Permutative Conversions in Generalised Multiary -calculus Jos e - PowerPoint PPT Presentation

✁ � ✂ Permutative Conversions in Generalised Multiary -calculus Jos´ e Esp´ ırito Santo and Lu´ ıs Pinto jes,luis @math.uminho.pt. Departamento de Matem´ atica, Universidade do Minho Braga, Portugal First APPSEM-II Workshop (26-28 March 2003, Nottingham) – p.1/8

� ✠ ✔ ✗ ✗ ✜ ✗ ✸ ✙ ✗ ✬ ✒ ✴ ✭ ✙ ✗ ✬ ✙ ✗ ✬ ✒ ✠ ✴ ☛ ✗ ✴ ✮ ✱ ✴ ✷ ✔ ✜ ☞ ✄ ✸ ✴ ★✩ ✗ ✸ ✠ ✔ ✬ ✗ ✬ ✱ ✲ ✭ ✴ ✜ ✗ ✸ ✱ ✲ ✭ ✲ ✰ ✠ ✙ ✗ ✸ ✒ ✱ ✲ ✠ ✴ ✖ ✗ ✹ ✴ ✗ ☛ ✆ ✜ ✒ ✆ ✙ ✑ ✖ ✑ ✗ ✹ ✸ ✜ ✠ ☛ ✴ ✝ ☛ ✗ ✬ ✮ ✭ ✱ ✲ ✠ ✴ ✗ ✴ ✬ ✮ ✭ ✱ ✲ ✠ ✴ ✔ ✗ ✬ ✭ ✶ ✏ ✧ ✜ ✥ ✆ ✝ ✞ ✜ ✧ ✏ ☛ ✏ ✑ ✗ ✢ ✗ ✘ ☛ ✗ ✗ ✪✫ ✚★✩ ✆ ☞ ✏ ✣✤ ✑ ✬ ✘ ☎ � ✆ ✝ ✞ ✎✏ ✑ ✖ ✗ ✗ ✙ ✖ ✚ ✝ ☛ ✚ ☛ ✆ ✜ ✒ ✆ ✙ ✑ ✑ ✜ ✒ ☛ ✠ ✴ ☛ ✗ ✭ ✆ ✜ ✧ ✏ ✏ ☞✰ ☞✯ ✔ ☞✰ ✑ ✭ ✴ ✜ ✗ ✬ ✵ ✑ ✬ ✔ ✭ ✗ ✗ ☞✯ ✘ ✏ ✎ ✍ ✫ ☞ ✚ ☛ ✬ ✆ ✮ First APPSEM-II Workshop (26-28 March 2003, Nottingham) – p.2/8 gm-Elim gm-application Lft Right -calculus ✔✕✒ ✙✕✛ ✙✕✛ ✱✳✲ ✱✳✲ ✱✳✲ : the generalised multiary ✱✳✲ ✱✳✲ ✔✕✒ ✔✕✒ ☛✓✒ Axiom ✱✳✲ ☛✌✏ ☛✌✏ Ax ☛✌☞✍ ✟✡✠ ✟✦✠ ☛✌✪✫ ✱✳✲ ✱✳✲

❑ ❃ ❅ ❆ ❅ ❍ ■ ✺ ❏ � ❉❊ ✽ ✾ ✿ ❁ ❁ ✾ ✾ ❅ ❄ ❅ ✽ ✽ ◆ ❃ ❅ ▲ ✼ ▼ ❇ ❄ ❁ ✺ ❇ ✄ ☎ ✺ ✻ ✼ ▼ ✽ ▲ ❃ ❁ ❊ ✿ ❅ ✼ ✽ ❈ ✺ ✻ ❉ ✽ ✾ ✽ ❉❊ ❁ ❅ ❄ ❅ ❅ ❈ Subsystems of gm-application g-application ✾❀✿❂❁ ✾❀❄ ❅❀❆ ✾❀❄ ❅❀❆ ✿❂❁ Abbreviation: ✾❀✿●❋ m-application application ✾❀❄ ✾❀❄ ✿❂❁ Abbreviation: ✾❀✿●❋ Abbreviation: First APPSEM-II Workshop (26-28 March 2003, Nottingham) – p.3/8

☛ ✴ ✝ ✞ ✑ ✱ ✲ ✠ ☛ ★ ✗ ✬ ✮ ✭ ✱ ✲ ✠ ✆ ✸ ✔ � ✖ ✗ ✸ ✔ ✠ ✴ ✆ ✗ ✔ ❖ ✆ ✙ ✑ ✖ ✑ ✴ ✗ ✠ ☛ P ◗ ✑ ✬ ✠ ✴ ✗ ✆ ✬ ✮ ✭ ✱ ✲ ✠ ✴ ✝ ✠ ✬ ✠ ☛ ✭ ✴ ✜ ✗ ✸ ✴ ✴ ✸ ☛ ✆ ✔ ❖ ✜ ✑ ✗ ✴ ✩ ✗ ✸ ✴ ✜ ✗ ✸ ✙ ✗ ✒ ❘ ✝ ✠ ✴ ✖ ✗ ✹ ✆ ✭ ✬ ✴ ✗ ✄ ☎ ✑ ✠ ✴ ☛ ✬ ✗ ✮ ✭ ✱ ✲ ✠ ✴ ✔ ✠ ☛ ✒ ✲ ☛ ✗ ✬ ✮ ✭ ✱ ✠ ✠ ✴ ✔ ✗ ✬ ✙ ✗ ✭ ✴ ✲ ✆ ✑ ✭ ✜ ✒ ✆ ✙ ✑ ✖ ✗ ✱ ✹ ✗ ✆ ✝ ✞ ✟ ✑ ✔ First APPSEM-II Workshop (26-28 March 2003, Nottingham) – p.4/8 gm-Elim g-Elim m-Elim Elim Elimination rules for the subsystems of ✱✳✲ ✱✳✲ ✱✳✲ ✱✳✲ ✔✕✒ ✱✳✲ ✱✳✲ ✱✳✲ ✱✳✲ ✱✳✲ ✱✳✲

❬ ✒ ❴ ❵ ❛ ✆ ✜ ✒ ✔ ❬ ✒ ✜ ❬ ✑ ✆ ❫ ✪ ✑ ✖ ❬ ✑ ✒ ✙ ✒ ✔ ❬ ✒ ✜ ❫ ❪ ✒ � ✑ ☛ ✆ ✪ ✜ ✒ ✆ ✙ ✑ ✙ ✆ ✔ ✒ ✆ ✜ ❬ ✒ ✆ ✪ ✑ ✖ ❬ ✑ ✑ ❱ ☛ ✆ ❬ ✖ ✑ ✚★✩ ✪ ✑ ✖ ✑ ✜ ✗ ✗ ✘ ✔ ✗ ✗ ✜ ✆ ✒ ✔ ✑ ✒ ✆ ❙ ❳ ✑ ✒ ✆ ✑ ✒ ✪ ✑ ✆ ✜ ❬ ☛ ✆ ❬ ✑ ✒ ✆ ❙ ❳ ✑ ✒ ✜ ✑ ✒ ✒ ❬ ✔ ✗ ✗ ✘ ✙ ✚ ✝ ✙ ✛ ☛ ✚ ✙ ✆ ❬ ✑ ✖ ✆ ✪ ✒ ✖ ✑ ✆ ❬ ✑ ✆ ✝ ✔ ☛ ✑ ✪ ✑ ✖ ✗ ✗ ✜ ✒ ✆ ✪ ✑ ✖ ✑ ❱ ❲ ✒ ☛ ✑ ★ ✄ ☎ ✆ ❬ ✑ ✆ ✝ ✖ ☛ ✑ ✆ ❬ ✩ ✒ ✒ ✆ ✪ ✑ ✖ ✑ ❱ ❲ ✆ ❲ ✆ ✜ ✙ ✆ ✙ ❬ ✆ ✆ ✪ ✑ ✖ ❬ ✑ ❱ ☛ ✆ ✑ ✜ ✒ ✙ ❬ ✑ ✖ ✆ ✔ ❬ ✒ ✜ ❬ ✒ ✆ ✪ ✑ ✖ ✒ ✜ ✒ ❬ ☛ ✑ ✆ ✖ ✒ ✜ ✒ ✆ ✪ ✑ ✖ ✑ ✑ ✒ ☛ ✆ ✖ ✜ ✒ ✆ ✙ ✑ ✖ ✑ ✆ ✔ ❬ ✘ First APPSEM-II Workshop (26-28 March 2003, Nottingham) – p.5/8 ❜❞❝ or , must occur either in ✔✕✒ -normal forms: as above, with proviso ✔✕✒ ✔✕✒ ✔✕✒ ✔✕✒ , Reduction rules for -normal forms: ✔✕✒ ✔✕✒ ✙✕✛ if ✙✕✛ ✆✕❭ ✔✕✒ ✔✕✒ ✆❩❨ ✆❩❨ ✆❩❨ ✆✕❭ ❙❯❚ ❙❯❳ ❙❯❚ ❙❡❚

❫ ✖ ✖ ✑ ✆ ✙ ✒ ✖ ✒ ✜ ❫ ✒ ✆ ✆ ✪ ✜ ✒ ✙ ✒ ★✩ ✑ ✘ ★✩ ❫ ✩ ✆ ✔ ✑ ✆ ✜ ☛ ✒ ✆ ✙ ✑ ✔ ❳ ✑ ✒ ☛ ✙ ✆ ❚ ✒ ✒ ✔ ❚ ✒ ✜ ❚ ✒ ✙ ✒ ✜ ❳ ✑ ★ ✒ ✜ ✑ ✒ ✜ ❬ ✑ ☛ ✆ ✖ ★✩ ✒ ✆ ✙ ✖ ✒ ✑ ✆ ✯ ✑ ☛ ✆ ✑ ✖ ✗ ✗ ✜ ✒ ✙ ✜ ✙ ✜ ✒ ✔ ❬ ✗ ✗ ✜ ❬ ✑ ✘ ☛ ✆ ☛ ✒ ❬ ✆ ✙ ✑ ✔ ❬ ✑ ✗ ✗ ❫ ❱ ✆ ✑ ❚ ✒ ✙ ✙ ✪ ✆ ✫ ❳ ✑ ☛ ✆ ✖ ✜ ✒ ✆ ✑ ❜ ✝ ✪ ✛ ✖ ✑ ❱ ✝ ✪ ✛ ☛ ✆ ✑ ✘ ✙ ✒ ❚ ✆ ✑ ✜ ✒ ✆ ✙ ✑ ✙ ✑ ✆ ✫ ✑ ✒ ☛ ✆ ❬ ✜ ✒ ✆ ✙ ✑ ✪ ✑ ❱ ✪ ✜ ✆ ❚ ✜ ✑ ✖ ✑ ✑ ❱ ☛ ❚ ✆ ✔ ❚ ✒ ❚ ✆ ✒ ✆ ✙ ✑ ☛ ❳ ✑ ✆ ☛ ❚ ✆ ✔ ✪ ✒ ✙ ✒ ✑ ✖ ✑ ✆ ✫ ❢ ✑ ☛ ❚ ✆ ✔ ❚ ✜ ❳ ❚ ✒ ✆ ✙ ✑ ☛ ❳ ✆ ✔ ❳ ✒ ✜ ✆ First APPSEM-II Workshop (26-28 March 2003, Nottingham) – p.6/8 ❜❞❝ if ✔✕✒ ☛✓✒ ❬❤❣ ✔✕✒ ❬❤❣ ✔✕✒ Permutative conversions ✔✕✒ ✔✕✒ ✔✕✒ ☛✓✒ ✔✕✒ ✔✕✒ ❬❤❣ ✔✕✒ ☛✓✒ ✔✕✒ ❬❤❣ where

✽ ❧ ✉ ✉ ❁ ✺ t ✻ ❧ ✽ s ♥ ♠ ❇ ✾ ❅ ✺ ❏ ✾ ❇ ❦ ❅ ✐ ✽ ✾ ❇ ✽ ✻ ✺ ✉ ❅ ❁ ✺ ❑ ✽ ✽ ① ✇ ❦ ❅ ✽ ✾ ▲ ✾ ✼ ✻ ✺ ✉ ✽ t ❅ ✽ ✽ ✾ s ❅ ③ ② ① ✇ ❦ ❅ ✽ ✾ ❍ t ✐ t ♦ ❇ ✺ ✾ ❍ ❦ ❅ ❏ ✾ ❍ ❑ ▼ ▲ ◆ ✺ ❈ ▼ ✉ ✼ ▲ ❑ ❏ ✺ ■ ❍ ✻ ✺ ❈ ✼ ❇ ✼ ✻ ❅ ✽ ❦ ✾ ✇ ① ♠ ④ ⑤ ❅ ❧ ✽ ✾ ▲ ❦ ❅ ✐ ✽ ▲ ♠ s ✼ ✻ ✺ ✉ ✾ ❁ ✽ t ❅ ❧ ✽ s t ② First APPSEM-II Workshop (26-28 March 2003, Nottingham) – p.7/8 . . Representation Thms. . , , , ♦rq . . . ✽❥❧ Permutability Thms. ✽✈❧ ✽❥❧ ✽❥✐ ✽❥✐ , , , ✽✈❧ ♥♣♦ ♥♣♦rq Main results ✽✈✐ ✽❥✐ ✽❥✐ iff iff iff ✽❥❧ ✽❥✐

✝ ⑥ ✞ ⑥ ✞ ✝ ✝ P ◗ ✝ ✟ ✞ ✝ ✝ ⑥ ❘ ✟ Conclusion Permutability study on a multiary sequent calculus with cuts Defined the calculus and the notion of generalised multiary application Computational interpretation for fragments of sequent calculus obtained via their correspondence to extended -calculi ( , , , ) First APPSEM-II Workshop (26-28 March 2003, Nottingham) – p.8/8