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  1. ❆ π ✲❈❛❧❝✉❧✉s ■♥t❡r♥❛❧ ❉♦♠❛✐♥✲❙♣❡❝✐✜❝ ▲❛♥❣✉❛❣❡ ❢♦r ❙❝❛❧❛ P❡❞r♦ ▼❛t✐❡❧❧♦ ♣♠❛t✐❡❧❧♦❅❣♠❛✐❧✳❝♦♠ ❆♥❛ ❈✳ ❱✳ ❞❡ ▼❡❧♦ ❛❝✈♠❅✐♠❡✳✉s♣✳❜r

  2. ❆❣❡♥ts ✵ ◆✐❧ P P Pr❡✜① P P ❙✉♠ P P P❛r❛❧❧❡❧ ① P ❘❡str✐❝t✐♦♥ ① ② P ▼❛t❝❤ ① ② P ▼✐s♠❛t❝❤ ❞❡❢ P ❉❡✜♥✐t✐♦♥s ❆ ① ✶ ① ♥ π ✲❈❛❧❝✉❧✉s ✖ ❙②♥t❛① Pr❡✜①❡s α ::= ②① ❖✉t♣✉t ¯ ② ( ① ) ■♥♣✉t ❙✐❧❡♥t τ

  3. ❞❡❢ P ❉❡✜♥✐t✐♦♥s ❆ ① ✶ ① ♥ π ✲❈❛❧❝✉❧✉s ✖ ❙②♥t❛① Pr❡✜①❡s α ::= ②① ❖✉t♣✉t ¯ ② ( ① ) ■♥♣✉t ❙✐❧❡♥t τ ❆❣❡♥ts P ::= ✵ ◆✐❧ α . P Pr❡✜① P + P ❙✉♠ P | P P❛r❛❧❧❡❧ ( ν ① ) P ❘❡str✐❝t✐♦♥ [ ① = ② ] . P ▼❛t❝❤ [ ① � = ② ] . P ▼✐s♠❛t❝❤

  4. π ✲❈❛❧❝✉❧✉s ✖ ❙②♥t❛① Pr❡✜①❡s α ::= ②① ❖✉t♣✉t ¯ ② ( ① ) ■♥♣✉t ❙✐❧❡♥t τ ❆❣❡♥ts P ::= ✵ ◆✐❧ α . P Pr❡✜① P + P ❙✉♠ P | P P❛r❛❧❧❡❧ ( ν ① ) P ❘❡str✐❝t✐♦♥ [ ① = ② ] . P ▼❛t❝❤ [ ① � = ② ] . P ▼✐s♠❛t❝❤ ❆ ( ① ✶ , ..., ① ♥ ) ❞❡❢ ❉❡✜♥✐t✐♦♥s = P

  5. π ✲❈❛❧❝✉❧✉s ✖ ❚r❛♥s✐t✐♦♥ ❘✉❧❡s Pr❡✜① α α . P − → P α → P ′ , ① �∈ α P − ❘❡str✐❝t✐♦♥ α → ( ν ① ) . P ′ ( ν ① ) . P − α − → P α . P ▼❛t❝❤ α [ ① = ① ] . P − → P α − → P , ① � = ② α . P ▼✐s♠❛t❝❤ α − → P [ ① � = ② ] . P

  6. π ✲❈❛❧❝✉❧✉s ✖ ❚r❛♥s✐t✐♦♥ ❘✉❧❡s α → P ′ , ❜♥ ( α ) ∩ ❢♥ ( ◗ )= ∅ P − P❛r❛❧❧❡❧ α → P ′ | ◗ P | ◗ − α ( ① ) → P ′ , ◗ ¯ α ✉ → ◗ ′ P − − ❈♦♠♠✉♥✐❝❛t✐♦♥ α → P ′ { ✉ / ① }| ◗ ′ P | ◗ − α → P ′ P − ❙✉♠♠❛t✐♦♥ α → P ′ P + ◗ −

  7. ❙❝❛❧❛ ✖ ❱❛r✐❛❜❧❡s ❛♥❞ ▼❡t❤♦❞s ✈❛❧ str✐♥❣✿❙tr✐♥❣ ❂ ✧◆♦t r❡❛ss✐❣♥❛❜❧❡✧ ✈❛r str✐♥❣✿❙tr✐♥❣ ❂ ✧❘❡❛ss✐❣♥❛❜❧❡✧ ❞❡❢ ♠❛①✭①✿■♥t ✱ ②✿■♥t✮✿■♥t ❂ ④ ✐❢ ✭① ❃ ②✮ ① ❡❧s❡ ② ⑥

  8. ❙❝❛❧❛ ✖ ❚②♣❡ ■♥❢❡r❡♥❝❡ ✈❛❧ str✐♥❣ ❂ ✧◆♦t r❡❛ss✐❣♥❛❜❧❡✧ ✈❛r str✐♥❣ ❂ ✧❘❡❛ss✐❣♥❛❜❧❡✧ ❞❡❢ ♠❛①✭①✿■♥t ✱ ②✿■♥t✮ ❂ ④ ✐❢ ✭① ❃ ②✮ ① ❡❧s❡ ② ⑥

  9. ❙❝❛❧❛ ✖ ❈❧❛ss❡s ❛♥❞ ❚r❛✐ts tr❛✐t P❡rs♦♥ ④ ❞❡❢ s❧❡❡♣ ④ ❚❤r❡❛❞ s❧❡❡♣ ✶✵✵✵ ⑥ ❞❡❢ t❛❧❦✿❯♥✐t ⑥ ❝❧❛ss ◆✐❝❡P❡rs♦♥ ❡①t❡♥❞s P❡rs♦♥ ④ ❞❡❢ t❛❧❦ ④ ♣r✐♥t❧♥ ✭✧ ❍❡❧❧♦ ✧✮ ⑥ ⑥

  10. ❙❝❛❧❛ ✖ ❖❜❥❡❝ts ♦❜❥❡❝t P✐♥❡❛♣♣❧❡ ④ ❞❡❢ ❡❛t ④ ♣r✐♥t❧♥ ✭✧ t❛st② ✧✮ ⑥ ⑥ s❝❛❧❛ ❃ P✐♥❡❛♣♣❧❡✳❡❛t t❛st②✦

  11. ❙❝❛❧❛ ✖ ❈❛s❡ ❈❧❛ss❡s ❛♥❞ P❛tt❡r♥ ▼❛t❝❤✐♥❣ tr❛✐t ❍✉♠❛♥ ❝❛s❡ ❝❧❛ss ▼❛♥✭♥❛♠❡✿❙tr✐♥❣✮ ❡①t❡♥❞s ❍✉♠❛♥ ❝❛s❡ ❝❧❛ss ❲♦♠❛♥✭♥❛♠❡✿❙tr✐♥❣✮ ❡①t❡♥❞s ❍✉♠❛♥ ❞❡❢ ✇❤♦■s✭❤✉♠❛♥✿❍✉♠❛♥✮ ④ ❤✉♠❛♥ ♠❛t❝❤ ④ ❝❛s❡ ▼❛♥✭♥❛♠❡✮ ❂❃ ♣r✐♥t❧♥ ✭✧❍❡ ✐s ✧✰ ♥❛♠❡✮ ❝❛s❡ ❲♦♠❛♥✭♥❛♠❡✮ ❂❃ ♣r✐♥t❧♥ ✭✧❙❤❡ ✐s ✧✰ ♥❛♠❡✮ ⑥ ⑥ s❝❛❧❛ ❃ ✇❤♦■s✭▼❛♥✭✧❏♦❡ ❉♦❡✧✮✮ ❍❡ ✐s ❏♦❡ ❉♦❡

  12. ❙❝❛❧❛ ✖ ■♠♣❧✐❝✐t ❈♦♥✈❡rs✐♦♥s ✐♠♣❧✐❝✐t ❞❡❢ ■♥t✷❙tr✐♥❣✭✐♥t✿■♥t✮ ❂ ✐♥t✳t♦❙tr✐♥❣ ❞❡❢ ❧❡♥✭str✿❙tr✐♥❣✮ ❂ str✳s✐③❡ s❝❛❧❛ ❃ ❧❡♥ ✭✶✷✸✹✮ r❡s✵✿ ■♥t ❂ ✹

  13. P✐st❛❝❤❡ ✖ ❆P■ ❆❣❡♥t ❞❡✜♥✐t✐♦♥✿ ✈❛❧ P ❂ ❆❣❡♥t ✭✳✳✳✮ ❧❛③② ✈❛❧ r❡❝P✿❆❣❡♥t ❂ ❆❣❡♥t ✭✳✳✳✮ ✈❛❧ r❡strP ❂ ❆❣❡♥t ④ ✈❛❧ r❡str✐❝t❡❞◆❛♠❡ ❂ ◆❛♠❡ ✭✳✳✳✮ ✳✳✳ ⑥ ❞❡❢ ❛r❣P✭❛r❣✶✿❚②♣❡✶ ✱ ✳✳✳✱ ❛r❣◆✿❚②♣❡◆✮✿❆❣❡♥t ❂ ❆❣❡♥t ④ ✳✳✳ ⑥

  14. P✐st❛❝❤❡ ✖ ❆P■ ◆❛♠❡s✿ ✈❛❧ ♥❛♠❡ ❂ ◆❛♠❡✭s♦♠❡❴♦❜❥❡❝t✮ ✈❛❧ ♥❛♠❡ ❂ ◆❛♠❡❬❚②♣❡❪ ♥❛♠❡ ✿❂ ♦t❤❡r❴♦❜❥❡❝t ✈❛❧✉❡ ❂ ♥❛♠❡✳✈❛❧✉❡

  15. P✐st❛❝❤❡ ✖ ❆P■ ▲✐♥❦s✿ ✈❛❧ ❧✐♥❦ ❂ ▲✐♥❦❬❚②♣❡❪ ❧✐♥❦⑦♥❛♠❡ ❧✐♥❦✭❛♥♦t❤❡r❴♥❛♠❡✮ ❙✐❧❡♥t ❚r❛♥s✐t✐♦♥s✿ ✈❛❧ s✐❧❡♥t ❂ ❆❝t✐♦♥④ ❞♦❙♦♠❡t❤✐♥❣ ✭✮ ⑥

  16. P✐st❛❝❤❡ ✖ ❆P■ Pr❡✜① ❈♦♥❝❛t❡♥❛t✐♦♥✿ ✈❛❧ P ❂ ❆❣❡♥t ④ ♣✶ ✯ ♣✷ ✯ ◗ ⑥ ▼❛t❝❤✐♥❣✿ ✈❛❧ P ❂ ❆❣❡♥t✭■❢ ✭❝♦♥❞✐t✐♦♥✮ ④◗⑥✮

  17. P✐st❛❝❤❡ ✖ ❆P■ ❆❣❡♥t ❈♦♠♣♦s✐t✐♦♥✿ ✈❛❧ P ❂ ❆❣❡♥t ④ ◗✶ ⑤ ◗✷ ⑤ ◗✸ ⑥ ❙✉♠♠❛t✐♦♥✿ ✈❛❧ P ❂ ❆❣❡♥t ④ ✭♣✶ ✿✿ ◗✶✮ ✰ ✭♣✷ ✿✿ ◗✷✮ ✰ ✭♣✸ ✿✿ ◗✸✮ ⑥

  18. π ✲❈❛❧❝✉❧✉s ✖ ❊①❛♠♣❧❡ ❚❤❡ ❛❣❡♥ts✿ ❈ = ( ν ♣ )( ν ① ) ❛ ( ♣ ) .¯ ♣① P C P = ( ν ② ) ❜ ( ② ) . τ . P b a ❙ = ¯ ❛❜ . ❙ S ❚❤❡ ❝♦♠♣♦s✐t✐♦♥✿ ❈ | P | ❙

  19. P✐st❛❝❤❡ ✖ ❊①❛♠♣❧❡ ♦❜❥❡❝t Pr✐♥ts❡r✈❡r ④ ❞❡❢ ♠❛✐♥ ✭❛r❣s✿❆rr❛②❬❙tr✐♥❣ ❪✮ ④ ✈❛❧ ❛ ❂ ▲✐♥❦❬▲✐♥❦❬❙tr✐♥❣ ❪❪ ✈❛❧ ❜ ❂ ▲✐♥❦❬❙tr✐♥❣❪ ⑥ ⑥

  20. P✐st❛❝❤❡ ✖ ❊①❛♠♣❧❡ ♦❜❥❡❝t Pr✐♥ts❡r✈❡r ④ ❞❡❢ ♠❛✐♥ ✭❛r❣s✿❆rr❛②❬❙tr✐♥❣ ❪✮ ④ ✈❛❧ ❛ ❂ ▲✐♥❦❬▲✐♥❦❬❙tr✐♥❣ ❪❪ ✈❛❧ ❜ ❂ ▲✐♥❦❬❙tr✐♥❣❪ ✈❛❧ ❈ ❂ ❆❣❡♥t ④ ✈❛❧ ♣ ❂ ◆❛♠❡❬▲✐♥❦❬❙tr✐♥❣ ❪❪ ❛✭♣✮ ✯ ♣⑦✧ ♠❡ss❛❣❡✧ ⑥ ⑥ ⑥

  21. P✐st❛❝❤❡ ✖ ❊①❛♠♣❧❡ ♦❜❥❡❝t Pr✐♥ts❡r✈❡r ④ ❞❡❢ ♠❛✐♥ ✭❛r❣s✿❆rr❛②❬❙tr✐♥❣ ❪✮ ④ ✈❛❧ ❛ ❂ ▲✐♥❦❬▲✐♥❦❬❙tr✐♥❣ ❪❪ ✈❛❧ ❜ ❂ ▲✐♥❦❬❙tr✐♥❣❪ ✈❛❧ ❈ ❂ ❆❣❡♥t ④ ✈❛❧ ♣ ❂ ◆❛♠❡❬▲✐♥❦❬❙tr✐♥❣ ❪❪ ❛✭♣✮ ✯ ♣⑦✧ ♠❡ss❛❣❡✧ ⑥ ❧❛③② ✈❛❧ ❙✿❆❣❡♥t ❂ ❆❣❡♥t ④ ❛⑦❜✯❙ ⑥ ⑥ ⑥

  22. P✐st❛❝❤❡ ✖ ❊①❛♠♣❧❡ ♦❜❥❡❝t Pr✐♥ts❡r✈❡r ④ ❞❡❢ ♠❛✐♥ ✭❛r❣s✿❆rr❛②❬❙tr✐♥❣ ❪✮ ④ ✈❛❧ ❛ ❂ ▲✐♥❦❬▲✐♥❦❬❙tr✐♥❣ ❪❪ ✈❛❧ ❜ ❂ ▲✐♥❦❬❙tr✐♥❣❪ ✈❛❧ ❈ ❂ ❆❣❡♥t ④ ✈❛❧ ♣ ❂ ◆❛♠❡❬▲✐♥❦❬❙tr✐♥❣ ❪❪ ❛✭♣✮ ✯ ♣⑦✧ ♠❡ss❛❣❡✧ ⑥ ❧❛③② ✈❛❧ ❙✿❆❣❡♥t ❂ ❆❣❡♥t ④ ❛⑦❜✯❙ ⑥ ❧❛③② ✈❛❧ P✿❆❣❡♥t ❂ ❆❣❡♥t ④ ✈❛❧ ♠s❣ ❂ ◆❛♠❡❬❙tr✐♥❣❪ ✈❛❧ ❛❝t ❂ ❆❝t✐♦♥ ④ ♣r✐♥t❧♥✭♠s❣✳✈❛❧✉❡✮ ⑥ ❜✭♠s❣✮ ✯ ❛❝t ✯ P ⑥ ⑥ ⑥

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