|
|---|
| 2229 | 1952 y Fa(n)2532 1937 y Fi(\()p Ff(n)28 b(>)g Fi(1\))k(suc)m(h)i(that)
|
|---|
| 2230 | 42 2058 y Fg(\000)p Fi(\()p Ff(a)208 2073 y Fa(i)237
|
|---|
| 2231 | 2058 y Ff(;)17 b(a)332 2073 y Fa(i)p Fb(+1)450 2058 y
|
|---|
| 2232 | Fi(\))28 b Fg(2)g Ff(g)36 b Fi(for)c Ff(i)c Fg(2)g(f)p
|
|---|
| 2233 | Fi(1)p Ff(;)17 b Fi(2)p Ff(;)g(:)g(:)g(:)e(;)i(n)22 b
|
|---|
| 2234 | Fg(\000)h Fi(1)p Fg(g)p Fi(,)42 2178 y Fg(\000)p Ff(a)170
|
|---|
| 2235 | 2193 y Fb(1)238 2178 y Fi(=)28 b Ff(a)p Fi(,)42 2299
|
|---|
| 2236 | y Fg(\000)p Ff(a)170 2314 y Fa(n)246 2299 y Fi(=)f Ff(b)p
|
|---|
| 2237 | Fi(.)-104 2419 y(An)33 b Fc(acyclic)h(p)-5 b(ath)33 b
|
|---|
| 2238 | Fi(is)g(a)f(path)h(consisting)g(of)g(distinct)g(no)s(des.)15
|
|---|
| 2239 | 2626 y(1.)49 b(W)-8 b(e)28 b(b)s(egin)g(with)g(the)g(problem)g(of)f
|
|---|
| 2240 | (computing)i(the)f(transitiv)m(e)h(closure)g(of)e(a)g(D)m(A)m(G.)g(The)
|
|---|
| 2241 | i(transitiv)m(e)140 2746 y(closure)37 b(of)f(a)f(D)m(A)m(G)h(is)g
|
|---|
| 2242 | (obtained)h(b)m(y)g(adding)f(all)g(arcs)g(to)g(the)g(graph)g(that)g
|
|---|
| 2243 | (are)g(com)m(binations)h(of)140 2866 y(other)e(arcs,)g(for)f(example)i
|
|---|
| 2244 | (if)f([A,B],)g([B,C],)g([C,D])g(are)f(in)h(the)g(list)g(then)g(also)g
|
|---|
| 2245 | ([A,D].)f(Implemen)m(t)140 2987 y(the)f(follo)m(wing)g(predicate)h(b)m
|
|---|
| 2246 | (y)f(means)h(of)e(a)g(Prolog)g(program.)190 3182 y Fh(trans)p
|
|---|
| 2247 | 451 3182 31 4 v 38 w(dag\(X,)53 b(Y,)f(Graph\))101 b(:-)f
|
|---|
| 2248 | Fi(the)33 b(pair)g([X,Y])g(is)g(in)g(the)g(transitiv)m(e)1605
|
|---|
| 2249 | 3302 y(closure)h(of)e(the)h(D)m(A)m(G)f(Graph.)15 3493
|
|---|
| 2250 | y(2.)49 b(Next,)40 b(consider)g(the)f(general)f(case,)j(and)d(implemen)
|
|---|
| 2251 | m(t)i(the)f(follo)m(wing)f(predicate)i(b)m(y)f(means)g(of)f(a)140
|
|---|
| 2252 | 3614 y(Prolog)32 b(program.)190 3809 y Fh(trans\(X,)53
|
|---|
| 2253 | b(Y,)f(Graph\))101 b(:-)f Fi(the)33 b(pair)g([X,Y])g(is)g(in)g(the)g
|
|---|
| 2254 | (transitiv)m(e)1414 3929 y(closure)h(of)e(Graph.)140
|
|---|
| 2255 | 4126 y Fe(Hin)m(t:)40 b Fi(De\014ne)26 b(a)g(predicate)h
|
|---|
| 2256 | Fh(trans\(X,)53 b(Y,)f(Graph,)h(Avoids\))28 b Fi(that)e(uses)h(the)f
|
|---|
| 2257 | (argumen)m(t)h(Av)m(oids)140 4246 y(to)35 b(collect)h(the)f(list)h(of)e
|
|---|
| 2258 | (elemen)m(ts)k(that)d(should)h(b)s(e)f(a)m(v)m(oided)h(when)h(searc)m
|
|---|
| 2259 | (hing)f(a)f(path)g(from)g(X)g(to)140 4367 y(Y.)15 4565
|
|---|
| 2260 | y(3.)49 b(Finally)-8 b(,)50 b(consider)d(the)f(problem)h(of)e
|
|---|
| 2261 | (generating,)50 b(for)c(eac)m(h)g(pair)g(of)g(no)s(des)g(b)s(elonging)g
|
|---|
| 2262 | (to)g(the)140 4685 y(transitiv)m(e)40 b(closure,)i(a)c(path)h(whic)m(h)
|
|---|
| 2263 | h(connects)g(them.)63 b(In)39 b(general,)i(it)d(cannot)h(b)s(e)g
|
|---|
| 2264 | (claimed)h(that)140 4805 y(this)30 b(path)f(will)g(alw)m(a)m(ys)i(b)s
|
|---|
| 2265 | (e)e(acyclic,)i(b)s(ecause)f(pairs)g(of)e(the)i(form)e([)p
|
|---|
| 2266 | Ff(a;)17 b(a)p Fi(])29 b(can)h(b)s(elong)f(to)f(the)i(graph.)140
|
|---|
| 2267 | 4926 y(Ho)m(w)m(ev)m(er,)i(for)27 b(eac)m(h)j(pair)e(of)g(no)s(des)h(w)
|
|---|
| 2268 | m(e)h(can)e(alw)m(a)m(ys)i(\014nd)f(a)g(connecting)g(path)g
|
|---|
| 2269 | Ff(a)3139 4941 y Fb(1)3178 4926 y Ff(;)17 b(:)g(:)g(:)f(;)h(a)3448
|
|---|
| 2270 | 4941 y Fa(n)3523 4926 y Fi(\()p Ff(n)28 b(>)g Fi(1\),)140
|
|---|
| 2271 | 5046 y(whose)43 b Fc(tail)f Ff(a)667 5061 y Fb(2)706
|
|---|
| 2272 | 5046 y Ff(;)17 b(:)g(:)g(:)f(;)h(a)976 5061 y Fa(n)1065
|
|---|
| 2273 | 5046 y Fi(is)42 b(acyclic.)73 b(\(F)-8 b(or)41 b Ff(n)j
|
|---|
| 2274 | Fi(=)f(2)f(w)m(e)h(stipulate)f(here)h(that)f(a)f(sequence)k(of)c(one)
|
|---|
| 2275 | 140 5167 y(elemen)m(t)30 b(is)g(acyclic.\))43 b(Call)29
|
|---|
| 2276 | b(suc)m(h)i(a)d(path)h Fc(semi-acyclic)p Fi(.)41 b(Implemen)m(t)31
|
|---|
| 2277 | b(the)e(follo)m(wing)g(predicate)h(b)m(y)140 5287 y(means)k(of)e(a)g
|
|---|
| 2278 | (Prolog)g(program.)190 5482 y Fh(path\(X,)53 b(Y,)f(Graph,)h(Path\))100
|
|---|
| 2279 | b(:-)h Fi(P)m(ath)33 b(is)g(a)f(semi-acyclic)j(path)e(whic)m(h)1671
|
|---|
| 2280 | 5602 y(connects)h(X)e(and)h(Y)g(in)g(the)g(graph)f(Graph)140
|
|---|
| 2281 | 5799 y Fe(Hin)m(t:)40 b Fi(a)27 b(program)f(that)h(solv)m(es)i(the)e
|
|---|
| 2282 | (ab)s(o)m(v)m(e)h(problem)f(can)h(b)s(e)f(obtained)g(b)m(y)h(a)e(sligh)
|
|---|
| 2283 | m(t)i(mo)s(di\014cation)140 5920 y(of)45 b(the)h(relation)g
|
|---|
| 2284 | Fh(trans\(X,)53 b(Y,)f(Graph,)h(Z\))46 b Fi(obtained)g(b)m(y)h(adding)f
|
|---|
| 2285 | (an)g(argumen)m(t)g(to)f(it)h(\(i.e.,)140 6040 y Fh(trans\(X,)53
|
|---|
| 2286 | b(Y,)f(Graph,)h(Z,)f(Path\))p Fi(\))34 b(that)e(is)h(used)h(to)e
|
|---|
| 2287 | (incremen)m(tally)j(construct)f(a)e(path.)1856 6129 y(2)p
|
|---|
| 2288 | eop
|
|---|
| 2289 | %%Page: 3 3
|
|---|
| 2290 | 3 2 bop -104 -150 a Fd(Exercise)46 b(2:)60 b(Binary)45
|
|---|
| 2291 | b(Searc)l(h)g(T)-11 b(rees)-104 35 y Fi(W)j(e)34 b(shall)g(consider)h
|
|---|
| 2292 | (binary)f(trees)h(whose)g(no)s(des)f(are)f(\(lab)s(elled)i(with\))f
|
|---|
| 2293 | (natural)f(n)m(um)m(b)s(ers,)j(and)e(use)g(the)-104 156
|
|---|
| 2294 | y(term)g Ff(v)t(oid)f Fi(to)g(denote)h(the)g(empt)m(y)g(tree,)h(and)e
|
|---|
| 2295 | (the)h(term)g Fc(tr)-5 b(e)g(e\(x,)35 b(left,)h(right\))p
|
|---|
| 2296 | Fi(.)45 b(to)33 b(denote)h(the)g(tree)g(with)-104 276
|
|---|
| 2297 | y(ro)s(ot)k Ff(x)p Fi(,)k(left)d(subtree)i Fc(left)48
|
|---|
| 2298 | b Fi(and)40 b(righ)m(t)f(subtree)h Fc(right)p Fi(.)64
|
|---|
| 2299 | b(F)-8 b(or)38 b(example,)43 b(the)c(term)h Fc(tr)-5
|
|---|
| 2300 | b(e)g(e\(1,)42 b(tr)-5 b(e)g(e\(2,)42 b(void,)-104 396
|
|---|
| 2301 | y(void\),)34 b(tr)-5 b(e)g(e\(3,)35 b(void,)f(void\)\))h
|
|---|
| 2302 | Fi(represen)m(ts)g(the)e(tree)g(with)g(ro)s(ot)f Fc(1)46
|
|---|
| 2303 | b Fi(and)33 b(c)m(hildren)h Fc(2)46 b Fi(and)32 b Fc(3)p
|
|---|
| 2304 | Fi(.)42 517 y(W)-8 b(e)33 b(call)g(a)f(binary)i(tree)f
|
|---|
| 2305 | Fc(tr)-5 b(e)g(e\(x,)34 b(left,)h(right\))h(nic)-5 b(e)40
|
|---|
| 2306 | b Fi(if:)15 720 y(1.)49 b(if)32 b Fc(left)42 b Fi(is)33
|
|---|
| 2307 | b(not)g(empt)m(y)h(then)f Fc(x)44 b Fi(is)34 b(greater)e(than)h(all)f
|
|---|
| 2308 | (the)h(elemen)m(ts)i(in)e Fc(left)9 b Fi(;)15 923 y(2.)49
|
|---|
| 2309 | b(if)32 b Fc(right)42 b Fi(is)33 b(not)g(empt)m(y)h(then)f
|
|---|
| 2310 | Fc(x)44 b Fi(is)34 b(less)f(than)g(all)g(the)g(elemen)m(ts)h(in)f
|
|---|
| 2311 | Fc(right)p Fi(.)42 1127 y(A)i(binary)h(tree)g(is)f(called)h(a)f
|
|---|
| 2312 | Fc(se)-5 b(ar)g(ch)36 b(tr)-5 b(e)g(e)43 b Fi(if)34 b(ev)m(ery)j
|
|---|
| 2313 | (subtree)g(of)d(it)h(is)h(nice.)51 b(W)-8 b(rite)36 b(a)e(program)h
|
|---|
| 2314 | (whic)m(h)-104 1247 y(tests)f(whether)g(a)e(ground)h(term)g(is)g(a)f
|
|---|
| 2315 | (searc)m(h)i(tree.)42 1417 y Fe(Hin)m(t:)43 b Fi(Use)34
|
|---|
| 2316 | b(the)f(follo)m(wing)g(predicate)g Fh(is)p 1636 1417
|
|---|
| 2317 | 31 4 v 38 w(search)p 1980 1417 V 38 w(tree\(T\))i Fi(in)e(the)g
|
|---|
| 2318 | (de\014nition)g(of)f(searc)m(h)i(tree:)-54 1579 y Fh(is)p
|
|---|
| 2319 | 54 1579 V 37 w(search)p 397 1579 V 39 w(tree\(void\))p
|
|---|
| 2320 | Fi(.)-54 1700 y Fh(is)p 54 1700 V 37 w(search)p 397 1700
|
|---|
| 2321 | V 39 w(tree\(T\))282 b(:-)100 b(is)p 1379 1700 V 38 w(search)p
|
|---|
| 2322 | 1723 1700 V 38 w(tree\(T,)53 b(Min,)g(Max\))p Fi(.)42
|
|---|
| 2323 | 1863 y(where)29 b Fh(Min)f Fi(and)f Fh(Max)h Fi(are)f(the)h(minim)m(um)
|
|---|
| 2324 | h(and)e(maxim)m(um)i(elemen)m(t)g(of)d(the)i(tree)g Fh(T)p
|
|---|
| 2325 | Fi(.)f(Then)h(implemen)m(t)-104 1983 y(the)33 b(predicate)h
|
|---|
| 2326 | Fh(is)p 595 1983 V 37 w(search)p 938 1983 V 39 w(tree\(T,)53
|
|---|
| 2327 | b(Min,)f(Max\))p Fi(.)-104 2316 y Fj(3)161 b(Ho)l(w)53
|
|---|
| 2328 | b(to)g(submit)-104 2535 y Fi(Y)-8 b(our)30 b(programs)g(should)i(b)s(e)
|
|---|
| 2329 | e(submitted)i(together)e(with)h(a)f(written)h(rep)s(ort)f(in)h(whic)m
|
|---|
| 2330 | (h)h(y)m(ou)f(explain)g(y)m(our)-104 2656 y(programs,)i(to)f(Thijs)i(v)
|
|---|
| 2331 | -5 b(an)32 b(Ommen)i(\(m)m(v)m(ommen@liacs.nl\).)-104
|
|---|
| 2332 | 2988 y Fj(References)-104 3207 y Fi([1])49 b Fh(http://www.swi-)t
|
|---|
| 2333 | (prolog.org)q(/)1856 6129 y Fi(3)p eop
|
|---|
| 2334 | %%Trailer
|
|---|
| 2335 | end
|
|---|
| 2336 | userdict /end-hook known{end-hook}if
|
|---|
| 2337 | %%EOF
|
|---|
