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t t rrs rss r ts tr s 1 r 1

  1. ❖♥ t❤❡ ❈♦♠♣❧❡①✐t② ♦❢ ●r❛♠♠❛r✲❇❛s❡❞ ❈♦♠♣r❡ss✐♦♥ ♦✈❡r ❋✐①❡❞ ❆❧♣❤❛❜❡ts ❑❛tr✐♥ ❈❛s❡❧ 1 ✱ ❍❡♥♥✐♥❣ ❋❡r♥❛✉ 1 ✱ ❙❡r❣❡ ●❛s♣❡rs 2 ✱ ❇❡♥❥❛♠✐♥ ●r❛s 3 ✱ ▼❛r❦✉s ▲✳ ❙❝❤♠✐❞ 1 1 ❚r✐❡r ❯♥✐✈❡rs✐t②✱ ●❡r♠❛♥② 2 ❉❛t❛✻✶✱ ❈❙■❘❖✱ ❆✉str❛❧✐❛ 3 ➱❝♦❧❡ ◆♦r♠❛❧❡ ❙✉♣❡r✐❡✉r❡ ❞❡ ▲②♦♥✱ ❋r❛♥❝❡ ■❈❆▲P ✷✵✶✻

  2. ✏str❛✐❣❤t✲❧✐♥❡ ♣r♦❣r❛♠✑ ✭♦r s✐♠♣❧② ❣r❛♠♠❛r✮ ❈♦♥t❡①t✲❋r❡❡ ●r❛♠♠❛rs G = ( N, Σ , R, S ) ✱ N s❡t ♦❢ ♥♦♥t❡r♠✐♥❛❧s ✱ Σ t❡r♠✐♥❛❧ ❛❧♣❤❛❜❡t ✱ S ∈ N st❛rt s②♠❜♦❧ ✱ R ⊆ N × ( N ∪ Σ) + s❡t ♦❢ r✉❧❡s r✉❧❡s ✳

  3. ❈♦♥t❡①t✲❋r❡❡ ●r❛♠♠❛rs G = ( N, Σ , R, S ) ✱ N s❡t ♦❢ ♥♦♥t❡r♠✐♥❛❧s ✱ Σ t❡r♠✐♥❛❧ ❛❧♣❤❛❜❡t ✱ S ∈ N st❛rt s②♠❜♦❧ ✱ R ⊆ N × ( N ∪ Σ) + s❡t ♦❢ r✉❧❡s r✉❧❡s ✳ G ✏str❛✐❣❤t✲❧✐♥❡ ♣r♦❣r❛♠✑ ✭♦r s✐♠♣❧② ❣r❛♠♠❛r✮ ⇔ | L ( G ) | = 1

  4. ❈♦♥t❡①t✲❋r❡❡ ●r❛♠♠❛rs G = ( N, Σ , R, S ) ✱ N s❡t ♦❢ ♥♦♥t❡r♠✐♥❛❧s ✱ Σ t❡r♠✐♥❛❧ ❛❧♣❤❛❜❡t ✱ S ∈ N st❛rt s②♠❜♦❧ ✱ R ⊆ N × ( N ∪ Σ) + s❡t ♦❢ r✉❧❡s r✉❧❡s ✳ G ✏str❛✐❣❤t✲❧✐♥❡ ♣r♦❣r❛♠✑ ✭♦r s✐♠♣❧② ❣r❛♠♠❛r✮ ⇔ | L ( G ) | = 1 | G | = � A → α ∈ R | α |

  5. ❲❡ ♠❛② ❛s❦ ❢♦r ❛ s❤♦rt❡st ❣r❛♠♠❛r✳ ❛ s❤♦rt ❣r❛♠♠❛r ✭❜✉t ❝♦♠♣✉t❡❞ ❢❛st✮✳ ❛ ❣r❛♠♠❛r t❤❛t ✐s s❤♦rt❡st ✭s❤♦rt✮ ❛♠♦♥❣ ❛❧❧ ❣r❛♠♠❛rs ✇✐t❤ ♦♥❧② ♥♦♥✲t❡r♠✐♥❛❧s ✭✐✳ ❡✳✱ ♦♥❧② r✉❧❡s✮✱ ❛ ❞❡r✐✈❛t✐♦♥s tr❡❡ ✇✐t❤ ❛t ♠♦st ❧❡✈❡❧s✱ r✉❧❡s t❤❛t ❤❛✈❡ r✐❣❤t s✐❞❡s ♦❢ s✐③❡ ❛t ♠♦st ✱ ●r❛♠♠❛r✲❇❛s❡❞ ❈♦♠♣r❡ss✐♦♥ ●❡♥❡r❛❧ ✐❞❡❛ ■♥♣✉t✿ ✇♦r❞ w ✱ ❖✉t♣✉t✿ ❣r❛♠♠❛r ❢♦r w ✳

  6. ●r❛♠♠❛r✲❇❛s❡❞ ❈♦♠♣r❡ss✐♦♥ ●❡♥❡r❛❧ ✐❞❡❛ ■♥♣✉t✿ ✇♦r❞ w ✱ ❖✉t♣✉t✿ ❣r❛♠♠❛r ❢♦r w ✳ ❲❡ ♠❛② ❛s❦ ❢♦r ❛ s❤♦rt❡st ❣r❛♠♠❛r✳ ❛ s❤♦rt ❣r❛♠♠❛r ✭❜✉t ❝♦♠♣✉t❡❞ ❢❛st✮✳ ❛ ❣r❛♠♠❛r t❤❛t ✐s s❤♦rt❡st ✭s❤♦rt✮ ❛♠♦♥❣ ❛❧❧ ❣r❛♠♠❛rs ✇✐t❤ ◮ ♦♥❧② k ♥♦♥✲t❡r♠✐♥❛❧s ✭✐✳ ❡✳✱ ♦♥❧② k r✉❧❡s✮✱ ◮ ❛ ❞❡r✐✈❛t✐♦♥s tr❡❡ ✇✐t❤ ❛t ♠♦st k ❧❡✈❡❧s✱ ◮ r✉❧❡s t❤❛t ❤❛✈❡ r✐❣❤t s✐❞❡s ♦❢ s✐③❡ ❛t ♠♦st k ✱ ◮ . . .

  7. ●r❛♠♠❛rs ❝♦✈❡r ♠❛♥② ❝♦♠♣r❡ss✐♦♥ s❝❤❡♠❡s ❢r♦♠ ♣r❛❝t✐❝❡ ✭❡✳ ❣✳✱ ▲❡♠♣❡❧✲❩✐✈✮✱ ❛r❡ ♠❛t❤❡♠❛t✐❝❛❧❧② ❡❛s② t♦ ❤❛♥❞❧❡✱ ❛❧❧♦✇ s♦❧✈✐♥❣ ♦❢ ❜❛s✐❝ ♣r♦❜❧❡♠s ✭❝♦♠♣❛r✐s♦♥✱ ♣❛tt❡r♥ ♠❛t❝❤✐♥❣✱ ♠❡♠❜❡rs❤✐♣ ✐♥ ❛ r❡❣✉❧❛r ❧❛♥❣✉❛❣❡✱ r❡tr✐❡✈✐♥❣ s✉❜✇♦r❞s✮ ❡✣❝✐❡♥t❧②✳ ●r❛♠♠❛r✲❜❛s❡❞ ❝♦♠♣r❡ss✐♦♥ ❤❛s ❛♣♣❧✐❝❛t✐♦♥s ✐♥ ❝♦♠❜✐♥❛t♦r✐❛❧ ❣r♦✉♣ t❤❡♦r②✱ ❝♦♠♣✉t✳ t♦♣♦❧♦❣②✱ ❜❡❡♥ ❡①t❡♥❞❡❞ t♦ ♠♦r❡ ❝♦♠♣❧✐❝❛t❡❞ ♦❜❥❡❝ts✱ ❡✳ ❣✳✱ tr❡❡s✱ ✇♦r❞s✳ ❆❧❣♦r✐t❤♠✐❝s ♦♥ ❈♦♠♣r❡ss❡❞ ❙tr✐♥❣s ❘❡q✉✐r❡♠❡♥t ❢♦r ❝♦♠♣r❡ss✐♦♥ s❝❤❡♠❡s✿ ❧✐♥❡❛r ♦r ♥❡❛r ❧✐♥❡❛r t✐♠❡ ❝♦♠♣r❡ss✐♦♥ ❛♥❞ ❞❡❝♦♠♣r❡ss✐♦♥✱ s✉✐t❛❜✐❧✐t② ❢♦r s♦❧✈✐♥❣ ♣r♦❜❧❡♠s ❞✐r❡❝t❧② ♦♥ t❤❡ ❝♦♠♣r❡ss❡❞ ❞❛t❛✳ ⇒ ❆❧❣♦r✐t❤♠✐❝s ♦♥ ❝♦♠♣r❡ss❡❞ str✐♥❣s

  8. ●r❛♠♠❛r✲❜❛s❡❞ ❝♦♠♣r❡ss✐♦♥ ❤❛s ❛♣♣❧✐❝❛t✐♦♥s ✐♥ ❝♦♠❜✐♥❛t♦r✐❛❧ ❣r♦✉♣ t❤❡♦r②✱ ❝♦♠♣✉t✳ t♦♣♦❧♦❣②✱ ❜❡❡♥ ❡①t❡♥❞❡❞ t♦ ♠♦r❡ ❝♦♠♣❧✐❝❛t❡❞ ♦❜❥❡❝ts✱ ❡✳ ❣✳✱ tr❡❡s✱ ✇♦r❞s✳ ❆❧❣♦r✐t❤♠✐❝s ♦♥ ❈♦♠♣r❡ss❡❞ ❙tr✐♥❣s ❘❡q✉✐r❡♠❡♥t ❢♦r ❝♦♠♣r❡ss✐♦♥ s❝❤❡♠❡s✿ ❧✐♥❡❛r ♦r ♥❡❛r ❧✐♥❡❛r t✐♠❡ ❝♦♠♣r❡ss✐♦♥ ❛♥❞ ❞❡❝♦♠♣r❡ss✐♦♥✱ s✉✐t❛❜✐❧✐t② ❢♦r s♦❧✈✐♥❣ ♣r♦❜❧❡♠s ❞✐r❡❝t❧② ♦♥ t❤❡ ❝♦♠♣r❡ss❡❞ ❞❛t❛✳ ⇒ ❆❧❣♦r✐t❤♠✐❝s ♦♥ ❝♦♠♣r❡ss❡❞ str✐♥❣s ●r❛♠♠❛rs ❝♦✈❡r ♠❛♥② ❝♦♠♣r❡ss✐♦♥ s❝❤❡♠❡s ❢r♦♠ ♣r❛❝t✐❝❡ ✭❡✳ ❣✳✱ ▲❡♠♣❡❧✲❩✐✈✮✱ ❛r❡ ♠❛t❤❡♠❛t✐❝❛❧❧② ❡❛s② t♦ ❤❛♥❞❧❡✱ ❛❧❧♦✇ s♦❧✈✐♥❣ ♦❢ ❜❛s✐❝ ♣r♦❜❧❡♠s ✭❝♦♠♣❛r✐s♦♥✱ ♣❛tt❡r♥ ♠❛t❝❤✐♥❣✱ ♠❡♠❜❡rs❤✐♣ ✐♥ ❛ r❡❣✉❧❛r ❧❛♥❣✉❛❣❡✱ r❡tr✐❡✈✐♥❣ s✉❜✇♦r❞s✮ ❡✣❝✐❡♥t❧②✳

  9. ❆❧❣♦r✐t❤♠✐❝s ♦♥ ❈♦♠♣r❡ss❡❞ ❙tr✐♥❣s ❘❡q✉✐r❡♠❡♥t ❢♦r ❝♦♠♣r❡ss✐♦♥ s❝❤❡♠❡s✿ ❧✐♥❡❛r ♦r ♥❡❛r ❧✐♥❡❛r t✐♠❡ ❝♦♠♣r❡ss✐♦♥ ❛♥❞ ❞❡❝♦♠♣r❡ss✐♦♥✱ s✉✐t❛❜✐❧✐t② ❢♦r s♦❧✈✐♥❣ ♣r♦❜❧❡♠s ❞✐r❡❝t❧② ♦♥ t❤❡ ❝♦♠♣r❡ss❡❞ ❞❛t❛✳ ⇒ ❆❧❣♦r✐t❤♠✐❝s ♦♥ ❝♦♠♣r❡ss❡❞ str✐♥❣s ●r❛♠♠❛rs ❝♦✈❡r ♠❛♥② ❝♦♠♣r❡ss✐♦♥ s❝❤❡♠❡s ❢r♦♠ ♣r❛❝t✐❝❡ ✭❡✳ ❣✳✱ ▲❡♠♣❡❧✲❩✐✈✮✱ ❛r❡ ♠❛t❤❡♠❛t✐❝❛❧❧② ❡❛s② t♦ ❤❛♥❞❧❡✱ ❛❧❧♦✇ s♦❧✈✐♥❣ ♦❢ ❜❛s✐❝ ♣r♦❜❧❡♠s ✭❝♦♠♣❛r✐s♦♥✱ ♣❛tt❡r♥ ♠❛t❝❤✐♥❣✱ ♠❡♠❜❡rs❤✐♣ ✐♥ ❛ r❡❣✉❧❛r ❧❛♥❣✉❛❣❡✱ r❡tr✐❡✈✐♥❣ s✉❜✇♦r❞s✮ ❡✣❝✐❡♥t❧②✳ ●r❛♠♠❛r✲❜❛s❡❞ ❝♦♠♣r❡ss✐♦♥ ❤❛s ❛♣♣❧✐❝❛t✐♦♥s ✐♥ ❝♦♠❜✐♥❛t♦r✐❛❧ ❣r♦✉♣ t❤❡♦r②✱ ❝♦♠♣✉t✳ t♦♣♦❧♦❣②✱ ❜❡❡♥ ❡①t❡♥❞❡❞ t♦ ♠♦r❡ ❝♦♠♣❧✐❝❛t❡❞ ♦❜❥❡❝ts✱ ❡✳ ❣✳✱ tr❡❡s✱ 2 D ✇♦r❞s✳

  10. ❣r❛♠♠❛r ✇✐t❤ ❊①❛♠♣❧❡s ✭ 1 / 2 ✮ w = � n i =1 10 i ✱ ✇✐t❤ n = 2 k ✳ 10 100 1000 10000 100000 1000000 10000000 100000000 . . .

  11. ❣r❛♠♠❛r ✇✐t❤ ❊①❛♠♣❧❡s ✭ 1 / 2 ✮ w = � n i =1 10 i ✱ ✇✐t❤ n = 2 k ✳ 10 100 1000 10000 100000 1000000 10000000 100000000 . . .

  12. ❣r❛♠♠❛r ✇✐t❤ ❊①❛♠♣❧❡s ✭ 1 / 2 ✮ w = � n i =1 10 i ✱ ✇✐t❤ n = 2 k ✳ A 1 100 1000 10000 100000 1000000 10000000 100000000 . . . A 1 → 10

  13. ❣r❛♠♠❛r ✇✐t❤ ❊①❛♠♣❧❡s ✭ 1 / 2 ✮ w = � n i =1 10 i ✱ ✇✐t❤ n = 2 k ✳ A 1 100 1000 10000 100000 1000000 10000000 100000000 . . . A 1 → 10

  14. ❣r❛♠♠❛r ✇✐t❤ ❊①❛♠♣❧❡s ✭ 1 / 2 ✮ w = � n i =1 10 i ✱ ✇✐t❤ n = 2 k ✳ A 1 A 2 1000 10000 100000 1000000 10000000 100000000 . . . A 1 → 10 A 2 → A 1 0

  15. ❣r❛♠♠❛r ✇✐t❤ ❊①❛♠♣❧❡s ✭ 1 / 2 ✮ w = � n i =1 10 i ✱ ✇✐t❤ n = 2 k ✳ A 1 A 2 1000 10000 100000 1000000 10000000 100000000 . . . A 1 → 10 A 2 → A 1 0

  16. ❣r❛♠♠❛r ✇✐t❤ ❊①❛♠♣❧❡s ✭ 1 / 2 ✮ w = � n i =1 10 i ✱ ✇✐t❤ n = 2 k ✳ A 1 A 2 A 3 10000 100000 1000000 10000000 100000000 . . . A 1 → 10 A 2 → A 1 0 A 3 → A 2 0

  17. ❣r❛♠♠❛r ✇✐t❤ ❊①❛♠♣❧❡s ✭ 1 / 2 ✮ w = � n i =1 10 i ✱ ✇✐t❤ n = 2 k ✳ A 1 A 2 A 3 10000 100000 1000000 10000000 100000000 . . . A 1 → 10 A 2 → A 1 0 A 3 → A 2 0

  18. ❣r❛♠♠❛r ✇✐t❤ ❊①❛♠♣❧❡s ✭ 1 / 2 ✮ w = � n i =1 10 i ✱ ✇✐t❤ n = 2 k ✳ A 1 A 2 A 3 A 4 100000 1000000 10000000 100000000 . . . A 1 → 10 A 2 → A 1 0 A 3 → A 2 0 A 4 → A 3 0

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