HAL Id: hal-00995238
https://hal.archives-ouvertes.fr/hal-00995238
Submitted on 23 May 2014
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
Approximation of relations by propositional formulas:
complexity and semantics
Bruno Zanuttini
To cite this version:
Bruno Zanuttini. Approximation of relations by propositional formulas: complexity and semantics.
5th Symposium on Abstraction, Reformulation and Approximation (SARA 2002), 2002, Canada.
pp.242–255. �hal-00995238�
✡✄✂✝✏☞✞✌ ✑ ✒✄✝✁☞☛☎✆✟✎ ✞✠✓ ✌☛✝✞✠✟✆✒✌
✔✕✖✗✘ ✙✚✗✖✛✛✜✗✜✢
✣✤✥✦✧ ★ ✩✪✫✬✭✮✯✫✰✭✱✲✭ ✧✳✭✪
✴✲✵✶✳✷ ✸✹✫✪✺✻★✼✽✾✿❀✧✳✭✪ ✧✭✲✭❁✺✻ ✮✳✪❂✭
❃❄❅❆❇❇❈❉❈❅❊❋●❆❅❈❍❄■❅●❊❏
❑▲▼◆❖P◗◆❘ ❙✭✷❚✳✪ ✳✪✲ ❯✳✹✰❱ ✫✪✰✮❲✲✹❂✭✲ ✰❳✭ ✪❲✰✫❲✪ ❲❨ ✳❩❩✮❲❁✫❚✳✰✫❲✪
❲❨ ✳ ✰❳✭❲✮❬ ✳✪✲ ✯❳❲❭✭✲ ✫✰✯ ✹✯✭❨✹✷✪✭✯✯ ❨❲✮ ❪✪❲❭✷✭✲❫✭ ❂❲❚❩✫✷✳✰✫❲✪ ✳✪✲
❲✪★✷✫✪✭ ✮✭✳✯❲✪✫✪❫✵ ❴✭ ✯✰✹✲❬ ❳✭✮✭ ✰❳✭ ❂❲❚❩✷✭❁✫✰❬ ❲❨ ✰❳✭ ❚✳✫✪ ❂❲❚❩✹✰✳★
✰✫❲✪✳✷❩✮❲✴✷✭❚✯✮✭✷✳✰✭✲✰❲✰❳✭✳❩❩✮❲❁✫❚✳✰✫❲✪ ❲❨✮✭✷✳✰✫❲✪✯❵✯✭✰✯❲❨❩❲✯✯✫✴✷✭
❭❲✮✷✲✯❛
✴
❬ ❩✮❲❩❲✯✫✰✫❲✪✳✷ ❨❲✮❚✹✷✳✯✺ ✳✪✲ ✰❳✭ ✯✭❚✳✪✰✫❂✯❲❨ ✮✭✳✯❲✪✫✪❫ ❭✫✰❳
✰❳✭✯✭ ✳❩❩✮❲❁✫❚✳✰✫❲✪✯ ❵ ✲✭✲✹❂✰✫❲✪ ✳✪✲ ✳✴✲✹❂✰✫❲✪❛✵❜❳✭ ❂✷✳✯✯✭✯ ❲❨ ❨❲✮❚✹★
✷✳✯ ✰❳✳✰ ❭✭ ❂❲✪✯✫✲✭✮ ✳✮✭ ✰❳❲✯✭ ❲❨ ❵✮✭✬✭✮✯✭★❛❝❲✮✪✺ ✴✫❞✹✪❂✰✫✬✭ ✳✪✲ ✳❡✪✭
❨❲✮❚✹✷✳✯✺❭❳✫❂❳ ✳✮✭ ✰❳✭ ❚❲✯✰✫✪✰✭✮✭✯✰✫✪❫❨❲✮ ✯✰❲✮✫✪❫ ❪✪❲❭✷✭✲❫✭✵
✧❲✪❂✭✮✪✫✪❫✰❳✭❂❲❚❩✹✰✳✰✫❲✪❲❨✳❩❩✮❲❁✫❚✳✰✫❲✪✯✺❭✭✯✹✮✬✭❬✳✪✲ ❂❲❚❩✷✭✰✭
✰❳✭ ✮✭✯✹✷✰✯✰❳✳✰❂✳✪ ✴✭❨❲✹✪✲ ✫✪ ✰❳✭✷✫✰✭✮✳✰✹✮✭✺✰✮❬✫✪❫✰❲ ✳✲❲❩✰✳✹✪✫❢✭✲
❩❲✫✪✰❲❨ ✬✫✭❭✵❣✪ ✰❳✭ ❂❲✪✰✮✳✮❬✺✳✯❨✳✮ ✳✯❭✭ ❪✪❲❭ ✰❳✫✯❩✳❩✭✮ ✫✯✰❳✭ ❢✮✯✰
✮✭✳✷ ✳✰✰✭❚❩✰ ✰❲ ✯✰✹✲❬ ✰❳✭ ✯✭❚✳✪✰✫❂✯❲❨ ✳✴✲✹❂✰✫❲✪ ❭✫✰❳ ✰❳✭ ✴❲✹✪✲✯ ❲❨ ✳
✰❳✭❲✮❬✵
❤ ✐❥❦❧♠♥♦♣❦q♠❥
rsts✗✛✉✈ ✛✇s✕s ✇✚① ②ss✗ ✚ ③✕s✚✛ ✚④✘✖✗✛ ✘⑤ ✕s①s✚✕t✇ ✚②✘✖✛ ⑥⑦⑧⑨⑩❶❷❸❶ ❹
❺❺❻
⑧❼❽
❾❿
❹➀❾⑧⑦ ➁➂➃➄➅➆➇➈➈➉➅➊➇➋✚✉➅➊➇➉➈➅➌➇➈r➅➌➍ ⑤✘✕ ✜✗①✛✚✗ts➎➏ ➐✗ ✚➑➑✕✘➒✜④✚✛✜✘✗ ✘⑤
✚ ✛✇s✘✕✈ ➓ ✕s➑✕s①s✗✛✜✗③ ①✘④s ➔✗✘→✉s➣③s ✜① ✚✗✘✛✇s✕ ✛✇s✘✕✈ ✛✇✚✛ ✚➑➑✕✘✚t✇s① ✜✛
✛✇s ②s①✛ ➁✜✗ ①✘④s➑✕st✜①s ①s✗①s➎②✖✛ ✚✉✉✘→① ⑤✘✕ s↔t✜s✗✛ ✕s✚①✘✗✜✗③ ➁⑤✘✕ ✜✗①✛✚✗ts➇
➣s➣✖t✛✜✘✗➎ →✇s✗ ➓ ➣✘s① ✗✘✛➏ ↕✇s ✜➣s✚ ✜① ✛✘ ①✛✘✕s ✛✇s ✚➑➑✕✘➒✜④✚✛✜✘✗ →✜✛✇ ✛✇s
✛✇s✘✕✈✜✛①s✉⑤➇✚✗➣✛✘✖①s✜✛⑤✘✕①➑ss➣✜✗③✖➑⑤✖✕✛✇s✕✕s✚①✘✗✜✗③➏↕✇✖①t✘④➑✖✛✜✗③ ✚➑➙
➑✕✘➒✜④✚✛✜✘✗①t✚✗②s➛✜s→s➣✚①✚ ⑥⑦⑧⑨⑩❶❷❸❶ ➜⑧❿
❺
❾⑩❹➀❾⑧⑦ ✛✚①➔ ➂➉➈➅➌➍ ➝➑✕✚t✛✜t✚✉✉✈
①➑s✚➔✜✗③➇ ✛✇✜① t✘④➑✜✉✚✛✜✘✗ ✜① ④✘①✛ ✘⑤ ✛✇s ✛✜④s ✜✗✛s✗➣s➣ ✚① ✚✗
⑧➞
❽⑩❾⑦❶ ➑✕✘ts①①
➑s✕⑤✘✕④s➣ ✜✗ ✘✕➣s✕✛✘ ①➑ss➣ ✖➑ ⑤✖✕✛✇s✕ ⑧⑦❽⑩❾⑦❶ ✕s✚①✘✗✜✗③➏
➟s ✚✕s ✜✗✛s✕s①✛s➣ ✇s✕s ✜✗
❺❻
⑧
❺
⑧➠❾➀❾⑧⑦❹⑩ ✛✇s✘✕✜s①➏➡✚✗✈ ➑✚➑s✕① ✇✚➛s ⑤✘t✖①s➣
✘✗✛✇s④ ➂➃➄➅➆➇➈➈➉➅➊➇➉➈➅➌➇✔✘✖➅➢➇➤➉➥➥➍➇④✚✈✛✇s✘✕✜③✜✗✚✉✛✇s✘✕✜s①②s③✜➛s✗✚①
✕s✉✚✛✜✘✗①➂➃➄➅➆➇➈➈➉➅➊➍✘✕✚①⑤✘✕④✖✉✚①➂➋✚✉➅➊➇ ➉➈➅➌➇➤➉➥➥➍➏➦✗✛✇✜①➑✚➑s✕➇→st✘✗➙
①✜➣s✕✛✇st✚①s→✇s✗✛✇s✘✕✜③✜✗✚✉✛✇s✘✕✈➓ ✜①③✜➛s✗✚①✚✕s✉✚✛✜✘✗➇→✇✜t✇t✘✕✕s①➑✘✗➣①
✛✘ ✛✇st✚①s→✇s✗ ➔✘✗→✉s➣③s✜① ①✛✘✕s➣ ✚①✚ ①s✛ ✘⑤ ✘②①s✕➛✚✛✜✘✗① ➂➃➄➅➆➇✙➧➥➆✚➍ ➁✘✕
❺
⑧➠➠❾➨⑩❶⑨⑧
❻
⑩❷➠ ➂➈➈➉➅➊➍➎➝✜✗✛✇✜①⑤✕✚④s→✘✕➔➇→s①✛✖➣✈✛✇st✘④➑✉s➒✜✛✈✘⑤t✘④➑✖✛➙
✜✗③ ✚
❺❻
⑧
❺
⑧➠❾➀❾⑧⑦❹⑩➩⑧
❻
❿
➫⑩❹ ✚➑➑✕✘➒✜④✚✛✜✗③ ➓➏↕✇s④✘✛✜➛✚✛✜✘✗⑤✘✕✚➑➑✕✘➒✜④✚✛✜✗③
②✈ ⑤✘✕④✖✉✚①✜①✛✇✚✛ ④✘①✛✘⑤✛✇s✛✜④s⑤✘✕④✖✉✚①✚✕s①④✚✉✉s✕✛✇✚✗✛✇s✜✕ ①s✛ ✘⑤ ④✘➣➙
s✉①➇✚✗➣ ①✜✗ts→✚ ✚✜④ ✚✛ ①✛✘✕✜✗③✛✇s✚➑➑✕✘➒✜④✚✛✜✘✗✚✗➣✕s✚①✘✗✜✗③→✜✛✇✜✛➇✜✛①①✜➭s
✜① ✚ ➛s✕✈ ✜④➑✘✕✛✚✗✛➑✚✕✚④s✛s✕ ✛✘ ✛✚➔s ✜✗✛✘ ✚tt✘✖✗✛➏
✛✇s✘✕✈ ✜✗ ✛s✕④① ✘⑤ ①s✛① ✘⑤ ④✘➣s✉① ➝→s ✛✇s✗ ✛✚✉➔ ✚②✘✖✛ ✁
❺❺
❶
❻ ✂
⑧➫⑦❷➠ →✇s✗ ✛✇s
✚➑➑✕✘➒✜④✚✛✜✘✗ ✜① ⑩⑧❸❾➜❹⑩⑩✄ ⑨❶❹⑥❶
❻
✛✇✚✗ ✛✇s ✛✇s✘✈➇ ✚✗➣ ✘⑤ ☎⑧⑨❶
❻ ✂
⑧➫⑦❷➠ →✇s✗
✜✛ ✜① ⑩⑧❸❾➜❹⑩⑩✄ ➠➀
❻
⑧⑦❸❶
❻
➏ ✜✗✚✉✉✈➇ →s ✚✕s ✜✗✛s✕s①✛s➣ ✜✗ ✚➑➑✕✘➒✜④✚✛✜✘✗① ✜✗✛✘ ✛✇s
t✉✚①①s① ✘⑤ ➁✕s➛s✕①s➙➎➧✘✕✗➇ ②✜✆✖✗t✛✜➛s ✚✗➣ ✚↔✗s ⑤✘✕④✖✉✚①➇ ⑤✘✕ ✛✇s①s t✉✚①①s① ✚✕s
✛✇s ④✘①✛ ✜④➑✘✕✛✚✗✛ ✘✗s① ⑤✘✕ ✕s✚①✘✗✜✗③ ✚✗➣ ✕s➑✕s①s✗✛✜✗③ ➔✗✘→✉s➣③s➇ ✜✗ ④✚✗✈
①s✗①s①➏➦✗➣ss➣➇ ✛✇s✈ ✚✕s➇✚④✘✗③ ✘✛✇s✕ ➑✕✘➑s✕✛✜s①➇ ✛✕✚t✛✚②✉s ⑤✘✕ ✛✇s ①✚✛✜①✝✚②✜✉✜✛✈
➑✕✘②✉s④➇ ①✛✚②✉s ②✈ t✘✗✆✖✗t✛✜✘✗➇ ②✈ t✘✗✆✖✗t✛✜✘✗ ✘⑤ ⑤✚t✛① ➁✖✗✜✛ t✉✚✖①s①➎ ✚✗➣ ②✈
✖✗✜✛ ✕s①✘✉✖✛✜✘✗➇✚✗➣ ✛✕✚t✛✚②✉s⑤✘✕ ➣s➣✖t✛✜✘✗ ✘⑤t✉✚✖①s①➏
✞✖✕ t✘✗✛✕✜②✖✛✜✘✗ ✜① ✛✇s ⑤✘✉✉✘→✜✗③➏ ➟s ✝✕①✛ ①✖✕➛s✈ ✚✗➣ t✘④➑✉s✛s ✛✇s ✕s①✖✉✛①
t✘✗ts✕✗✜✗③ ✛✇s t✘④➑✉s➒✜✛✈ ✘⑤ t✘④➑✖✛✜✗③ ✟s✚①✛ ✠➑➑s✕ ✔✘✖✗➣① ✚✗➣ ✡✕s✚✛s①✛
✟✘→s✕ ✔✘✖✗➣① ✘⑤ ✕s✉✚✛✜✘✗① ✜✗✛✘ ✛✇s①s t✉✚①①s①✘⑤⑤✘✕④✖✉✚①➇✛✕✈✜✗③ ✛✘ ✚➣✘➑✛ ✚ ✖✗✜➙
✝s➣ ➑✘✜✗✛ ✘⑤ ➛✜s→ ➝ ④✘✕s✘➛s✕➇→s ③✜➛s s➒➑✉✜t✜✛ s➒✚④➑✉s① ✚✗➣ t✘✖✗✛s✕➙s➒✚④➑✉s①➇
✘✕ ➣s✛✚✜✉ ✛✇s ➑✕✘✘⑤① ✛✇✚✛ t✚✗ ②s ⑤✘✖✗➣ ✜✗ ✛✇s ✉✜✛s✕✚✛✖✕s →✇s✗ ✖①s⑤✖✉➏ ➉st✘✗➣✉✈➇
→s ①✛✖➣✈ ✛✇s ①s④✚✗✛✜t① ✘⑤ ✕s✚①✘✗✜✗③ →✜✛✇ ①✖t✇ ②✘✖✗➣①➇ ✜➏s➏➇ ✛✇s ✉✜✗➔① ②s✛→ss✗
✛✇s✚✗①→s✕①✘②✛✚✜✗s➣→✇s✗✕s✚①✘✗✜✗③→✜✛✇✛✇s②✘✖✗➣①✚✗➣✛✇✘①s✛✇✚✛t✘✖✉➣✇✚➛s
②ss✗ ✘②✛✚✜✗s➣ →✜✛✇ ✛✇s ✘✕✜③✜✗✚✉ ✛✇s✘✕✈➏ ➟✇✜✉s ➣s➣✖t✛✜✘✗ ✇✚① ②ss✗ s➒✛s✗①✜➛s✉✈
①✛✖➣✜s➣ ✜✗ ➂➋✚✉➅➊➇ ➉➈➅➌➍➇✚①⑤✚✕ ✚①→s➔✗✘→ ✛✇s ①s④✚✗✛✜t①✘⑤✚②➣✖t✛✜✘✗ ✇✚➛s ✗✘✛
✕s✚✉✉✈ ②ss✗ ①✛✖➣✜s➣ ✜✗ ✛✇s✉✜✛s✕✚✛✖✕s➏
↕✇s ➑✚➑s✕ ✜① ①✛✕✖t✛✖✕s➣ ✚① ⑤✘✉✉✘→①➏ ➉st✛✜✘✗ ➆ ✜① ➣s➛✘✛s➣ ✛✘ ✛✇s s➒➑✉✚✗✚✛✜✘✗
✘⑤ ✘✖✕ ➣s✝✗✜✛✜✘✗① ✚✗➣ t✘✗➛s✗✛✜✘✗①➏ ➦✗ ①st✛✜✘✗ ☛ →s ➣✜①t✖①① ✛✇s t✘④➑✖✛✚✛✜✘✗ ✘⑤
➫
❺❺
❶
❻
➨⑧➫⑦❷➠➇✜✗ ①st✛✜✘✗ ☞ ✛✇✚✛✘⑤ ⑩⑧⑨❶
❻
②✘✖✗➣①➇✚✗➣ ✜✗ ①st✛✜✘✗ ➊ →s ➣✜①t✖①①✛✇s
①s④✚✗✛✜t①✘⑤✕s✚①✘✗✜✗③→✜✛✇✛✇s①s②✘✖✗➣①➏➟ss✗➣ →✜✛✇ ✚ ①✖④④✚✕✈✘⑤✛✇s✕s①✖✉✛①
✚✗➣ ⑤✖✕✛✇s✕ ➣✜✕st✛✜✘✗① ⑤✘✕ ✕s①s✚✕t✇ ✜✗ ①st✛✜✘✗ ➌➏
✌ ✍❧✎✏q✑q❥✒❧q✎✓
➟s ✚①①✖④s✚ t✘✖✗✛✚②✉s ✗✖④②s✕ ✘⑤➑✕✘➑✘①✜✛✜✘✗✚✉ ➛✚✕✜✚②✉s① ✔
✢✕
✔✖
✕✗✗✗
➏ ➐
❻
❶⑩❹➀❾⑧⑦
✘✗ ✘➥
✕✙✚
✜① ✚ ①✖②①s✛ ✘⑤ ✘➥
✕✙✚✛
➝⑤✘✕ ✜✗①✛✚✗ts➇ ✜ ✢ ✘➥➥
✙✕
➥
✙✙✕✙
➥
✙✚
✜① ✚ ✕s✉✚✛✜✘✗➏
➟s➛✜s→ ✚✗✈➛st✛✘✕✣ ✘⑤✘➥
✕✙✚✛
✚①✚ ➥✤
✙
✚①①✜③✗④s✗✛✛✘ ✥➛✚✕✜✚②✉s① ➁✜✗ ✛✇s✘✕➣s✕
✘⑤ ✛✇s✜✕ ✜✗➣✜ts①➎➇ ✚✗➣ ✣✦ ➣s✗✘✛s① ✛✇s ➛✚✉✖s ✚①①✜③✗s➣ ②✈ ✣ ✛✘ ✔✦➏ ➟s →✕✜✛s ✧✜✧
✛✇s ✗✖④②s✕ ✘⑤ ➛st✛✘✕①✜✗ ✚ ✕s✉✚✛✜✘✗ ✜➏
➐ ⑩❾➀❶
❻
❹⑩ ✜① s✜✛✇s✕ ✚ ➛✚✕✜✚②✉s✔✦ ➁
❺
⑧➠❾➀❾★❶✉✜✛s✕✚✉➎✘✕✜✛① ✗s③✚✛✜✘✗ ✩✔✦ ➁⑦❶❸❹➀❾★❶
✉✜✛s✕✚✉➎➏➐ ➜⑩❹➫➠❶✜①✚ ✝✗✜✛s➣✜①✆✖✗t✛✜✘✗✘⑤✉✜✛s✕✚✉①➇✚✗➣✚ ⑤✘✕④✖✉✚✜①✜✗ ➜⑧⑦✪➫⑦➜➀❾★❶
⑦⑧
❻
❿❹⑩➩⑧
❻
❿ ➁➤✫ ⑤✘✕①✇✘✕✛➎✜⑤✜✛✜①→✕✜✛✛s✗ ✚①✚✝✗✜✛s t✘✗✆✖✗t✛✜✘✗ ✘⑤ t✉✚✖①s①➏➐
➥✤
✙
✚①①✜③✗④s✗✛✣ ✛✘✛✇s➛✚✕✜✚②✉s①✘⑤✚⑤✘✕④✖✉✚✬ ✜✗ ➤✫ ✜①✚ ❿⑧❷❶⑩✘⑤✬ ➁→✕✜✛✛s✗
✣ ✧✢ ✬➎ ✜⑤ ✜✛ ①✚✛✜①✝s① ✚✛ ✉s✚①✛ ✘✗s ✉✜✛s✕✚✉ ✜✗ s✚t✇ t✉✚✖①s ✘⑤ ✬➏ ✘✕ ✜✗①✛✚✗ts➇ ✛✇s
✚①①✜③✗④s✗✛ ✣ ✢ ➥➥
✙
✛✘ ✛✇s ➛✚✕✜✚②✉s① ✔
✢✕
✔
✖
✕
✔✭ ✜① ✚ ④✘➣s✉ ✘⑤ ✛✇s ➤✫ ✬ ✢
➁✔
✢✮
✩✔✖➎✯ ➁ ✩✔
✢✮
✔✖
✮
✩✔✭➎➏ ➐✗ ❹✰⑦❶ ⑤✘✕④✖✉✚ ➂➉t✇✱➢➇➈➉➅➢➍ ✜① ✚ t✘✗✆✖✗t✛✜✘✗
✘⑤ ✉✜✗s✚✕ s✲✖✚✛✜✘✗① ④✘➣✖✉✘ ➆➏➐
❿
⑧❷❶⑩ ✘⑤ ✚✗ ✚↔✗s ⑤✘✕④✖✉✚ ✬ ✜① ✚✗ ✚①①✜③✗④s✗✛
✣ ✛✘ ✜✛①➛✚✕✜✚②✉s①✛✇✚✛ ①✚✛✜①✝s①✚✉✉✛✇ss✲✖✚✛✜✘✗① ➁→✕✜✛✛s✗ ✣ ✧✢ ✬➎➏ ✘✕ ✜✗①✛✚✗ts➇
✣ ✢ ➥➥
✙
✜① ✚ ④✘➣s✉ ✘⑤ ✛✇s ✚↔✗s ⑤✘✕④✖✉✚ ✬ ✢ ➁✔
✢✳
✔✭ ✢
✙
➎✯ ➁✔
✢✳
✔✖ ✢ ➥➎➏
➐①t✚✗②s ①ss✗➇s✲✖✚✛✜✘✗① ➑✉✚✈ ✛✇s①✚④s✕✘✉s⑤✘✕ ✚↔✗s ⑤✘✕④✖✉✚①✛✇✚✗ t✉✚✖①s①➣✘
⑤✘✕ ➤✫ ①➏➦⑤ ✬ ✜① ✚ ➤✫ ✘✕✚✗ ✚↔✗s ⑤✘✕④✖✉✚➇→s➣s✗✘✛s ②✈ ✴➁✬➎ ✛✇s ①s✛ ✘⑤✜✛①
④✘➣s✉①➏
❻
✉✜✛s✕✚✉ ➑s✕ t✉✚✖①s ➝ ✜✛ ✜① ✕s➛s✕①s➙➧✘✕✗ ✜⑤ ✜✛ ✇✚① ✚✛ ④✘①✛ ✘✗s ✗s③✚✛✜➛s ✉✜✛s✕✚✉ ➑s✕
t✉✚✖①s➏ ✘✕ ✜✗①✛✚✗ts➇ ✛✇s ⑤✘✕④✖✉✚ ✬
✢
✢ ➁✔
✢✮
✩✔✖
✮
✩✔✭➎✯ ➁✔✖➎✯ ➁ ✩✔
✢✮
✩✔✭➎
✜① ➧✘✕✗➇ ✚✗➣ ✬✖ ✢ ➁✔
✢ ✮
✩✔✖➎ ✯ ➁✔✭➎ ✜① ✕s➛s✕①s➙➧✘✕✗➏ ✫✘✛s ✛✇✚✛ ✜⑤ ✚ ➤✫ ✜①
✕s➛s✕①s➙➧✘✕✗➇✛✇s✗ ✕s➑✉✚t✜✗③ s✚t✇ ➛✚✕✜✚②✉s →✜✛✇ ✜✛① ✗s③✚✛✜✘✗ ✜✗ ✜✛ ✈✜s✉➣① ✚ ➧✘✕✗
⑤✘✕④✖✉✚➇ ✚✗➣ t✘✗➛s✕①s✉✈ ➝ ✛✇s t✘✕✕s①➑✘✗➣✜✗③ ④✘➣s✉① ✚✕s ✚✉①✘ ①✛✕✘✗③✉✈ ✉✜✗➔s➣➇ ✜✗
✛✇s ①s✗①s ✛✇✚✛ ✛✇s ④✘➣s✉① ✘⑤ ✘✗s✚✕s✛✇s ✘✛✇s✕①✁→✇s✕s ➥ ✜① ✕s➑✉✚ts➣ →✜✛✇
✙
✚✗➣
✙
→✜✛✇ ➥➏ ✘✕ ✛✇✚✛ ✕s✚①✘✗➇ →s ✘✗✉✈ ①✛✖➣✈ ➧✘✕✗ ⑤✘✕④✖✉✚① ✜✗ ✛✇s ➑✚➑s✕ ➝ ✚✉✉ ✛✇s
✕s①✖✉✛① ⑤✘✕ ✕s➛s✕①s➙➧✘✕✗ ⑤✘✕④✖✉✚① ✚✕s ➣✖✚✉➏ ✜✗✚✉✉✈➇ ✚ ➤✫ ✜① ➨❾✪➫⑦➜➀❾★❶ ✜⑤✜✛ ✇✚①
✚✛ ④✘①✛ ✛→✘ ✉✜✛s✕✚✉① ➑s✕ t✉✚✖①s➏ ✘✕ ✜✗①✛✚✗ts➇ ✬
✢
✚②✘➛s ✜① ✗✘✛ ②✜✆✖✗t✛✜➛s →✇✜✉s
✬✖ ✜①➏ ➟s ➣s✗✘✛s ②✈ ✂✄☎✆ ➁✕s①➑➏ ✝✞✟✞☎✠✞✡✂✄☎✆➇ ☛☞✌✍✆✎✏☞✟✞➎ ✛✇s t✉✚①① ✘⑤
✚✉✉ ➧✘✕✗ ➁✕s①➑➏ ✕s➛s✕①s➙➧✘✕✗➇ ②✜✆✖✗t✛✜➛s➎ ➤✫ ①➇ ✚✗➣ ②✈ ✑✒✒☞✆✞ ✛✇s t✉✚①① ✘⑤ ✚✉✉
✚↔✗s⑤✘✕④✖✉✚①➏✠✗✉s①① ①✛✚✛s➣✘✛✇s✕→✜①s➇→✇s✗ →s t✘✗①✜➣s✕ ✚ t✉✚①① ✘⑤ ⑤✘✕④✖✉✚①✓
✜✛ ✜① ✚✉→✚✈①✘✗s✘⑤✛✇s①s➏➦✗ ✛✇s➑✚➑s✕➇✛✇s✛s✕④ ➩⑧
❻
❿
➫⑩❹ ③✕✘✖➑①②✘✛✇ ➤✫ ①✚✗➣
✚↔✗s⑤✘✕④✖✉✚①➏
✞✖✕ ④✘✛✜➛✚✛✜✘✗ ⑤✘✕ ①✛✖➣✈✜✗③ ✛✇s①s t✉✚①①s① ✘⑤⑤✘✕④✖✉✚①t✘④s①⑤✕✘④ ✛✇s✜✕➛s✕✈
③✘✘➣t✘④➑✖✛✚✛✜✘✗✚✉➑✕✘➑s✕✛✜s①⑤✘✕✕s✚①✘✗✜✗③➏ ✜✕①✛✘⑤✚✉✉➇✛✇s✈✚✕s✚✉✉➑✘✉✈✗✘④✜✚✉
⑤✘✕ ✛✇s①✚✛✜①✝✚②✜✉✜✛✈➑✕✘②✉s④ ✔✉✜✗s✚✕⑤✘✕ ➁✕s➛s✕①s➙➎➧✘✕✗ ✚✗➣ ②✜✆✖✗t✛✜➛s⑤✘✕④✖✉✚①➇
✚✗➣ ✲✖✚➣✕✚✛✜t⑤✘✕✚↔✗s ➁④✘✕s➑✕st✜①s✉✈➇➉➐↕ ⑤✘✕✚↔✗s⑤✘✕④✖✉✚①t✚✗②s①✘✉➛s➣ ✜✗
✛✜④s✕➁✖
✖
✥➎✜⑤✛✇s⑤✘✕④✖✉✚t✘✗①✜①✛①✘⑤✖ s✲✖✚✛✜✘✗①✘➛s✕✥➛✚✕✜✚②✉s①➇→✜✛✇③✚✖①①✜✚✗
s✉✜④✜✗✚✛✜✘✗➇ ①ss ⑤✘✕ ✜✗①✛✚✗ts ➂ ➤✖✕➢☞➍➎➏➉st✘✗➣✉✈➇✜✛ ⑤✘✉✉✘→① ⑤✕✘④ ✛✇s✜✕ ➣s✝✗✜✛✜✘✗①
✛✇✚✛ ✚✉✉ ✛✇s①s t✉✚①①s① ✚✕s ①✛✚②✉s ②✈ t✘✗✆✖✗t✛✜✘✗➇→✇✜t✇ ✚✉✉✘→①⑤✘✕✜✗①✛✚✗ts✛✘ ✖➑➙
➣✚✛s✚➧✘✕✗➔✗✘→✉s➣③s②✚①s②✈✚➣➣✜✗③ ✗s→ ➧✘✕✗✕✖✉s①→✇✜✉s➑✕s①s✕➛✜✗③✛✇s➧✘✕✗
⑤✘✕④➏ ✕✘④ ✛✇s✜✕ ➣s✝✗✜✛✜✘✗① ✜✛ ✚✉①✘ ⑤✘✉✉✘→① ✛✇✚✛ ✛✇s✈ ✚✕s ①✛✚②✉s ②✈ t✘✗✆✖✗t✛✜✘✗
✘⑤ ⑤✚t✛① ➁✖✗✜✛ t✉✚✖①s①➎ ✚✗➣ ②✈ ✖✗✜✛ ✕s①✘✉✖✛✜✘✗➇ →✇✜t✇ ✚✉✉✘→① ⑤✘✕ ✜✗①✛✚✗ts ✘✗s ✛✘
①➑st✜⑤✈ ✚ ➛✚✉✖s ⑤✘✕ ✚ ➛✚✕✜✚②✉s ✚✗➣ ➑✕✘➑✚③✚✛s ✜✛ →✇✜✉s ➑✕s①s✕➛✜✗③ ✛✇s t✉✚①①➇ ✚✗➣
→✇✜t✇ ✚✉①✘ ✜④➑✉✜s①✛✇✚✛ ➣s➣✖t✛✜✘✗ ✘⑤ t✉✚✖①s①➇✘✗s✘⑤✛✇s④✘①✛t✘④④✘✗ ✕s✚①✘✗✜✗③
➑✕✘②✉s④①➇✜① ✛✕✚t✛✚②✉s ➝ ✜✗➣ss➣➇ ➣st✜➣✜✗③ →✇s✛✇s✕ ➓ ✜④➑✉✜s① ✗
✢✮ ✘✘✘✮
✗✙ ✜① ✛✇s
①✚④s ✚① ➣st✜➣✜✗③ →✇s✛✇s✕ ➓✯✗
✢
✯
✘✘✘
✯✗✙ ✜①✖✗①✚✛✜①✝✚②✉s➇✚✗➣ ✛✇✖① ①✛✚②✜✉✜✛✈②✈
t✘✗✆✖✗t✛✜✘✗ ✘⑤✖✗✜✛ t✉✚✖①s① ✚✗➣ ②✈✖✗✜✛ ✕s①✘✉✖✛✜✘✗➇✜✗ ✚➣➣✜✛✜✘✗ ✘⑤✛✕✚t✛✚②✜✉✜✛✈⑤✘✕
➉➐↕➇④✚➔s①➣s➣✖t✛✜✘✗✘⑤t✉✚✖①s①✛✕✚t✛✚②✉s➏➡✘✕s✘➛s✕➇✜✗ ①✘④s①s✗①s✛✇s①st✉✚①①s①
✚✕s ✛✇s ✘✗✉✈ ✘✗s① ✛✘ ➑✘①①s①① ✛✕✚t✛✚②✜✉✜✛✈ ⑤✘✕ ➉➐↕ ✚✗➣ ①✛✚②✜✉✜✛✈ ②✈ t✘✗✆✖✗t✛✜✘✗➇
→✇✜t✇ ⑤✘✉✉✘→① ⑤✕✘④ ➉t✇✚s⑤s✕✁① ➣✜t✇✘✛✘④✈ ✛✇s✘✕s④ ➂➉t✇✱➢➍➏ ✜✗✚✉✉✈➇✛✇s①s t✉✚①①s①
✚✉✉✘→ s↔t✜s✗✛ ✕s①✛✕✜t✛s➣ ⑤✘✕④① ✘⑤ ✚②➣✖t✛✜✘✗ ➂➋✚✉➥➥➇✙➥➆②➍ →✇✜✉s ✛✇✜① ➑✕✘②✉s④➇
✚✉①✘ ✘⑤ ④✚✆✘✕ ✜④➑✘✕✛✚✗ts ⑤✘✕ ✕s✚①✘✗✜✗③➇✜① ➓✚
✖
➙t✘④➑✉s✛s ✜✗ ③s✗s✕✚✉ ➂✛✡➅➊➍➏ ✘✕
✚✉✉ ✛✇s①s ✕s✚①✘✗①→s ✚✕③✖s ✛✇✚✛ ✛✇s①s ⑤✘✖✕ t✉✚①①s① ✚✕s✛✇s ④✘①✛ ✜✗✛s✕s①✛✜✗③ ✘✗s①
⑤✘✕ ✕s✚①✘✗✜✗③ ✚✗➣ ✕s➑✕s①s✗✛✜✗③ ➑✕✘➑✘①✜✛✜✘✗✚✉➔✗✘→✉s➣③s➏
➐ ⑤✘✕④✖✉✚ ✬ ✜① ①✚✜➣ ✛✘ ❷❶➠➜
❻
❾➨❶ ✚ ✕s✉✚✛✜✘✗ ✜ ✜⑤ ✜ ✢ ✴➁ ✬➎ ➝ ✬ ✜① t✚✉✉s➣ ✚
❷❶➠➜
❻
❾
❺
➀❾⑧⑦ ✘⑤✜➏ ✘✕✣
✕
✣✜ ✢ ✘➥
✕✙✚✛
➇✉s✛✖①→✕✜✛s✣✯✣✜✛✇s➛st✛✘✕✣✜✜ ✢ ✘➥
✕✙✚✛
①✖t✇ ✛✇✚✛✣✤ ✢
✙✕✗✗✗✕
✥
✕
✣✜✜
✦
✢✣✦✯✣✜
✦
➇✚✗➣①✜④✜✉✚✕✉✈⑤✘✕
✮
✚✗➣
✳
➏↕✇s⑤✘✉✉✘→✜✗③
✕s①✖✉✛➇➑✕✘➛s➣ ②✈ ➉t✇✚s⑤s✕➇✜① s①①s✗✛✜✚✉ ⑤✘✕ ✛✇s ➑✚➑s✕➏
✥✦✧★✧✩✪✫✪✧✬ ✭ ✮✯✰✱✲✳✴✵✶✷ ☎❶➀ ✜ ➨❶ ❹
❻
❶⑩❹➀❾⑧⑦✸ ✹✺❶⑦ ✜ ✺❹➠ ❹➀ ⑩❶❹➠➀ ⑧⑦❶ ❷❶❽
➠➜
❻
❾
❺
➀❾⑧⑦ ❾⑦
✻❾✼
✂✄☎✆ ❾➩ ❹⑦❷ ⑧⑦⑩✄ ❾➩ ✣✣ ✣✜ ✢✜ ✣ ✯✣✜ ✢✜✽
