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❲✐▼ ⑤ ❛ s✐♠♣❧❡ ❛❜str❛❝t ♠❛❝❤✐♥❡ ❢♦r ❧♦❣✐❝❛❧ ❧❛♥❣✉❛❣❡s

❚❤❡ ❧❛♥❣✉❛❣❡ Pr♦❧❧

❲❡ ❝♦♥s✐❞❡r ❛ s♠❛❧❧ ❧♦❣✐❝ ♣r♦❣r❛♠♠✐♥❣ ❧❛♥❣✉❛❣❡ Pr♦❧❧ ✭✧Pr♦❧♦❣✲❧✐❣❤t✧✮✳ ❈♦♠♣❛r❡❞ ✇✐t❤ Pr♦❧♦❣✱ ✇❡ ❞♦ ♥♦t tr❡❛t✿ ❛r✐t❤♠❡t✐❝ ♦♣❡r❛t✐♦♥s❀ t❤❡ ❝✉t ♦♣❡r❛t♦r❀ s❡❧❢✲♠♦❞✐☞❝❛t✐♦♥ ♦❢ ♣r♦❣r❛♠s ✉s✐♥❣ assert ❛♥❞ retract✳

❚❤❡ ❧❛♥❣✉❛❣❡ Pr♦❧❧

❆ ♣r♦❣r❛♠ ♣ ❤❛s t❤❡ ❢♦❧❧♦✇✐♥❣ s②♥t❛①✿ t ✿✿❂ ❛ ❥ ❳ ❥ ❥ ❢✭t✶❀ ✿ ✿ ✿ ❀ t♥✮ ❣ ✿✿❂ ♣✭t✶❀ ✿ ✿ ✿ ❀ t❦✮ ❥ ❳ ❂ t ❝ ✿✿❂ ♣✭❳✶❀ ✿ ✿ ✿ ❀ ❳❦✮ ✥ ❣✶❀ ✿ ✿ ✿ ❀ ❣r ♣ ✿✿❂ ❝✶ ✿ ✿ ✿ ❝♠ ❄❣ ❆ t❡r♠ t ✐s ❡✐t❤❡r ❛♥ ❛t♦♠ ✭✐❡✳ ❝♦♥st❛♥t✮✱ ❛ ✈❛r✐❛❜❧❡✱ ❛♥ ❛♥♦♥②♠♦✉s ✈❛r✐❛❜❧❡✱ ♦r ❛ ❝♦♥str✉❝t♦r ❛♣♣❧✐❝❛t✐♦♥✳ ❆ ❣♦❛❧ ❣ ✐s ❡✐t❤❡r ❧✐t❡r❛❧✱ ✐❡✳ ❛ ♣r❡❞✐❝❛t❡ ❝❛❧❧✱ ♦r ❛ ✉♥✐☞❝❛t✐♦♥✳ ❆ ❝❧❛✉s❡ ❝ ❤❛s ❛ ❤❡❛❞ ♣✭❳✶❀ ✿ ✿ ✿ ❀ ❳❦✮ ❛♥❞ ❛ ❜♦❞② ✭✐❡✳ ❛ s❡q✉❡♥❝❡ ♦❢ ❣♦❛❧s✮✳ ❆ ♣r♦❣r❛♠ ❝♦♥s✐sts ♦❢ s❡q✉❡♥❝❡ ♦❢ ❝❧❛✉s❡s t♦❣❡t❤❡r ✇✐t❤ ❛ q✉❡r② ✭✐❡✳ ❛ s✐♥❣❧❡ t♦♣✲❧❡✈❡❧ ❣♦❛❧✮✳

❚❤❡ ❧❛♥❣✉❛❣❡ Pr♦❧❧

❊①❛♠♣❧❡✿ bigger✭❳❀ ❨ ✮ ✥ ❳ ❂ ❡❧❡♣❤❛♥t❀ ❨ ❂ ❤♦rs❡ bigger✭❳❀ ❨ ✮ ✥ ❳ ❂ ❤♦rs❡❀ ❨ ❂ ❞♦♥❦❡② bigger✭❳❀ ❨ ✮ ✥ ❳ ❂ ❞♦♥❦❡②❀ ❨ ❂ ❞♦❣ bigger✭❳❀ ❨ ✮ ✥ ❳ ❂ ❞♦♥❦❡②❀ ❨ ❂ ♠♦♥❦❡② is bigger✭❳❀ ❨ ✮ ✥ bigger✭❳❀ ❨ ✮ is bigger✭❳❀ ❨ ✮ ✥ bigger✭❳❀ ❩✮❀ is bigger✭❩❀ ❨ ✮ ❄is bigger✭❡❧❡♣❤❛♥t❀ ❞♦❣✮

❚❤❡ ❧❛♥❣✉❛❣❡ Pr♦❧❧

❊①❛♠♣❧❡✿ app✭❳❀ ❨❀ ❩✮ ✥ ❳ ❂ ❬ ❪❀ ❨ ❂ ❩ app✭❳❀ ❨❀ ❩✮ ✥ ❳ ❂ ❬❍ ❥ ❳✵❪❀ ❩ ❂ ❬❍ ❥ ❩✵❪❀ app✭❳✵❀ ❨❀ ❩✵✮ ❄app✭❳❀ ❬❨❀ ❝❪❀ ❬❛❀ ❜❀ ❩❪✮ ❬ ❪ t❤❡ ❛t♦♠ ❞❡♥♦t✐♥❣ ❛♥ ❡♠♣t② ❧✐st❀ ❬❍ ❥ ❩❪ ❛ ❜✐♥❛r② ❧✐st ❝♦♥str✉❝t♦r ❛♣♣❧✐❝❛t✐♦♥❀ ❬❛❀ ❜❀ ❩❪ ✐s ❛ s❤♦rt❤❛♥❞ ♦❢ ❬❛ ❥ ❬❜ ❥ ❬❩ ❥ ❬ ❪❪❪❪✳

❲✐▼ ❛r❝❤✐t❡❝t✉r❡

❈♦❞❡✿

P❈

✶ ✵

❈ ❈ ❂ ❈♦❞❡✲st♦r❡ ⑤ ♠❡♠♦r② ❛r❡❛ ❢♦r ❛ ♣r♦❣r❛♠ ❝♦❞❡❀ ❡❛❝❤ ❝❡❧❧ ❝♦♥t❛✐♥s ❛ s✐♥❣❧❡ ❆▼ ✐♥str✉❝t✐♦♥✳ P❈ ❂ Pr♦❣r❛♠ ❈♦✉♥t❡r ⑤ r❡❣✐st❡r ❝♦♥t❛✐♥✐♥❣ ❛♥ ❛❞❞r❡ss ♦❢ t❤❡ ✐♥str✉❝t✐♦♥ t♦ ❜❡ ❡①❡❝✉t❡❞ ♥❡①t✳ ■♥✐t✐❛❧❧②✱ P❈ ❝♦♥t❛✐♥s t❤❡ ❛❞❞r❡ss ✵❀ ✐❡✳ ❈❬✵❪ ❝♦♥t❛✐♥s t❤❡ ☞rst ✐♥str✉❝t✐♦♥ ♦❢ t❤❡ ♣r♦❣r❛♠✳

❲✐▼ ❛r❝❤✐t❡❝t✉r❡

❙t❛❝❦✿

❙P ❋P

✶ ✵

❙ ❙ ❂ ❙t❛❝❦ ⑤ ❡❛❝❤ ❝❡❧❧ ❝♦♥t❛✐♥s ❛ ♣r✐♠✐t✐✈❡ ✈❛❧✉❡ ♦r ❛♥ ❛❞❞r❡ss❀ ❙P ❂ ❙t❛❝❦✲P♦✐♥t❡r ⑤ ♣♦✐♥ts t♦ t♦♣ ♦❢ t❤❡ st❛❝❦❀ ❋P ❂ ❋r❛♠❡✲P♦✐♥t❡r ⑤ ♣♦✐♥ts t♦ t❤❡ ❝✉rr❡♥t❧② ❛❝t✐✈❡ ❢r❛♠❡✳

❲✐▼ ❛r❝❤✐t❡❝t✉r❡

❍❡❛♣✿

❍P

✶ ✵

❍ ❍ ❂ ❍❡❛♣ ⑤ ♠❡♠♦r② ❛r❡❛ ❢♦r ❞②♥❛♠✐❝❛❧❧② ❛❧❧♦❝❛t❡❞ ❞❛t❛❀ ❍P ❂ ❍❡❛♣✲P♦✐♥t❡r ⑤ ♣♦✐♥ts t♦ t❤❡ ☞rst ❢r❡❡ ❝❡❧❧✳ ❚❤❡ ✐♥str✉❝t✐♦♥ new ❝r❡❛t❡s ❛ ♥❡✇ ♦❜❥❡❝t ✐♥ t❤❡ ❤❡❛♣✳ ❖❜❥❡❝ts ❛r❡ t❛❣❣❡❞ ✇✐t❤ t❤❡✐r t②♣❡s ✭❧✐❦❡ ✐♥ ▼❛▼❛✮✳

❲✐▼ ❛r❝❤✐t❡❝t✉r❡

❍❡❛♣ ♠❛② ❝♦♥t❛✐♥ ❢♦❧❧♦✇✐♥❣ ♦❜❥❡❝ts✿

❆t♦♠ ❱❛r✐❛❜❧❡ ❯♥❜♦✉♥❞ ❱❛r✐❛❜❧❡ ❙tr✉❝t✉r❡ ✶ ❝❡❧❧ ✶ ❝❡❧❧ ✶ ❝❡❧❧ ♥✰✶ ❝❡❧❧s

❆ ❛ ❘ ❘ ❙ ❢✴♥

❈♦♥str✉❝t✐♦♥ ♦❢ ❚❡r♠s

❇❡❢♦r❡ ♣❛r❛♠❡t❡rs ❛r❡ ♣❛ss❡❞ t♦ ❣♦❛❧s✱ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ t❡r♠s ❛r❡ ❝♦♥str✉❝t❡❞ ✐♥ t❤❡ ❤❡❛♣✳ ❚❤❡ ❛❞❞r❡ss ❡♥✈✐r♦♥♠❡♥t ✚ ❜✐♥❞s ❡❛❝❤ ❝❧❛✉s❡ ✈❛r✐❛❜❧❡ ❳ ✇✐t❤ ✐ts ❛❞❞r❡ss ✐♥ t❤❡ st❛❝❦ ✭r❡❧❛t✐✈❡ ♦❢ ❋P✮✳ ❈♦♥str✉❝t✐♦♥ ♦❢ t❡r♠s ✐s ♣❡r❢♦r♠❡❞ ❜② ❢✉♥❝t✐♦♥ ❝♦❞❡❆ t ✚✱ ✇❤✐❝❤✿ ❝r❡❛t❡s ❛ tr❡❡ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ t❡r♠ t ✐♥ t❤❡ ❤❡❛♣ ❀ r❡t✉r♥s ❛ ♣♦✐♥t❡r t♦ ✐t ♦♥ t♦♣ ♦❢ t❤❡ st❛❝❦✳

❈♦♥str✉❝t✐♦♥ ♦❢ ❚❡r♠s

❊①❛♠♣❧❡✿ ❘❡♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ t❡r♠ t ✑ ❢✭❣✭❳❀ ❨ ✮❀ ❛❀ ❩✮✱ ✇❤❡r❡ ❳ ✐s ❛♥ ✐♥✐t✐❛❧✐③❡❞ ✈❛r✐❛❜❧❡✱ ❛♥❞ ❨ ❛♥❞ ❩ ❛r❡ ♥♦t ②❡t ✐♥✐t✐❛❧✐③❡❞✳

❙ ❢✴✸ ❘ ❆ ❛ ❙ ❣✴✷ ❘ ❘ ❳

❈♦♥str✉❝t✐♦♥ ♦❢ ❚❡r♠s

❝♦❞❡❆ ❛ ✚ ❂ putatom a ❝♦❞❡❆ ❳ ✚ ❂ putvar ✭✚ ❳✮ ❝♦❞❡❆ ✖ ❳ ✚ ❂ putref ✭✚ ❳✮ ❝♦❞❡❆ ✚ ❂ putanon ❝♦❞❡❆ ❢✭t✶❀ ✿ ✿ ✿ ❀ t♥✮ ✚ ❂ ❝♦❞❡❆ t✶ ✚ ✳ ✳ ✳ ❝♦❞❡❆ t♥ ✚ putstruct f/n ✇❤❡r❡ ❳ ✐s ❛♥ ✉♥✐♥✐t✐❛❧✐③❡❞ ❛♥❞ ✖ ❳ ✐s ❛♥ ✐♥✐t✐❛❧✐③❡❞ ✈❛r✐❛❜❧❡✳ ❊①❛♠♣❧❡✿ ❧❡t t ✑ ❢✭❣✭ ✖ ❳❀ ❨ ✮❀ ❛❀ ❩✮ ❛♥❞ ✚ ❂ ❢❳ ✼✦ ✶❀ ❨ ✼✦ ✷❀ ❩ ✼✦ ✸❣✱ t❤❡♥ ❝♦❞❡❆ t ✚ ❡♠✐ts t❤❡ ❝♦❞❡✿ putref 1 putatom a putvar 2 putvar 3 putstruct g/2 putstruct f/3

❈♦♥str✉❝t✐♦♥ ♦❢ ❚❡r♠s

♣✉t❛t♦♠ ❛ ❛ ❆

SP++; S[SP] = new (A,a);

❈♦♥str✉❝t✐♦♥ ♦❢ ❚❡r♠s

✐

♣✉t✈❛r ✐ ❋P ❋P ❘

SP++; S[SP] = new (R,HP); S[FP+i] = S[SP];

❈♦♥str✉❝t✐♦♥ ♦❢ ❚❡r♠s

♣✉t❛♥♦♥ ❘

SP++; S[SP] = new (R,HP);

❈♦♥str✉❝t✐♦♥ ♦❢ ❚❡r♠s

✐

♣✉tr❡❢ ✐ ❋P ❋P

SP++; S[SP] = deref (S[FP+i]);

❚❤❡ ❛✉①✐❧✐❛r② ❢✉♥❝t✐♦♥ deref ❝♦♥tr❛❝ts ❝❤❛✐♥s ♦❢ r❡❢❡r❡♥❝❡s✿ ref deref (ref v) { if (H[v] ❂ (R,w) && v ✻❂ w) return deref(w); else return v; }

❈♦♥str✉❝t✐♦♥ ♦❢ ❚❡r♠s

♣✉tstr✉❝t ❢✴♥

♥

❙ ❢✴♥

v = new (S,f,n); SP = SP - n + 1; for (i=1; i✔n; i++) H[v+i] = SP[SP+i-1]; S[SP] = v;

❈♦♥str✉❝t✐♦♥ ♦❢ ❚❡r♠s

❘❡♠❛r❦s✿ ❚❤❡ ✐♥str✉❝t✐♦♥ putref i ♥♦t ♦♥❧② ❝♦♣✐❡s ❛ r❡❢❡r❡♥❝❡ ❢r♦♠ S[FP+i]✱ ❜✉t ❛❧s♦ ❞❡r❡❢❡r❡♥❝❡s ✐t ❛s ♠✉❝❤ ❛s ♣♦ss✐❜❧❡✳ ❉✉r✐♥❣ t❡r♠ ❝♦♥str✉❝t✐♦♥ r❡❢❡r❡♥❝❡s ❛❧✇❛②s ♣♦✐♥t t♦ s♠❛❧❧❡r ❤❡❛♣ ❛❞❞r❡ss❡s✳ ❊✈❡♥ t❤♦✉❣❤✱ t❤✐s ✐s ❛❧s♦ ❝❛s❡ ✐♥ ♠❛♥② ♦t❤❡r s✐t✉❛t✐♦♥s✱ ✐t ✐s ♥♦t ❣✉❛r❛♥t❡❡❞ ✐♥ ❣❡♥❡r❛❧✳

❚r❛♥s❧❛t✐♦♥ ♦❢ ●♦❛❧s

• ♦❛❧s ❝♦rr❡s♣♦♥❞ t♦ ♣r♦❝❡❞✉r❡ ❝❛❧❧s✳

❚❤❡✐r tr❛♥s❧❛t✐♦♥ ✐s ♣❡r❢♦r♠❡❞ ❜② t❤❡ ❢✉♥❝t✐♦♥ ❝♦❞❡●✳ ❋✐rst ❝r❡❛t❡ ❛ st❛❝❦ ❢r❛♠❡✳ ❚❤❡♥ ❝♦♥str✉❝t t❤❡ ❛❝t✉❛❧ ♣❛r❛♠❡t❡rs ✐♥ t❤❡ ❤❡❛♣ ✳ ✳ ✳ ❛♥❞ st♦r❡ r❡❢❡r❡♥❝❡s t♦ t❤❡s❡ ✐♥t♦ t❤❡ st❛❝❦ ❢r❛♠❡✳ ❋✐♥❛❧❧②✱ ❥✉♠♣ t♦ t❤❡ ❝♦❞❡ ♦❢ t❤❡ ♣r❡❞✐❝❛t❡✳

❚r❛♥s❧❛t✐♦♥ ♦❢ ●♦❛❧s

❝♦❞❡● ♣✭t✶❀ ✿ ✿ ✿ ❀ t❦✮ ✚ ❂ mark A ❝♦❞❡❆ t✶ ✚ ✳ ✳ ✳ ❝♦❞❡❆ t❦ ✚ call p/k ❆✿ ✳ ✳ ✳ ❊①❛♠♣❧❡✿ ❧❡t ❣ ✑ ♣✭❛❀ ❳❀ ❣✭ ✖ ❳❀ ❨ ✮✮ ❛♥❞ ✚ ❂ ❢❳ ✼✦ ✶❀ ❨ ✼✦ ✷❣✱ t❤❡♥ ❝♦❞❡● ❣ ✚ ❡♠✐ts t❤❡ ❝♦❞❡✿ mark A putatom a putvar 1 putref 1 putvar 2 putstruct g/2 call p/3 ❆✿ ✳ ✳ ✳

❚r❛♥s❧❛t✐♦♥ ♦❢ ●♦❛❧s

❙tr✉❝t✉r❡ ♦❢ ❛ ❢r❛♠❡✿ ❋P ❙P

✲✶ ✲✷

❛r❣✉♠❡♥ts ❧♦❝❛❧ st❛❝❦

✵ ✲✺ ✲✹ ✲✸

♦r❣❛♥✐③❛t✐♦♥❛❧ ❝❡❧❧s

❋P♦❧❞ P♦s❈♦♥t

❚r❛♥s❧❛t✐♦♥ ♦❢ ●♦❛❧s

❘❡♠❛r❦s✿ ❚❤❡ ♣♦s✐t✐✈❡ ❝♦♥t✐♥✉❛t✐♦♥ ❛❞❞r❡ss PosCont r❡❝♦r❞s ✇❤❡r❡ t♦ ❝♦♥t✐♥✉❡ ❛❢t❡r s✉❝❝❡ss❢✉❧ tr❡❛t♠❡♥t ♦❢ t❤❡ ❣♦❛❧✳ ❆❞❞✐t✐♦♥❛❧ ♦r❣❛♥✐③❛t✐♦♥❛❧ ❝❡❧❧s ❛r❡ ♥❡❝❡ss❛r② ❢♦r ❜❛❝❦tr❛❝❦✐♥❣✳

❚r❛♥s❧❛t✐♦♥ ♦❢ ●♦❛❧s

♠❛r❦ ❆

✻

❋P ❋P ❆

SP = SP + 6; S[SP] = A; S[SP-1] = FP;

❚r❛♥s❧❛t✐♦♥ ♦❢ ●♦❛❧s

❝❛❧❧ ♣✴♥

♥

♣✴♥

❋P P❈

FP = SP - n; PC = p/n;

❯♥✐☞❝❛t✐♦♥

❲❡ ❞❡♥♦t❡ ♦❝❝✉rr❡♥❝❡s ♦❢ ❛ ✈❛r✐❛❜❧❡ ❳ ❜② ⑦ ❳✳ ■t ✇✐❧❧ ❜❡ tr❛♥s❧❛t❡❞ ❞✐☛❡r❡♥t❧② ❞❡♣❡♥❞✐♥❣ ✇❤❡t❤❡r ✐t✬s ✐♥✐t✐❛❧✐③❡❞ ♦r ♥♦t✳ ❲❡ ✐♥tr♦❞✉❝❡ t❤❡ ♠❛❝r♦ ♣✉t ⑦ ❳ ✚✿ ♣✉t ❳ ✚ ❂ putvar ✭✚❳✮ ♣✉t ✖ ❳ ✚ ❂ putref ✭✚❳✮ ♣✉t ✚ ❂ putanon

❯♥✐☞❝❛t✐♦♥

❚r❛♥s❧❛t✐♦♥ ♦❢ t❤❡ ✉♥✐☞❝❛t✐♦♥ ⑦ ❳ ❂ t✿ ♣✉s❤ ❛ r❡❢❡r❡♥❝❡ t♦ ❳ ♦♥t♦ t❤❡ st❛❝❦❀ ❝♦♥str✉❝t t❤❡ t❡r♠ t ✐♥ t❤❡ ❤❡❛♣❀ ✐♥tr♦❞✉❝❡ ❛ ♥❡✇ ✐♥str✉❝t✐♦♥ ✇❤✐❝❤ ✐♠♣❧❡♠❡♥ts t❤❡ ✉♥✐☞❝❛t✐♦♥✳ ❝♦❞❡● ✭ ⑦ ❳ ❂ t✮ ✚ ❂ ♣✉t ⑦ ❳ ✚ ❝♦❞❡❆ t ✚ unify

❯♥✐☞❝❛t✐♦♥

❊①❛♠♣❧❡✿ ❝♦♥s✐❞❡r t❤❡ ❡q✉❛t✐♦♥ ✖ ❯ ❂ ❢✭❣✭ ✖ ❳❀ ❨ ✮❀ ❛❀ ❩✮ ❚❤❡♥✱ ❣✐✈❡♥ ❛♥ ❛❞❞r❡ss ❡♥✈✐r♦♥♠❡♥t ✚ ❂ ❢❳ ✼✦ ✶❀ ❨ ✼✦ ✷❀ ❩ ✼✦ ✸❀ ❯ ✼✦ ✹❣ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦❞❡ ✐s ❣❡♥❡r❛t❡❞✿ putref 4 putref 1 putvar 2 putstruct g/2 putatom a putvar 3 putstruct f/3 unify

❯♥✐☞❝❛t✐♦♥

■♥str✉❝t✐♦♥ unify ❛♣♣❧✐❡s t❤❡ r✉♥✲t✐♠❡ ❢✉♥❝t✐♦♥ unify() t♦ t❤❡ t♦♣♠♦st t✇♦ r❡❢❡r❡♥❝❡s✿ ✉♥✐❢②

unify (S[SP-1], S[SP-2]); SP = SP - 2;

❯♥✐☞❝❛t✐♦♥

❋✉♥❝t✐♦♥ unify() ✳ ✳ ✳ t❛❦❡s t✇♦ ❤❡❛♣ ❛❞❞r❡ss❡s✳ ❋♦r ❡❛❝❤ ❝❛❧❧ ✇❡ ❣✉❛r❛♥t❡❡ t❤❛t t❤❡s❡ ❛r❡ ♠❛①✐♠❛❧❧② ❞❡r❡❢❡r❡♥❝❡❞✳ ✳ ✳ ✳ ❝❤❡❝❦s ✇❤❡t❤❡r t❤❡ t✇♦ ❛❞❞r❡ss❡s ❛r❡ ❛❧r❡❛❞② ✐❞❡♥t✐❝❛❧✳ ■♥ t❤❛t ❝❛s❡ ❞♦❡s ♥♦t❤✐♥❣ ❛♥❞ t❤❡ ✉♥✐☞❝❛t✐♦♥ s✉❝❝❡❡❞❡❞✳ ✳ ✳ ✳ ❜✐♥❞s ②♦✉♥❣❡r ✈❛r✐❛❜❧❡s ✭❧❛r❣❡r ❛❞❞r❡ss❡s✮ t♦ ♦❧❞❡r ✈❛r✐❛❜❧❡s ✭s♠❛❧❧❡r ❛❞❞r❡ss❡s✮✳ ✳ ✳ ✳ ✇❤❡♥ ❜✐♥❞✐♥❣ ❛ ✈❛r✐❛❜❧❡ t♦ ❛ t❡r♠✱ ❝❤❡❝❦s ✇❤❡t❤❡r t❤❡ ✈❛r✐❛❜❧❡ ♦❝❝✉rs ✐♥s✐❞❡ t❤❡ t❡r♠ ✭♦❝❝✉r✲❝❤❡❝❦✮✳ ✳ ✳ ✳ r❡❝♦r❞s ♥❡✇❧② ❝r❡❛t❡❞ ❜✐♥❞✐♥❣s✳ ✳ ✳ ✳ ♠❛② ❢❛✐❧✱ ✐♥ ✇❤✐❝❤ ❝❛s❡ ✐♥✐t✐❛t❡s ❜❛❝❦tr❛❝❦✐♥❣✳

❯♥✐☞❝❛t✐♦♥

bool unify (ref u, ref v) { if (u == v) return true; if (H[u] == (R, )) { if (H[v] == (R, )) { if (u > v) { H[u] = (R,v); trail(u); return true; } else { H[v] = (R,u); trail(v); return true; } } else if (check (u,v)) { H[u] = (R,v); trail(u); return true; } else { backtrack(); return false; } } ...

❯♥✐☞❝❛t✐♦♥

... if (H[v] == (R, )) { if (check (v,u)) { H[v] = (R,u); trail(v); return true; } else { backtrack(); return false; } } if (H[u] == (A,a) && H[v] == (A,a)) return true; if (H[u] == (S,f/n) && H[v] == (S,f/n)) { for (int i=1; i<=n; i++) if (!unify (deref(H[u+i]), deref(H[v+i]))) return false; return true; } backtrack(); return false; }

❯♥✐☞❝❛t✐♦♥

❙ ❢✴✷ ❙ ❢✴✷ ❘ ❘ ❘ ❘ ❆ ❛

❯♥✐☞❝❛t✐♦♥

❙ ❢✴✷ ❘ ❘ ❘ ❘ ❆ ❛ ❙ ❢✴✷

❯♥✐☞❝❛t✐♦♥

❙ ❢✴✷ ❙ ❢✴✷ ❘ ❘ ❘ ❘ ❆ ❛

❯♥✐☞❝❛t✐♦♥

❙ ❢✴✷ ❙ ❢✴✷ ❘ ❘ ❘ ❘ ❆ ❛

❯♥✐☞❝❛t✐♦♥

❙ ❢✴✷ ❙ ❢✴✷ ❘ ❘ ❘ ❘ ❆ ❛

❯♥✐☞❝❛t✐♦♥

❚❤❡ ❢✉♥❝t✐♦♥ trail() r❡❝♦r❞s ♥❡✇ ❜✐♥❞✐♥❣s✳ ❚❤❡ ❢✉♥❝t✐♦♥ backtrack() ✐♥✐t✐❛t❡s ❜❛❝❦tr❛❝❦✐♥❣✳ ❚❤❡ ❢✉♥❝t✐♦♥ check() ♣❡r❢♦r♠s t❤❡ ♦❝❝✉r✲❝❤❡❝❦❀ ✐❡✳ t❡sts ✇❤❡t❤❡r ❛ ✈❛r✐❛❜❧❡ ✭✐ts ☞rst ❛r❣✉♠❡♥t✮ ♦❝❝✉rs ✐♥s✐❞❡ ❛ t❡r♠ ✭✐ts s❡❝♦♥❞ ❛r❣✉♠❡♥t✮✳ ❖❢t❡♥✱ t❤✐s ❝❤❡❝❦ ✐s s❦✐♣♣❡❞✿ bool check (ref u, ref v) { return true; }

❯♥✐☞❝❛t✐♦♥

❖t❤❡r✇✐s❡✱ ✇❡ ❝♦✉❧❞ ✐♠♣❧❡♠❡♥t check() ❛s ❢♦❧❧♦✇s✿ bool check (ref u, ref v) { if (u == v) return false; if (H[v] == (S,f/n)) for (int i=1; i<=n; i++) if (!check (u, deref (H[v+i]))) return false; return true; }

❯♥✐☞❝❛t✐♦♥

❚❤❡ tr❛♥s❧❛t✐♦♥ ♦❢ ❛♥ ❡q✉❛t✐♦♥ ⑦ ❳ ❂ t ✐s ✈❡r② s✐♠♣❧❡✱ ❜✉t ❛❧❧ t❤❡ ♦❜❥❡❝ts ❝♦♥str✉❝t❡❞ t♦ r❡♣r❡s❡♥t t ✇❤✐❝❤ ❤❛✈❡ ❝♦rr❡s♣♦♥❞✐♥❣ ♠❛t❝❤✐♥❣ ♦❜❥❡❝t r❡❛❝❤❛❜❧❡ ❢r♦♠ ❳ ❜❡❝♦♠❡s ✐♠♠❡❞✐❛t❡❧② ❣❛r❜❛❣❡✳ ■❞❡❛✿ ④ P✉s❤ ❛ r❡❢❡r❡♥❝❡ t♦ t❤❡ r✉♥✲t✐♠❡ ❜✐♥❞✐♥❣ ♦❢ ⑦ ❳ ♦♥t♦ t❤❡ st❛❝❦✳ ④ ❆✈♦✐❞ ❝♦♥str✉❝t✐♦♥ ♦❢ s✉❜t❡r♠s ♦❢ t ❛s ❧♦♥❣ ❛s ♣♦ss✐❜❧❡✳ ④ ■♥st❡❛❞✱ tr❛♥s❧❛t❡ ❡❛❝❤ ♥♦❞❡ ♦❢ t ✐♥t♦ ❛♥ ✐♥str✉❝t✐♦♥ ✇❤✐❝❤ ♣❡r❢♦r♠s t❤❡ ✉♥✐☞❝❛t✐♦♥ ✇✐t❤ t❤✐s ♥♦❞❡✦ ❝♦❞❡● ✭ ⑦ ❳ ❂ t✮ ✚ ❂ ♣✉t ⑦ ❳ ✚ ❝♦❞❡❯ t ✚

❯♥✐☞❝❛t✐♦♥

❯♥✐☞❝❛t✐♦♥ ♦❢ ❛t♦♠s ❛♥❞ ✈❛r✐❛❜❧❡s✿ ❝♦❞❡❯ ❛ ✚ ❂ uatom a ❝♦❞❡❯ ❳ ✚ ❂ uvar ✭✚❳✮ ❝♦❞❡❯ ✖ ❳ ✚ ❂ uref ✭✚❳✮ ❝♦❞❡❯ ✚ ❂ pop

❯♥✐☞❝❛t✐♦♥

■♥str✉❝t✐♦♥ uatom a ✐♠♣❧❡♠❡♥ts t❤❡ ✉♥✐☞❝❛t✐♦♥ ✇✐t❤ ❛♥ ❛t♦♠✿ ✉❛t♦♠ ❛ ❘ ❛ ❆ ❘

v = S[SP]; SP--; switch (H[v]) { case (A,a): break; case (R, ): H[v] = (R, new(A,a)); trail(v); break; default: backtrack(); }

❯♥✐☞❝❛t✐♦♥

■♥str✉❝t✐♦♥ uvar i ✐♠♣❧❡♠❡♥ts t❤❡ ✉♥✐☞❝❛t✐♦♥ ✇✐t❤ ❛♥ ✉♥✐♥✐t✐❛❧✐③❡❞ ✈❛r✐❛❜❧❡✿

✐

✉✈❛r ✐

✐

❋P ❋P

S[FP+i] = S[SP]; SP--;

❯♥✐☞❝❛t✐♦♥

■♥str✉❝t✐♦♥ pop ✐♠♣❧❡♠❡♥ts t❤❡ ✉♥✐☞❝❛t✐♦♥ ✇✐t❤ ❛♥ ❛♥♦♥②♠♦✉s ✈❛r✐❛❜❧❡✿ ♣♦♣

SP--;

❯♥✐☞❝❛t✐♦♥

■♥str✉❝t✐♦♥ uref i ✐♠♣❧❡♠❡♥ts t❤❡ ✉♥✐☞❝❛t✐♦♥ ✇✐t❤ ❛♥ ✐♥✐t✐❛❧✐③❡❞ ✈❛r✐❛❜❧❡✿

✐

✉r❡❢ ✐

✐

❋P ❋P ❨ ❳ ✒ ✒ ❂ ♠❣✉✭❳❀ ❨ ✮

unify (S[SP], deref(S[FP+i])); SP--;

❚❤❡ ♦♥❧② ♣❧❛❝❡✱ ✇❤❡r❡ t❤❡ r✉♥✲t✐♠❡ ❢✉♥❝t✐♦♥ unify() ✐s ❝❛❧❧❡❞✦

❯♥✐☞❝❛t✐♦♥

❯♥✐☞❝❛t✐♦♥ ♦❢ ❝♦♥str✉❝t♦r ❛♣♣❧✐❝❛t✐♦♥s✿ ❚❤❡ ✉♥✐☞❝❛t✐♦♥ ❝♦❞❡ ♣❡r❢♦r♠s ❛ ♣r❡✲♦r❞❡r tr❛✈❡rs❛❧ ♦✈❡r t✳ ❋✐rst ✐t ❝❤❡❝❦s ✇❤❡t❤❡r t❤❡ r♦♦t ♥♦❞❡ ✐s ✉♥✐☞❛❜❧❡✳ ■❢ ❜♦t❤ t❡r♠s ❤❛✈❡ t❤❡ s❛♠❡ t♦♣♠♦st ❝♦♥str✉❝t♦r✱ t❤❡♥ r❡❝✉rs✐✈❡❧② ❝❤❡❝❦s ❢♦r s✉❜t❡r♠s✳ ■♥ t❤❡ ❝❛s❡ ♦❢ ❛♥ ✉♥✐♥✐t✐❛❧✐③❡❞ ✈❛r✐❛❜❧❡ s✇✐t❝❤❡s ❢r♦♠ ❝❤❡❝❦✐♥❣ t♦ ❜✉✐❧❞✐♥❣✳

❯♥✐☞❝❛t✐♦♥

❯♥✐☞❝❛t✐♦♥ ♦❢ ❝♦♥str✉❝t♦r ❛♣♣❧✐❝❛t✐♦♥s✿ ❝♦❞❡❯ ✭❢✭t✶❀ ✿ ✿ ✿ ❀ t♥✮✮ ✚ ❂ ustruct f/n A son 1 ❝♦❞❡❯ t✶ ✚ ✳ ✳ ✳ son n ❝♦❞❡❯ t♥ ✚ up B ❆✿ ❝❤❡❝❦ ✐✈❛rs✭❢✭t✶❀ ✿ ✿ ✿ ❀ t♥✮✮ ✚ ❝♦❞❡❆ ✭❢✭t✶❀ ✿ ✿ ✿ ❀ t♥✮✮ ✚ bind ❇✿ ✳ ✳ ✳

❯♥✐☞❝❛t✐♦♥

❆ ✉str✉❝t ❢✴♥ ❆ ✉str✉❝t ❢✴♥ ❆ P❈ ❘ ❘ ❙ ❢✴♥ ❙ ❢✴♥

switch (H[S[SP]]) { case (S,f/n): break; case (R, ): PC = A; break; default: backtrack(); }

❯♥✐☞❝❛t✐♦♥

■♥str✉❝t✐♦♥ son i ♣✉s❤❡s t❤❡ r❡❢❡r❡♥❝❡ ♦❢ t❤❡ ✐✲t❤ s✉❜t❡r♠ ♦♥t♦ t❤❡ st❛❝❦✿ s♦♥ ✐

✐ ✐

❙ ❢✴♥ ❙ ❢✴♥

S[SP+1] = deref (H[S[SP]]+i); SP++;

❯♥✐☞❝❛t✐♦♥

■♥str✉❝t✐♦♥ up A ♣♦♣s ❛ r❡❢❡r❡♥❝❡ ❢r♦♠ t❤❡ st❛❝❦ ❛♥❞ ❥✉♠♣s t♦ t❤❡ ❝♦♥t✐♥✉❛t✐♦♥ ❛❞❞r❡ss✿ ❆ P❈ ✉♣ ❆

SP--; PC = A;

❯♥✐☞❝❛t✐♦♥

■♥ t❤❡ ❝❛s❡ ♦❢ ❛♥ ✉♥✐♥✐t✐❛❧✐③❡❞ ✈❛r✐❛❜❧❡ ✇❡ ♥❡❡❞ t♦ s✇✐t❝❤ ❢r♦♠ ❝❤❡❝❦✐♥❣ t♦ ❜✉✐❧❞✐♥❣✳ ❇❡❢♦r❡ ❝♦♥str✉❝t✐♥❣ t❤❡ ♥❡✇ t❡r♠ ✇❡ ♥❡❡❞ t♦ ❡①❝❧✉❞❡ t❤❛t ✐t ❝♦♥t❛✐♥s t❤❡ ✈❛r✐❛❜❧❡ ♦♥ t♦♣ ♦❢ t❤❡ st❛❝❦✿ ④ t❤❡ ❢✉♥❝t✐♦♥ ✐✈❛rs✭t✮ r❡t✉r♥s t❤❡ s❡t ♦❢ ✐♥✐t✐❛❧✐③❡❞ ✈❛r✐❛❜❧❡s ♦❢ t❀ ④ t❤❡ ♠❛❝r♦ ❝❤❡❝❦ ❢❨✶❀ ✿ ✿ ✿ ❀ ❨❞❣ ✚ ❣❡♥❡r❛t❡s t❤❡ ♥❡❝❡ss❛r② t❡sts✿ ❝❤❡❝❦ ❢❨✶❀ ✿ ✿ ✿ ❀ ❨❞❣ ✚ ❂ check ✭✚ ❨✶✮ ✳ ✳ ✳ check ✭✚ ❨❞✮

❯♥✐☞❝❛t✐♦♥

■♥str✉❝t✐♦♥ check i t❡sts ✇❤❡t❤❡r t❤❡ ✭✉♥✐♥✐t✐❛❧✐③❡❞✮ ✈❛r✐❛❜❧❡ ♦♥ t♦♣ ♦❢ t❤❡ st❛❝❦ ♦❝❝✉rs ✐♥s✐❞❡ t❤❡ t❡r♠ ❜♦✉♥❞ t♦ t❤❡ ✐✲t❤ ✈❛r✐❛❜❧❡✿

✐

❝❤❡❝❦ ✐

✐

❋P ❋P ❘ ❘

if (!check (S[SP], deref(S[FP+i]))) backtrack();

❯♥✐☞❝❛t✐♦♥

■♥str✉❝t✐♦♥ bind ❜✐♥❞s t❤❡ ✭✉♥✐♥✐t✐❛❧✐③❡❞✮ ✈❛r✐❛❜❧❡ t♦ t❤❡ ❝♦♥str✉❝t❡❞ t❡r♠✿ ❜✐♥❞ ❘ ❘

H[S[SP-1]] = (R, S[SP]); trail (S[SP-1]); SP = SP - 2;

❯♥✐☞❝❛t✐♦♥

❊①❛♠♣❧❡✿ ▲❡t t ✑ ❢✭❣✭ ✖ ❳❀ ❨ ✮❀ ❛❀ ❩✮ ✇✐t❤ ❡♥✈✐r♦♥♠❡♥t ✚ ❂ ❢❳ ✼✦ ✶❀ ❨ ✼✦ ✷❀ ❩ ✼✦ ✸❣✳ ❚❤❡♥ ❝♦❞❡❯ t ✚ ❣❡♥❡r❛t❡s t❤❡ ❝♦❞❡✿ ustruct f/3 A✶ son 1 ustruct g/2 A✷ son 1 uref 1 son 2 uvar 2 up B✷ ❆✷✿ check 1 putref 1 putvar 2 putstruct g/2 bind ❇✷✿ son 2 uatom a son 3 uvar 3 up B✶ ❆✶✿ check 1 putref 1 putvar 2 putstruct g/2 putatom a putvar 3 putstruct f/3 bind ❇✶✿ ✳ ✳ ✳

❈❧❛✉s❡s

❚❤❡ ❝♦❞❡ ❢♦r ❝❧❛✉s❡s ✇✐❧❧✿ ❛❧❧♦❝❛t❡ st❛❝❦ s♣❛❝❡ ❢♦r ❧♦❝❛❧s❀ ❡✈❛❧✉❛t❡ t❤❡ ❜♦❞②❀ ❢r❡❡ t❤❡ st❛❝❦ ❢r❛♠❡ ✭✐❢ ♣♦ss✐❜❧❡✮✳ ❲❡ ❞❡♥♦t❡ ❧♦❝❛❧ ✈❛r✐❛❜❧❡s ❜② ❢❳✶❀ ✿ ✿ ✿ ❀ ❳♠❣✱ ✇❤❡r❡ t❤❡ ☞rst ❦ ♦♥❡s ❛r❡ ❢♦r♠❛❧ ♣❛r❛♠❡t❡rs✳ ❝♦❞❡❈ ✭♣✭❳✶❀ ✿ ✿ ✿ ❀ ❳❦✮ ✥ ❣✶❀ ✿ ✿ ✿ ❀ ❣♥✮ ❂ pushenv m ❝♦❞❡● ❣✶ ✚ ✳ ✳ ✳ ❝♦❞❡● ❣♥ ✚ popenv

❈❧❛✉s❡s

■♥str✉❝t✐♦♥ pushenv m ❛❧❧♦❝❛t❡s st❛❝❦ s♣❛❝❡ ❢♦r ❧♦❝❛❧ ✈❛r✐❛❜❧❡s✿ ♣✉s❤❡♥✈ ♠

♠

❋P ❋P

SP = FP + m;

❈❧❛✉s❡s

❊①❛♠♣❧❡✿ ▲❡t r ✑ a✭❳❀ ❨ ✮ ✥ f✭ ✖ ❳❀ ❳✶✮❀ a✭ ✖ ❳✶❀ ✖ ❨ ✮ ❚❤❡♥ ❝♦❞❡❈ r ❣❡♥❡r❛t❡s t❤❡ ❝♦❞❡✿ pushenv 3 mark A putref 1 putvar 3 call f/2 ❆✿ mark B putref 3 putref 2 call a/2 ❇✿ popenv

Pr❡❞✐❝❛t❡s

❆ ♣r❡❞✐❝❛t❡ q/k ✐s ❞❡☞♥❡❞ ❜② ❛ s❡q✉❡♥❝❡ ♦❢ ❝❧❛✉s❡s rr ✑ r✶ ✿ ✿ ✿ r❢✳ ❚❤❡ tr❛♥s❧❛t✐♦♥ ♦❢ ♣r❡❞✐❝❛t❡s ✐s ♣❡r❢♦r♠❡❞ ❜② t❤❡ ❢✉♥❝t✐♦♥ ❝♦❞❡P ✳ ■❢ ❛ ♣r❡❞✐❝❛t❡ ❤❛s ❥✉st ❛ s✐♥❣❧❡ ❝❧❛✉s❡ ✭✐❡✳ ❢ ❂ ✶✮✱ ✇❡ ❤❛✈❡✿ ❝♦❞❡P r ❂ ❝♦❞❡❈ r ■❢ ❛ ♣r❡❞✐❝❛t❡ ❤❛s s❡✈❡r❛❧ ❝❧❛✉s❡s✱ t❤❡♥✿ ④ ✇❡ ☞rst ✧tr②✧ t❤❡ ☞rst ❝❧❛✉s❡❀ ④ ✐❢ ✐t ❢❛✐❧s✱ t❤❡♥ ✧tr②✧ t❤❡ s❡❝♦♥❞ ♦♥❡❀ ❡t❝✳

Pr❡❞✐❝❛t❡s

■❢ ✉♥✐☞❝❛t✐♦♥ ❢❛✐❧s✱ ✇❡ ❝❛❧❧ t❤❡ r✉♥✲t✐♠❡ ❢✉♥❝t✐♦♥ backtrack()✳ ❚❤❡ ❣♦❛❧ ✐s t♦ r♦❧❧ ❜❛❝❦ t❤❡ ✇❤♦❧❡ ❝♦♠♣✉t❛t✐♦♥ t♦ t❤❡ ❜❛❝❦tr❛❝❦ ♣♦✐♥t❀ ✐❡✳ t♦ t❤❡ ✭❞②♥❛♠✐❝❛❧❧②✮ ❧❛t❡st ❣♦❛❧ ✇❤❡r❡ t❤❡r❡ ✐s ❛♥♦t❤❡r ❝❧❛✉s❡ t♦ ✧tr②✧✳ ■♥ ♦r❞❡r t♦ r❡st♦r❡ ♣r❡✈✐♦✉s❧② ✈❛❧✐❞ ❜✐♥❞✐♥❣s✱ ✇❡ ❤❛✈❡ ✉s❡❞ t❤❡ r✉♥✲t✐♠❡ ❢✉♥❝t✐♦♥ trail() ✇❤✐❝❤ st♦r❡s ♥❡✇ ❜✐♥❞✐♥❣s ✐♥ ❛ s♣❡❝✐❛❧ ♠❡♠♦r② ❛r❡❛✳

Pr❡❞✐❝❛t❡s

❚r❛✐❧✿ ❚P

✶ ✵

❚ ❚ ❂ ❚r❛✐❧ ⑤ ♠❡♠♦r② ❛r❡❛ ❢♦r st♦r✐♥❣ ♥❡✇ ❜✐♥❞✐♥❣s❀ ❚P ❂ ❚❛✐❧✲P♦✐♥t❡r ⑤ ♣♦✐♥ts t♦ t❤❡ t♦♣♠♦st ✉s❡❞ ❝❡❧❧✳

Pr❡❞✐❝❛t❡s

❚❤❡r❡ ✐s ❛❧s♦ ❛ s♣❡❝✐❛❧ r❡❣✐st❡r ❇P ✇❤✐❝❤ ♣♦✐♥ts t♦ t❤❡ ❝✉rr❡♥t ❜❛❝❦tr❛❝❦ ♣♦✐♥t✳ ❙P ❋P ❇P

✶ ✵

❙

Pr❡❞✐❝❛t❡s

❆ ❜❛❝❦tr❛❝❦ ♣♦✐♥t ✐s ❛ st❛❝❦ ❢r❛♠❡ t♦ ✇❤✐❝❤ ♣r♦❣r❛♠ ❡①❡❝✉t✐♦♥ ♣♦ss✐❜❧② r❡t✉r♥s✿ ❋P

✲✶ ✲✷ ✵ ✲✺ ✲✹ ✲✸

❋P♦❧❞ P♦s❈♦♥t ❍P♦❧❞ ❚P♦❧❞ ❇P♦❧❞ ◆❡❣❈♦♥t ❲❡ ✇✐❧❧ ✉s❡ ❢♦❧❧♦✇✐♥❣ ♠❛❝r♦s t♦ ❞❡♥♦t❡ ♦r❣❛♥✐③❛t✐♦♥❛❧ ❝❡❧❧s✿ PosCont ✑ S[FP] FPold ✑ S[FP✶] HPold ✑ S[FP✷] TPold ✑ S[FP✸] BPold ✑ S[FP✹] NegCont ✑ S[FP✺]

Pr❡❞✐❝❛t❡s

❚❤❡ r✉♥✲t✐♠❡ ❢✉♥❝t✐♦♥ backtrack() r❡st♦r❡s r❡❣✐st❡rs ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ❢r❛♠❡ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ ❜❛❝❦tr❛❝❦ ♣♦✐♥t✿ void backtrack() { FP = BP; HP = HPold; reset (TPold,TP); TP = TPold; PC = NegCont; } ❚❤❡ ❢✉♥❝t✐♦♥ reset() r❡st♦r❡s ✈❛r✐❛❜❧❡ ❜✐♥❞✐♥❣s❀ ✐❡✳ ✉♥❞♦❡s ❛❧❧ ❜✐♥❞✐♥❣s ❝r❡❛t❡❞ ❛❢t❡r t❤❡ ❜❛❝❦tr❛❝❦ ♣♦✐♥t✳

Pr❡❞✐❝❛t❡s

❜❛❝❦tr❛❝❦✭✮ P❈ ❇P ❚P ❍P ❋P P❈ ❇P ❚P ❍P ❋P ✶✸ ✶✼ ✹✷ ✶✸ ✶✼ ✹✷ ✶✸ ✶✼ ✹✷

Pr❡❞✐❝❛t❡s

❚❤❡ ✈❛r✐❛❜❧❡s ✇❤✐❝❤ ❛r❡ ❝r❡❛t❡❞ s✐♥❝❡ t❤❡ ❧❛st ❜❛❝❦tr❛❝❦ ♣♦✐♥t ❝❛♥ ❜❡ r❡♠♦✈❡❞ t♦❣❡t❤❡r ✇✐t❤ t❤❡✐r ❜✐♥❞✐♥❣s s✐♠♣❧② ❜② r❡st♦r✐♥❣ t❤❡ ♦❧❞ ✈❛❧✉❡ ♦❢ t❤❡ r❡❣✐st❡r ❍P✳ ❚❤✐s ✇♦r❦s ☞♥❡ ✐❢ ②♦✉♥❣❡r ✈❛r✐❛❜❧❡s ❛❧✇❛②s ♣♦✐♥t t♦ ♦❧❞❡r ♦❜❥❡❝ts✳ ❇✐♥❞✐♥❣s ✇❤❡r❡ ♦❧❞❡r ✈❛r✐❛❜❧❡s ♣♦✐♥t t♦ ②♦✉♥❣❡r ♦❜❥❡❝ts ♠✉st ❜❡ r❡s❡t ✧♠❛♥✉❛❧❧②✧✳ ❚❤❡s❡ ❜✐♥❞✐♥❣s ❛r❡ r❡❝♦r❞❡❞ ✐♥ t❤❡ tr❛✐❧✳

Pr❡❞✐❝❛t❡s

❚❤❡ ❢✉♥❝t✐♦♥ trail() r❡❝♦r❞s ❛ ❜✐♥❞✐♥❣ ✐❢ t❤❡ ❛r❣✉♠❡♥t ♣♦✐♥ts t♦ ❛ ②♦✉♥❣❡r ♦❜❥❡❝t✿ void trail (ref u) { if (u < S[BP-2]) { TP = TP+1; T[TP] = u; } } ❚❤❡ ❝❡❧❧ S[BP-2] ❝♦♥t❛✐♥s t❤❡ ✈❛❧✉❡ ♦❢ ❍P ❜❡❢♦r❡ t❤❡ ❝r❡❛t✐♦♥ ♦❢ ❜❛❝❦tr❛❝❦ ♣♦✐♥t✳ ❚❤❡ ❢✉♥❝t✐♦♥ reset() r❡♠♦✈❡s ❛❧❧ ❜✐♥❞✐♥❣s ❝r❡❛t❡❞ ❛❢t❡r t❤❡ ❧❛st ❜❛❝❦tr❛❝❦ ♣♦✐♥t✿ void reset (ref x, ref y) { for (ref u=y; x<u; u--) H[T[u]] = (R,T[u]); }

Pr❡❞✐❝❛t❡s

❚r❛♥s❧❛t✐♦♥ ♦❢ ❛ ♣r❡❞✐❝❛t❡ q/k✱ ✇❤✐❝❤ ✐s ❞❡☞♥❡❞ ❜② ❝❧❛✉s❡s r✶❀ ✿ ✿ ✿ ❀ r❢ ✭❢ ❃ ✶✮✱ ❣❡♥❡r❛t❡s ❛ ❝♦❞❡ ✇❤✐❝❤✿ ❝r❡❛t❡s ❛ ❜❛❝❦tr❛❝❦ ♣♦✐♥t❀ s✉❝❝❡ss✐✈❡❧② ✧tr✐❡s✧ t❤❡ ❛❧t❡r♥❛t✐✈❡s❀ ❞❡❧❡t❡s t❤❡ ❜❛❝❦tr❛❝❦ ♣♦✐♥t✳

Pr❡❞✐❝❛t❡s

❝♦❞❡P ✭r✶❀ ✿ ✿ ✿ ❀ r❢✮ ❂ q✴❦✿ setbtp try A✶ ✳ ✳ ✳ try A❢✶ delbtp jump A❢ ❆✶✿ ❝♦❞❡❈ r✶ ✳ ✳ ✳ ❆❢✿ ❝♦❞❡❈ r❢ ◆❇✦ ❚❤❡ ❜❛❝❦tr❛❝❦ ♣♦✐♥t ✐s ❞❡❧❡t❡❞ ❜❡❢♦r❡ t❤❡ ❧❛st ❛❧t❡r♥❛t✐✈❡ ✐s ✧tr✐❡❞✧✳ ❋♦r t❤❡ ✧❧❛st tr②✧✱ t❤❡ ❝♦❞❡ ❥✉♠♣s ❞✐r❡❝t❧② t♦ t❤❡ ❛❧t❡r♥❛t✐✈❡ ❛♥❞ ♥❡✈❡r r❡t✉r♥s t♦ t❤❡ ♣r❡s❡♥t ❢r❛♠❡✳

Pr❡❞✐❝❛t❡s

❊①❛♠♣❧❡✿ s✭❳✮ ✥ t✭ ✖ ❳✮ s✭❳✮ ✥ ✖ ❳ ❂ ❛ ❚r❛♥s❧❛t✐♦♥ ♦❢ t❤❡ ♣r❡❞✐❝❛t❡ s❂1 r❡s✉❧ts✿ s✴✶✿ setbtp try A delbtp jump B ❆✿ pushenv 1 mark C putref 1 call t/1 ❈✿ popenv ❇✿ pushenv 1 putref 1 uatom a popenv

Pr❡❞✐❝❛t❡s

■♥str✉❝t✐♦♥ setbtp s❛✈❡s r❡❣✐st❡rs ❍P✱ ❚P✱ ❇P✿ s❡t❜t♣ ❋P ❋P ❇P ❚P ❍P ❇P ❚P ❍P ✹✷ ✶✼ ✹✷ ✶✼ ✶✼ ✹✷

HPold = HP; TPold = TP; BPold = BP; BP = FP;

Pr❡❞✐❝❛t❡s

■♥str✉❝t✐♦♥ try A s❛✈❡s t❤❡ ❝✉rr❡♥t P❈ ❛s t❤❡ ♥❡❣❛t✐✈❡ ❝♦♥t✐♥✉❛t✐♦♥ ❛❞❞r❡ss ❛♥❞ ❥✉♠♣s t♦ t❤❡ ❛❧t❡r♥❛t✐✈❡ t♦ ❜❡ ✧tr✐❡❞✧ ❛t ❛❞❞r❡ss ❆✿ tr② ❆ P❈ ❇P ❚P ❍P P❈ ❇P ❚P ❍P ❋P ❋P ✷✾ ✷✾ ❆

NegCont = PC; PC = A;

Pr❡❞✐❝❛t❡s

■♥str✉❝t✐♦♥ delbtp r❡st♦r❡s t❤❡ ✈❛❧✉❡ ♦❢ ❇P✿ ❞❡❧❜t♣ ❋P ❋P ❇P ❚P ❍P ❇P ❚P ❍P

BP = BPold;

Pr❡❞✐❝❛t❡s

■♥str✉❝t✐♦♥ popenv r❡st♦r❡s r❡❣✐st❡rs ❋P ❛♥❞ P❈✱ ❛♥❞ ✐❢ ♣♦ss✐❜❧❡ ♣♦♣s t❤❡ st❛❝❦ ❢r❛♠❡✿

✻

P❈ ❇P ❋P ♣♦♣❡♥✈ P❈ ❇P ❋P ✹✷ ✹✷

if (FP ❃ BP) SP = FP - 6; PC = PosCont; FP = FPold;

Pr❡❞✐❝❛t❡s

■❢ ❋P ✔ ❇P t❤❡ ❢r❛♠❡ ✐s ♥♦t ❞❡❛❧❧♦❝❛t❡❞✿

✻ ✻

P❈ ❇P ❋P ♣♦♣❡♥✈ P❈ ❇P ❋P ✹✷ ✹✷ ✹✷

if (FP ❃ BP) SP = FP - 6; PC = PosCont; FP = FPold;

◗✉❡r✐❡s ❛♥❞ Pr♦❣r❛♠s

❚r❛♥s❧❛t✐♦♥ ♦❢ ❛ ♣r♦❣r❛♠ ♣ ✑ rr✶ ✿ ✿ ✿ rr❤❄❣ ❣❡♥❡r❛t❡s✿ ④ ❝♦❞❡ ❢♦r ❡✈❛❧✉❛t✐♥❣ t❤❡ q✉❡r② ❣❀ ④ ❝♦❞❡ ❢♦r t❤❡ ♣r❡❞✐❝❛t❡ ❞❡☞♥✐t✐♦♥s rr✐✳ ◗✉❡r② ❡✈❛❧✉❛t✐♦♥ ✐s ♣r❡❝❡❞❡❞ ❜②✿ ④ ✐♥✐t✐❛❧✐③❛t✐♦♥ ♦❢ r❡❣✐st❡rs❀ ④ ❛❧❧♦❝❛t✐♦♥ ♦❢ s♣❛❝❡ ❢♦r ❣❧♦❜❛❧s✳ ◗✉❡r② ❡✈❛❧✉❛t✐♦♥ ✐s s✉❝❝❡❡❞❡❞ ❜②✿ ④ r❡t✉r♥✐♥❣ t❤❡ ✈❛❧✉❡s ♦❢ ❣❧♦❜❛❧s✳

◗✉❡r✐❡s ❛♥❞ Pr♦❣r❛♠s

❝♦❞❡ ✭rr✶ ✿ ✿ ✿ rr❤ ❄❣✮ ❂ init pushenv d ❝♦❞❡● ❣ ✚ halt d ❝♦❞❡P rr✶ ✳ ✳ ✳ ❝♦❞❡P rr❤ ✇❤❡r❡ ❢r❡❡✭❣✮ ❂ ❢❳✶❀ ✿ ✿ ✿ ❀ ❳❞❣ ❛♥❞ ✚ ❂ ❢❳✐ ✼✦ ✐ ❥ ✐ ❂ ✶ ✿ ✿ ✿ ❞❣✳ ■♥str✉❝t✐♦♥ halt d ✳ ✳ ✳ ✳ ✳ ✳ t❡r♠✐♥❛t❡s t❤❡ ♣r♦❣r❛♠ ❡①❡❝✉t✐♦♥❀ ✳ ✳ ✳ r❡t✉r♥s t❤❡ ✈❛❧✉❡s ♦❢ ❞ ❣❧♦❜❛❧s❀ ✳ ✳ ✳ ✐❢ ✉s❡r r❡q✉❡sts✱ ♣❡r❢♦r♠s ❜❛❝❦tr❛❝❦✐♥❣✳

◗✉❡r✐❡s ❛♥❞ Pr♦❣r❛♠s

■♥str✉❝t✐♦♥ init ❝r❡❛t❡s t❤❡ ✐♥✐t✐❛❧ ❜❛❝❦tr❛❝❦ ♣♦✐♥t✿ ✐♥✐t ❇P ❚P ❍P ❋P ❇P ❚P ❍P ❋P ✲✶ ✵ ✲✶ ✲✶ ❢ ✲✶ ✲✶ ✵ ✵ ✲✶

BP = FP = SP = 5; S[0] = f; S[1] = S[2] = -1; S[3] = 0; BP = FP;

■❢ t❤❡ q✉❡r② ❣ ❢❛✐❧s✱ t❤❡ ❝♦❞❡ ❛t ❛❞❞r❡ss ❢ ✇✐❧❧ ❜❡ ❡①❡❝✉t❡❞ ✭❡❣✳ ♣r✐♥ts ❛ ♠❡ss❛❣❡ t❡❧❧✐♥❣ ❛❜♦✉t t❤❡ ❢❛✐❧✉r❡✮✳

◗✉❡r✐❡s ❛♥❞ Pr♦❣r❛♠s

❊①❛♠♣❧❡✿ t✭❳✮ ✥ ✖ ❳ ❂ ❜ q✭❳✮ ✥ s✭ ✖ ❳✮ s✭❳✮ ✥ ✖ ❳ ❂ ❛ p ✥ q✭❳✮❀ t✭ ✖ ❳✮ s✭❳✮ ✥ t✭ ✖ ❳✮ ❄ p init pushenv 0 mark A call p/0 ❆✿ halt 0 t✴✶✿ pushenv 1 putref 1 uatom b popenv ♣✴✵✿ pushenv 1 mark B putvar 1 call q/1 ❇✿ mark C putref 1 call t/1 ❈✿ popenv q✴✶✿ pushenv 1 mark D putref 1 call s/1 ❉✿ popenv s✴✶✿ setbtp try E delbtp jump F ❊✿ pushenv 1 mark G putref 1 call t/1

• ✿ popenv

❋✿ pushenv 1 putref 1 uatom a popenv

▲❛st ❈❛❧❧ ❖♣t✐♠✐③❛t✐♦♥

❈♦♥s✐❞❡r t❤❡ ♣r❡❞✐❝❛t❡ app/3 ❞❡☞♥❡❞ ❛s ❢♦❧❧♦✇s✿ app✭❳❀ ❨❀ ❩✮ ✥ ❳ ❂ ❬ ❪❀ ❨ ❂ ❩ app✭❳❀ ❨❀ ❩✮ ✥ ❳ ❂ ❬❍ ❥ ❳✵❪❀ ❩ ❂ ❬❍ ❥ ❩✵❪❀ app✭❳✵❀ ❨❀ ❩✵✮ ❚❤❡ ❧❛st ❣♦❛❧ ♦❢ t❤❡ s❡❝♦♥❞ ❝❧❛✉s❡ ✐s ❛ r❡❝✉rs✐✈❡ ❝❛❧❧✿ ✇❡ ❝❛♥ ❡✈❛❧✉❛t❡ ✐t ✐♥ t❤❡ ❝✉rr❡♥t st❛❝❦ ❢r❛♠❡❀ ❛❢t❡r ✭s✉❝❝❡ss❢✉❧✮ ❝♦♠♣❧❡t✐♦♥✱ ✇❡ ✇✐❧❧ ♥♦t r❡t✉r♥ t♦ t❤❡ ❝✉rr❡♥t ❢r❛♠❡ ❜✉t ❣♦ ❞✐r❡❝t❧② ❜❛❝❦ t♦ t❤❡ ✧♣r❡❞❡❝❡ss♦r✧ ❢r❛♠❡✳

▲❛st ❈❛❧❧ ❖♣t✐♠✐③❛t✐♦♥

❈♦♥s✐❞❡r ❛ ❝❧❛✉s❡ r ✑ p✭❳✶❀ ✿ ✿ ✿ ❀ ❳❦✮ ✥ ❣✶❀ ✿ ✿ ✿ ❀ ❣♥✱ ✇❤✐❝❤ ❤❛s ♠ ❧♦❝❛❧ ✈❛r✐❛❜❧❡s ❛♥❞ ✇❤❡r❡ ❣♥ ✑ q✭t✶❀ ✿ ✿ ✿ ❀ t❤✮✳ ❝♦❞❡❈ r ❂ pushenv m ❝♦❞❡● ❣✶ ✚ ✳ ✳ ✳ ❝♦❞❡● ❣♥✶ ✚ mark B ❝♦❞❡❆ t✶ ✚ ✳ ✳ ✳ ❝♦❞❡❆ t❤ ✚ call q/h ❇✿ popenv

▲❛st ❈❛❧❧ ❖♣t✐♠✐③❛t✐♦♥

❈♦♥s✐❞❡r ❛ ❝❧❛✉s❡ r ✑ p✭❳✶❀ ✿ ✿ ✿ ❀ ❳❦✮ ✥ ❣✶❀ ✿ ✿ ✿ ❀ ❣♥✱ ✇❤✐❝❤ ❤❛s ♠ ❧♦❝❛❧ ✈❛r✐❛❜❧❡s ❛♥❞ ✇❤❡r❡ ❣♥ ✑ q✭t✶❀ ✿ ✿ ✿ ❀ t❤✮✳ ❝♦❞❡❈ r ❂ pushenv m ❝♦❞❡● ❣✶ ✚ ✳ ✳ ✳ ❝♦❞❡● ❣♥✶ ✚ lastmark ❝♦❞❡❆ t✶ ✚ ✳ ✳ ✳ ❝♦❞❡❆ t❤ ✚ lastcall (q/h,m)

▲❛st ❈❛❧❧ ❖♣t✐♠✐③❛t✐♦♥

■❢ t❤❡ ❝✉rr❡♥t ❝❧❛✉s❡ ✐s ♥♦t ❧❛st ♦r ❣♦❛❧s ❣✶❀ ✿ ✿ ✿ ❀ ❣♥✶ ❤❛✈❡ ❝r❡❛t❡❞ ❜❛❝❦tr❛❝❦ ♣♦✐♥ts✱ t❤❡♥ ❋P ✔ ❇P✳ ❚❤❡♥ t❤❡ ✐♥str✉❝t✐♦♥ lastmark ❝r❡❛t❡s ❛ ♥❡✇ ❢r❛♠❡ ❜✉t st♦r❡s ❛ r❡❢❡r❡♥❝❡ t♦ t❤❡ ♣r❡❞❡❝❡ss♦r ❢r❛♠❡✳ ❖t❤❡r✇✐s❡ ✭✐❡✳ ✐❢ ❋P ❃ ❇P✮✱ ✐t ❞♦❡s ♥♦t❤✐♥❣✳

▲❛st ❈❛❧❧ ❖♣t✐♠✐③❛t✐♦♥

✻

❧❛st♠❛r❦ ❋P ❇P ❇P ❋P ❆ ❆ ❆

if (FP ✔ BP) { SP = SP + 6; S[SP] = PosCont; S[SP-1] = FPold; }

▲❛st ❈❛❧❧ ❖♣t✐♠✐③❛t✐♦♥

■❢ ❋P ✔ ❇P✱ t❤❡♥ t❤❡ ✐♥str✉❝t✐♦♥ lastcall (q/h,m) ❜❡❤❛✈❡s ❧✐❦❡ call q/h✳ ❖t❤❡r✇✐s❡✱ t❤❡ ❝✉rr❡♥t st❛❝❦ ❢r❛♠❡ ✐s r❡✉s❡❞✿ ④ t❤❡ ❝❡❧❧s S[FP+1]✱ ✳ ✳ ✳ ✱ S[FP+h] ❣❡t ♥❡✇ ✈❛❧✉❡s❀ ④ ❛♥❞ t❤❡♥ ❞✐r❡❝t❧② ❥✉♠♣s t♦ t❤❡ ♣r❡❞✐❝❛t❡ q/h✳ lastcall (q/h,m) ❂ if (FP ✔ BP) call q/h; else { move (m,h); jump q/h; }

▲❛st ❈❛❧❧ ❖♣t✐♠✐③❛t✐♦♥

❧❛st❝❛❧❧✭q✴❤✱♠✮

❤ ♠

P❈ ❇P ❋P ❇P ❋P q✴❤

▲❛st ❈❛❧❧ ❖♣t✐♠✐③❛t✐♦♥

❈♦♥s✐❞❡r t❤❡ ❝❧❛✉s❡ a✭❳❀ ❨ ✮ ✥ f✭ ✖ ❳❀ ❳✶✮❀ a✭ ✖ ❳✶❀ ✖ ❨ ✮ ❚❤❡ ❧❛st ❝❛❧❧ ♦♣t✐♠✐③❛t✐♦♥ ②✐❡❧❞s✿ pushenv 3 mark A putref 1 putvar 3 call f/2 ❆✿ lastmark putref 3 putref 2 lastcall(a/2,3) ◆❇✦ ■❢ t❤❡ ❝❧❛✉s❡ ✐s ❧❛st ❛♥❞ ✐ts ❧❛st ❣♦❛❧ ✐s t❤❡ ♦♥❧② ♦♥❡✱ t❤❡♥ ✇❡ ❝❛♥ ♦♠✐t lastmark ❛♥❞ r❡♣❧❛❝❡ lastcall(q/h,m) ✇✐t❤ ✐♥str✉❝t✐♦♥s move(m,h) ❛♥❞ jump q/h✳

▲❛st ❈❛❧❧ ❖♣t✐♠✐③❛t✐♦♥

❚❤❡ ❧❛st ❝❛❧❧ ♦♣t✐♠✐③❛t✐♦♥ ❢♦r t❤❡ s❡❝♦♥❞ ❝❧❛✉s❡ ♦❢ app/3 ②✐❡❧❞s✿ ❆✿ pushenv 6 putref 1 ustruct [|]/2 B son 1 uvar 4 son 2 uvar 5 up C ❇✿ putvar 4 putvar 5 putstruct [|]/2 bind ❈✿ putref 3 ustruct [|]/2 D son 1 uref 4 son 2 uvar 6 up E ❉✿ check 4 putref 4 putvar 6 putstruct [|]/2 bind ❊✿ putref 5 putref 2 putref 6 move(6,3) jump app/3

❙t❛❝❦ ❋r❛♠❡ ❚r✐♠♠✐♥❣

❖r❞❡r ❧♦❝❛❧ ✈❛r✐❛❜❧❡s ❛❝❝♦r❞✐♥❣ t♦ t❤❡✐r ❧✐❢❡ t✐♠❡✳ ■❢ ♣♦ss✐❜❧❡✱ r❡♠♦✈❡ ❞❡❛❞ ✈❛r✐❛❜❧❡s✳ ❊①❛♠♣❧❡✿ a✭❳❀ ❩✮ ✥ p✶✭ ✖ ❳❀ ❳✶✮❀ p✷✭ ✖ ❳✶❀ ❳✷✮❀ p✸✭ ✖ ❳✷❀ ❳✸✮❀ p✹✭ ✖ ❳✸❀ ❩✮ ④ ❛❢t❡r t❤❡ ❣♦❛❧ p✷✭ ✖ ❳✶❀ ❳✷✮ t❤❡ ✈❛r✐❛❜❧❡ ❳✶ ✐s ❞❡❛❞❀ ④ ❛❢t❡r t❤❡ ❣♦❛❧ p✸✭ ✖ ❳✷❀ ❳✸✮ t❤❡ ✈❛r✐❛❜❧❡ ❳✷ ✐s ❞❡❛❞✳

❙t❛❝❦ ❋r❛♠❡ ❚r✐♠♠✐♥❣

❆❢t❡r ❡✈❡r② ♥♦♥✲❧❛st ❣♦❛❧ ✇❤✐❝❤ ❤❛s ❞❡❛❞ ✈❛r✐❛❜❧❡s ✐♥s❡rt ❛♥ ✐♥str✉❝t✐♦♥ trim✿ tr✐♠ ♠

♠

❋P ❋P

if (FP ✕ BP) SP = FP + m;

◆❇✦ ❲❡ ❝❛♥ r❡♠♦✈❡ ❞❡❛❞ ❧♦❝❛❧s ♦♥❧② ✐❢ t❤❡r❡ ❛r❡ ♥♦ ♥❡✇ ❜❛❝❦tr❛❝❦ ♣♦✐♥ts ❝r❡❛t❡❞✳

❙t❛❝❦ ❋r❛♠❡ ❚r✐♠♠✐♥❣

❊①❛♠♣❧❡✿ a✭❳❀ ❩✮ ✥ p✶✭ ✖ ❳❀ ❳✶✮❀ p✷✭ ✖ ❳✶❀ ❳✷✮❀ p✸✭ ✖ ❳✷❀ ❳✸✮❀ p✹✭ ✖ ❳✸❀ ❩✮ ❖r❞❡r✐♥❣ ♦❢ t❤❡ ✈❛r✐❛❜❧❡s✿ ✚ ❂ ❢❳ ✼✦ ✶❀ ❩ ✼✦ ✷❀ ❳✸ ✼✦ ✸❀ ❳✷ ✼✦ ✹❀ ❳✶ ✼✦ ✺❣ pushenv 5 mark A putref 1 putvar 5 call p✶/2 ❆✿ mark B putref 5 putvar 4 call p✷/2 ❇✿ trim 4 mark C putref 4 putvar 3 call p✸/2 ❈✿ trim 3 lastmark putref 3 putref 2 lastcall (p✹/2,3)

❈❧❛✉s❡ ■♥❞❡①✐♥❣

❖❢t❡♥✱ ♣r❡❞✐❝❛t❡s ❛r❡ ✐♠♣❧❡♠❡♥t❡❞ ❜② ❝❛s❡ ❞✐st✐♥❝t✐♦♥ ♦♥ t❤❡ ☞rst ❛r❣✉♠❡♥t✳ ❍❡♥❝❡✱ ❜② ✐♥s♣❡❝t✐♥❣ t❤❡ ☞rst ❛r❣✉♠❡♥t✱ ♠❛♥② ❛❧t❡r♥❛t✐✈❡s ❝❛♥ ❜❡ ❡①❝❧✉❞❡❞✳ ④ ❋❛✐❧✉r❡ ✐s ❞❡t❡❝t❡❞ ❡❛r❧✐❡r✳ ④ ❇❛❝❦tr❛❝❦ ♣♦✐♥ts ❛r❡ r❡♠♦✈❡❞ ❡❛r❧✐❡r✳ ④ ❙t❛❝❦ ❢r❛♠❡s ❛r❡ r❡♠♦✈❡❞ ❡❛r❧✐❡r✳

❈❧❛✉s❡ ■♥❞❡①✐♥❣

❊①❛♠♣❧❡✿ app✭❳❀ ❨❀ ❩✮ ✥ ❳ ❂ ❬ ❪❀ ❨ ❂ ❩ app✭❳❀ ❨❀ ❩✮ ✥ ❳ ❂ ❬❍ ❥ ❳✵❪❀ ❩ ❂ ❬❍ ❥ ❩✵❪❀ app✭❳✵❀ ❨❀ ❩✵✮ ■❢ t❤❡ ☞rst ❛r❣✉♠❡♥t ✐s []✱ t❤❡♥ ♦♥❧② t❤❡ ☞rst ❝❧❛✉s❡ ✐s ❛♣♣❧✐❝❛❜❧❡✳ ■❢ t❤❡ ☞rst ❛r❣✉♠❡♥t ❤❛s [|] ❛s ✐ts r♦♦t ❝♦♥str✉❝t♦r✱ t❤❡♥ ♦♥❧② t❤❡ s❡❝♦♥❞ ❝❧❛✉s❡ ✐s ❛♣♣❧✐❝❛❜❧❡✳ ❊✈❡r② ♦t❤❡r r♦♦t ❝♦♥str✉❝t♦r ♦❢ t❤❡ ☞rst ❛r❣✉♠❡♥t ✇✐❧❧ ❢❛✐❧✳ ❇♦t❤ ❛❧t❡r♥❛t✐✈❡s s❤♦✉❧❞ ❜❡ tr✐❡❞ ♦♥❧② ✐❢ t❤❡ ☞rst ❛r❣✉♠❡♥t ✐s ✉♥✐♥✐t✐❛❧✐③❡❞ ✈❛r✐❛❜❧❡✳

❈❧❛✉s❡ ■♥❞❡①✐♥❣

■♥tr♦❞✉❝❡ ❛ s❡♣❛r❛t❡ tr② ❝❤❛✐♥ ❢♦r ❡✈❡r② ♣♦ss✐❜❧❡ ❝♦♥str✉❝t♦r✳ ■♥s♣❡❝t t❤❡ r♦♦t ♥♦❞❡ ♦❢ t❤❡ ☞rst ❛r❣✉♠❡♥t✳ ❉❡♣❡♥❞✐♥❣ ♦♥ t❤❡ r❡s✉❧t✱ ♣❡r❢♦r♠ ❛♥ ✐♥❞❡①❡❞ ❥✉♠♣ t♦ t❤❡ ❛♣♣r♦♣r✐❛t❡ tr② ❝❤❛✐♥✳ ▲❡t t❤❡ ♣r❡❞✐❝❛t❡ p/k ❞❡☞♥❡❞ ❜② t❤❡ s❡q✉❡♥❝❡ ♦❢ ❝❧❛✉s❡s rr ✑ r✶ ✿ ✿ ✿ r♠✳ ❚❤❡ ♠❛❝r♦ t❝❤❛✐♥s rr ❞❡♥♦t❡s t❤❡ s❡q✉❡♥❝❡ ♦❢ tr② ❝❤❛✐♥s ✇❤✐❝❤ ❝♦rr❡s♣♦♥❞ t♦ t❤❡ r♦♦t ❝♦♥str✉❝t♦rs ♦❝❝✉rr✐♥❣ ✐♥ ✉♥✐☞❝❛t✐♦♥s ❳✶ ❂ t✳

❈❧❛✉s❡ ■♥❞❡①✐♥❣

❊①❛♠♣❧❡✿ ❈♦♥s✐❞❡r t❤❡ ♣r❡❞✐❝❛t❡ app/3✳ ▲❡t t❤❡ ❝♦❞❡ ❢♦r ✐ts t✇♦ ❝❧❛✉s❡s st❛rt ❛t ❛❞❞r❡ss❡s ❆✶ ❛♥❞ ❆✷✳ ❚❤❡♥ ✇❡ ❣❡t t❤❡ ❢♦❧❧♦✇✐♥❣ ❢♦✉r tr② ❝❤❛✐♥s✿ ❱❆❘✿ setbtb // variables try A✶ delbtp jump A✷ ◆■▲✿ jump A✶ // [] ❈❖◆❙✿ jump A✷ // [|] ❊▲❙❊✿ fail // default ■♥str✉❝t✐♦♥ fail ❤❛♥❞❧❡s ❛❧❧ ❝♦♥str✉❝t♦rs ❜❡s✐❞❡s [] ❛♥❞ [|]✳ fail ❂ backtrack()

❈❧❛✉s❡ ■♥❞❡①✐♥❣

❚❤❡♥ ✇❡ ❣❡♥❡r❛t❡ ❢♦r ❛ ♣r❡❞✐❝❛t❡ p/k✿ ❝♦❞❡P rr ❂ putref 1 getNode index p/k t❝❤❛✐♥s rr ❆✶✿ ❝♦❞❡❈ r✶ ✳ ✳ ✳ ❆♠✿ ❝♦❞❡❈ r♠

❈❧❛✉s❡ ■♥❞❡①✐♥❣

❣❡t◆♦❞❡ ❣❡t◆♦❞❡ ❘ ❙ ❢✴♥ ❘ ❢✴♥

switch (H[S[SP]]) { case (S,f/n): S[SP] = f/n; break; case (A,a): S[SP] = a; break; case (R, ): S[SP] = R; }

❈❧❛✉s❡ ■♥❞❡①✐♥❣

■♥str✉❝t✐♦♥ index p/k ♣❡r❢♦r♠s ❛♥ ✐♥❞❡①❡❞ ❥✉♠♣ t♦ t❤❡ ❛♣♣r♦♣r✐❛t❡ tr② ❝❤❛✐♥✿ ❆ ❛ ✐♥❞❡① ♣✴❦ P❈ ❆ ❂ ♠❛♣✭♣✴❦✱❛✮

PC = map(p/k,S[SP]); SP = SP - 1;

❚❤❡ ❢✉♥❝t✐♦♥ map() r❡t✉r♥s t❤❡ st❛rt ❛❞❞r❡ss ♦❢ t❤❡ ❛♣♣r♦♣r✐❛t❡ tr② ❝❤❛✐♥✳ ❈❛♥ ❜❡ ❞❡☞♥❡❞ ❡❣✳ t❤r♦✉❣❤ s♦♠❡ ❤❛s❤ t❛❜❧❡✳

❈✉t ❖♣❡r❛t♦r

❲❡ ❡①t❡♥❞ t❤❡ ❧❛♥❣✉❛❣❡ Pr♦❧❧ ✇✐t❤ t❤❡ ❝✉t ♦♣❡r❛t♦r ✧✦✧ ✇❤✐❝❤ ❡①♣❧✐❝✐t❧② ❛❧❧♦✇s t♦ ♣r✉♥❡ t❤❡ s❡❛r❝❤ s♣❛❝❡ ♦❢ ❜❛❝❦tr❛❝❦✐♥❣✳ ❊①❛♠♣❧❡✿ branch✭❳❀ ❨ ✮ ✥ p✭❳✮❀ ✦❀ q✶✭❳❀ ❨ ✮ branch✭❳❀ ❨ ✮ ✥ q✷✭❳❀ ❨ ✮ ■❢ ❛❧❧ t❤❡ q✉❡r✐❡s ❜❡❢♦r❡ t❤❡ ❝✉t ❤❛✈❡ s✉❝❝❡❡❞❡❞✱ t❤❡♥ t❤❡ ❝❤♦✐❝❡ ✐s ❝♦♠♠✐tt❡❞✿ ❜❛❝❦tr❛❝❦✐♥❣ ✇✐❧❧ r❡t✉r♥ ♦♥❧② t♦ ❜❛❝❦tr❛❝❦ ♣♦✐♥ts ♣r❡❝❡❞✐♥❣ t❤❡ ❝❛❧❧ t♦ t❤❡ ♣r❡❞✐❝❛t❡✳

❈✉t ❖♣❡r❛t♦r

❚❤❡ ❝✉t ♦♣❡r❛t♦r s❤♦✉❧❞✿ r❡st♦r❡ t❤❡ r❡❣✐st❡r ❇P ❜② ❛ss✐❣♥✐♥❣ t♦ ✐t ❇P♦❧❞ ❢r♦♠ t❤❡ ❝✉rr❡♥t ❢r❛♠❡❀ r❡♠♦✈❡ ❛❧❧ ❢r❛♠❡s ✇❤✐❝❤ ❛r❡ ♦♥ t♦♣ ♦❢ t❤❡ ❧♦❝❛❧ ✈❛r✐❛❜❧❡s✳ ❆❝❝♦r❞✐♥❣❧②✱ ✇❡ tr❛♥s❧❛t❡ t❤❡ ❝✉t ✐♥t♦ t❤❡ s❡q✉❡♥❝❡✿ prune pushenv m ✇❤❡r❡ ♠ ✐s t❤❡ ♥✉♠❜❡r ♦❢ ✭st✐❧❧ ❛❧✐✈❡✮ ❧♦❝❛❧ ✈❛r✐❛❜❧❡s ♦❢ t❤❡ ❝❧❛✉s❡✳

❈✉t ❖♣❡r❛t♦r

❊①❛♠♣❧❡✿ branch✭❳❀ ❨ ✮ ✥ p✭❳✮❀ ✦❀ q✶✭❳❀ ❨ ✮ branch✭❳❀ ❨ ✮ ✥ q✷✭❳❀ ❨ ✮ ❲❡ ♦❜t❛✐♥✿ setbtp try A delbtp jump B ❆✿ pushenv 2 mark C putref 1 call p/1 ❈✿ prune pushenv 2 lastmark putref 1 putref 2 lastcall(q✶/2,2) ❇✿ pushenv 2 putref 1 putref 2 move(2,2) jump q✷/2

❈✉t ❖♣❡r❛t♦r

❊①❛♠♣❧❡✿ branch✭❳❀ ❨ ✮ ✥ p✭❳✮❀ ✦❀ q✶✭❳❀ ❨ ✮ branch✭❳❀ ❨ ✮ ✥ q✷✭❳❀ ❨ ✮ ✳ ✳ ✳ ♦r✱ ✇❤❡♥ ✉s✐♥❣ ♦♣t✐♠✐③❛t✐♦♥s✿ setbtp try A delbtp jump B ❆✿ pushenv 2 mark C putref 1 call p/1 ❈✿ prune pushenv 2 putref 1 putref 2 move(2,2) jump q✶/2 ❇✿ pushenv 2 putref 1 putref 2 move(2,2) jump q✷/2

❈✉t ❖♣❡r❛t♦r

♣r✉♥❡ ❇P ❇P ❋P ❋P

BP = BPold;

❈✉t ❖♣❡r❛t♦r

Pr♦❜❧❡♠✿ ■❢ t❤❡ ♣r❡❞✐❝❛t❡ ✐s ❞❡☞♥❡❞ ❜② ❛ s✐♥❣❧❡ ❝❧❛✉s❡✱ t❤❡♥ ✇❡ ❤❛✈❡ ♥♦t st♦r❡❞ t❤❡ ♦❧❞ ❇P ✐♥s✐❞❡ t❤❡ st❛❝❦ ❢r❛♠❡✳ ❋♦r t❤❡ ❝✉t t♦ ✇♦r❦ ❛❧s♦ ✇✐t❤ s✐♥❣❧❡✲❝❧❛✉s❡ ♣r❡❞✐❝❛t❡s ♦r tr② ❝❤❛✐♥s ♦❢ ❧❡♥❣t❤ ✶✱ ✇❡ ✐♥s❡rt ❛♥ ❡①tr❛ ✐♥str✉❝t✐♦♥ setcut ❜❡❢♦r❡ t❤❡ ❝❧❛✉s❛❧ ❝♦❞❡ ✭♦r t❤❡ ❥✉♠♣✮✳

❈✉t ❖♣❡r❛t♦r

s❡t❝✉t ❇P ❇P ❋P ❋P

BPold = BP;

❈✉t ❖♣❡r❛t♦r

❋✐♥❛❧ ❡①❛♠♣❧❡✿ t❤❡ ♣r❡❞✐❝❛t❡ notP s✉❝❝❡❡❞s ✇❤❡♥❡✈❡r p ❢❛✐❧s ❛♥❞ ✈✐❝❡ ✈❡rs❛✿ notP✭❳✮ ✥ p✭❳✮❀ ✦❀ fail notP✭❳✮ ✥ ✇❤❡r❡ fail ❛❧✇❛②s ❢❛✐❧s✳ setbtp try A delbtp jump B ❆✿ pushenv 1 mark C putref 1 call p/1 ❈✿ prune pushenv 1 fail popenv ❇✿ pushenv 1 popenv