s strt r - PowerPoint PPT Presentation

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

❚r❛♥s❧❛t✐♦♥ ♦❢ ●♦❛❧s ❙tr✉❝t✉r❡ ♦❢ ❛ ❢r❛♠❡✿ ❙P ❧♦❝❛❧ st❛❝❦ ❛r❣✉♠❡♥ts ❋P P♦s❈♦♥t ✵ ❋P♦❧❞ ✲✶ ✲✷ ♦r❣❛♥✐③❛t✐♦♥❛❧ ❝❡❧❧s ✲✸ ✲✹ ✲✺

❚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 ⑦ ❳ ❂ t ✮ ✚ ❂ ❳ ✚ ❝♦❞❡ ❆ t ✚ unify

❯♥✐☞❝❛t✐♦♥ ❊①❛♠♣❧❡✿ ❝♦♥s✐❞❡r t❤❡ ❡q✉❛t✐♦♥ ❯ ❂ ❢ ✭ ❣ ✭ ✖ ✖ ❳❀ ❨ ✮ ❀ ❛❀ ❩ ✮ ❚❤❡♥✱ ❣✐✈❡♥ ❛♥ ❛❞❞r❡ss ❡♥✈✐r♦♥♠❡♥t ✚ ❂ ❢ ❳ ✼✦ ✶ ❀ ❨ ✼✦ ✷ ❀ ❩ ✼✦ ✸ ❀ ❯ ✼✦ ✹ ❣ t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦❞❡ ✐s ❣❡♥❡r❛t❡❞✿ putref 4 putatom a putref 1 putvar 3 putvar 2 putstruct f/3 putstruct g/2 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 ⑦ ❳ ❂ t ✮ ✚ ❂ ❳ ✚ ❝♦❞❡ ❯ 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✐❛❜❧❡✿ ✐ ✐ ❋P ❋P ✉✈❛r ✐ 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✐❛❜❧❡✿ ❳ ❨ ✒ ✐ ✐ ❋P ❋P ✒ ❂ ♠❣✉✭ ❳❀ ❨ ✮ ✉r❡❢ ✐ 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 up B ❆✿ ❝❤❡❝❦ ✐✈❛rs✭ ❢ ✭ t ✶ ❀ ✿ ✿ ✿ ❀ t ♥ ✮✮ ✚ son 1 ❝♦❞❡ ❯ t ✶ ✚ ❝♦❞❡ ❆ ✭ ❢ ✭ t ✶ ❀ ✿ ✿ ✿ ❀ t ♥ ✮✮ ✚ ✳ ✳ ✳ bind son n ❇✿ ✳ ✳ ✳ ❝♦❞❡ ❯ t ♥ ✚

❯♥✐☞❝❛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❤❡ ❝♦❞❡✿ ❆ ✶ ✿ check 1 ustruct f/3 A ✶ putref 1 son 1 putvar 2 putref 1 ustruct g/2 A ✷ putstruct g/2 putvar 2 son 1 bind putstruct g/2 ❇ ✷ ✿ son 2 uref 1 putatom a son 2 uatom a putvar 3 uvar 2 son 3 putstruct f/3 up B ✷ uvar 3 bind ❆ ✷ ✿ check 1 ❇ ✶ ✿ ✳ ✳ ✳ up B ✶

❈❧❛✉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✿ ♠ ❋P ❋P ♣✉s❤❡♥✈ ♠ SP = FP + m;

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

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

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

Pr❡❞✐❝❛t❡s ■♥str✉❝t✐♦♥ setbtp s❛✈❡s r❡❣✐st❡rs ❍P✱ ❚P✱ ❇P✿ ❋P ❋P ✹✷ s❡t❜t♣ ✶✼ ❍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 ❆✿ ❋P ❋P ❍P ❍P tr② ❆ ❚P ❚P ❇P ❇P P❈ P❈ ✷✾ ❆ ✷✾ NegCont = PC; PC = A;

Pr❡❞✐❝❛t❡s ■♥str✉❝t✐♦♥ delbtp r❡st♦r❡s t❤❡ ✈❛❧✉❡ ♦❢ ❇P✿ ❋P ❋P ❞❡❧❜t♣ ❍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 ❤ ❄ ❣ ✮ ❂ ❝♦❞❡ P rr ✶ init ✳ ✳ ✳ pushenv d ❝♦❞❡ ● ❣ ✚ ❝♦❞❡ P rr ❤ halt d ✇❤❡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✿ ❋P ❋P ✲✶ ❍P ❍P ✵ ✵ ❚P ✐♥✐t ❚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 1 q✴✶✿ pushenv 1 ❊✿ pushenv 1 pushenv 0 mark B mark D mark G mark A putvar 1 putref 1 putref 1 call p/0 call q/1 call s/1 call t/1 ❆✿ halt 0 ❇✿ mark C ❉✿ popenv ●✿ popenv t✴✶✿ pushenv 1 s✴✶✿ setbtp ❋✿ pushenv 1 putref 1 putref 1 call t/1 try E putref 1 ❈✿ popenv uatom b delbtp uatom a popenv jump F 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 ❝♦❞❡ ❆ t ✶ ✚ ❝♦❞❡ ● ❣ ✶ ✚ ✳ ✳ ✳ ✳ ✳ ✳ ❝♦❞❡ ❆ t ❤ ✚ ❝♦❞❡ ● ❣ ♥ � ✶ ✚ call q/h ❇✿ popenv mark B

▲❛st ❈❛❧❧ ❖♣t✐♠✐③❛t✐♦♥ ❈♦♥s✐❞❡r ❛ ❝❧❛✉s❡ r ✑ p ✭ ❳ ✶ ❀ ✿ ✿ ✿ ❀ ❳ ❦ ✮ ✥ ❣ ✶ ❀ ✿ ✿ ✿ ❀ ❣ ♥ ✱ ✇❤✐❝❤ ❤❛s ♠ ❧♦❝❛❧ ✈❛r✐❛❜❧❡s ❛♥❞ ✇❤❡r❡ ❣ ♥ ✑ q ✭ t ✶ ❀ ✿ ✿ ✿ ❀ t ❤ ✮✳ ❝♦❞❡ ❈ r ❂ pushenv m ❝♦❞❡ ❆ t ✶ ✚ ❝♦❞❡ ● ❣ ✶ ✚ ✳ ✳ ✳ ✳ ✳ ✳ ❝♦❞❡ ❆ t ❤ ✚ ❝♦❞❡ ● ❣ ♥ � ✶ ✚ lastcall (q/h,m) lastmark

▲❛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✐♦♥ ❆ ✻ ❇P ❇P ❧❛st♠❛r❦ ❋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 q✴❤ P❈

▲❛st ❈❛❧❧ ❖♣t✐♠✐③❛t✐♦♥ ❈♦♥s✐❞❡r t❤❡ ❝❧❛✉s❡ a ✭ ❳❀ ❨ ✮ ✥ f ✭ ✖ ❳❀ ❳ ✶ ✮ ❀ a ✭ ✖ ❳ ✶ ❀ ✖ ❨ ✮ ❚❤❡ ❧❛st ❝❛❧❧ ♦♣t✐♠✐③❛t✐♦♥ ②✐❡❧❞s✿ pushenv 3 putvar 3 putref 3 mark A call f/2 putref 2 putref 1 ❆✿ lastmark 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 putstruct [|]/2 ❉✿ check 4 putref 1 bind putref 4 ustruct [|]/2 B ❈✿ putref 3 putvar 6 son 1 ustruct [|]/2 D putstruct [|]/2 uvar 4 son 1 bind ❊✿ putref 5 son 2 uref 4 uvar 5 son 2 putref 2 up C uvar 6 putref 6 ❇✿ putvar 4 up E move(6,3) putvar 5 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 putvar 4 ❈✿ trim 3 mark A call p ✷ /2 lastmark putref 1 ❇✿ trim 4 putref 3 putvar 5 mark C putref 2 call p ✶ /2 putref 4 lastcall (p ✹ /2,3) ❆✿ mark B putvar 3 putref 5 call p ✸ /2

❈❧❛✉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 ◆■▲✿ jump A ✶ // [] ❈❖◆❙✿ jump A ✷ // [|] try A ✶ delbtp ❊▲❙❊✿ fail // default jump A ✷ ■♥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 ❆✿ pushenv 2 ❈✿ prune ❇✿ pushenv 2 try A mark C pushenv 2 putref 1 delbtp putref 1 lastmark putref 2 jump B call p/1 putref 1 move(2,2) putref 2 jump q ✷ /2 lastcall(q ✶ /2,2)

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