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Bounded arithmetic in free logic
Bounded arithmetic in free logic
Bounded arithmetic in free logic
Bounded arithmetic in free logic
Bounded arithmetic in free logic
Bounded arithmetic in free logic
Bounded arithmetic in free logic
Bounded arithmetic in free logic
Bounded arithmetic in free logic
Bounded arithmetic in free logic
Bounded arithmetic in free logic
Bounded arithmetic in free logic
Bounded arithmetic in free logic
Bounded arithmetic in free logic
Bounded arithmetic in free logic
Bounded arithmetic in free logic
Bounded arithmetic in free logic
Bounded arithmetic in free logic
Bounded arithmetic in free logic
Bounded arithmetic in free logic
Bounded arithmetic in free logic
Bounded arithmetic in free logic
Bounded arithmetic in free logic
Bounded arithmetic in free logic
Bounded arithmetic in free logic
Bounded arithmetic in free logic
Bounded arithmetic in free logic
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Bounded arithmetic in free logic

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  • 1. Bounded Arithmetic in Free Logic Yoriyuki Yamagata RIMS, 2012/09/12
  • 2. Resultsโ€ข Define ๐‘†2 ๐ธ, bounded arithmetic in free logic ๐‘–โ€ข โ€œBootstrappingโ€ ๐‘†2๐‘– ๐ธโ€ข Prove ๐‘–-consistency of ๐‘†2 ๐ธ in ๐‘†2 โˆ’1 ๐‘–
  • 3. Publicationsโ€ข Bounded Arithmetic in Free Logic Logical Methods in Computer Science Volume 8, Issue 3, Aug. 10, 2012
  • 4. Agendaโ€ข System ๐ธ ๐‘†2๐‘–โ€ข Bounded arithmetic and complexityโ€ข Consistency proof of ๐‘†2 ๐ธ โˆ’1
  • 5. BOUNDED ARITHMETIC ANDCOMPTATIONAL COMPLEXITY
  • 6. PH and Bussโ€™s theories ๐‘†2๐‘– ๐‘†2 ฮฃ2 3 ๐‘ โ€ฆ โ€ฆ ๐‘†2 2 NPโŠ† โŠ† ๐‘†2 1 ๐‘ƒโŠ† โŠ†
  • 7. PH and Bussโ€™s theories ๐‘†2๐‘– ๐‘†2 3 โŠข Tot(๐‘“) ๐‘“โˆˆ ๐‘ƒ ฮฃ2 ๐‘ โ€ฆ โ€ฆ ๐‘†2 2 ๐‘ƒ ๐‘๐‘โŠ† โŠ† ๐‘†2 1 ๐‘ƒโŠ† โŠ†
  • 8. Separation of ๐ผฮฃ ๐‘– ๐ผฮฃ3 โ€ฆ ๐ผฮฃ2โŠ† ๐ผฮฃ1โŠ†
  • 9. Separation of ๐ผฮฃ ๐‘– ๐ผฮฃ3 โŠข Con(Iฮฃ2 ) โ€ฆ ๐ผฮฃ2 โŠข Con Iฮฃ2โŠ† ๐ผฮฃ1โŠ†
  • 10. Separation of ๐‘†2๐‘–Problemโ€ข No truth definitionโ€ข No valuation of termsIn ๐‘†2 world, terms do not have values a priori. ๐‘– โ€ข E.g. 2#2#2#2#2#...#2โ€ข the predicate ๐ธ signifies the existence of a valueโ€ข We must prove the existence of values in proofs.
  • 11. SYSTEM ๐‘†2๐‘– ๐ธ
  • 12. Languageโ€ข =, โ‰ค, ๐ธPredicatesFunction symbolsโ€ข Finite number of polynomial functionsFormulasโ€ข ๐ด โˆจ ๐ต, ๐ด โˆง ๐ตโ€ข Atomic formula, negated atomic formulaโ€ข Bounded quantifiers
  • 13. E-axiomsโ€ข ๐ธ๐ธ ๐‘Ž1 , โ€ฆ , ๐‘Ž ๐‘› โ†’ ๐ธ๐‘Ž ๐‘—โ€ข ๐‘Ž1 = ๐‘Ž2 โ†’ ๐ธ๐‘Ž ๐‘—โ€ข ๐‘Ž1 โ‰  ๐‘Ž2 โ†’ ๐ธ๐‘Ž ๐‘—โ€ข ๐‘Ž1 โ‰ค ๐‘Ž2 โ†’ ๐ธ๐‘Ž ๐‘—โ€ข ยฌ๐‘Ž1 โ‰ค ๐‘Ž2 โ†’ ๐ธ๐‘Ž ๐‘—
  • 14. Equality axiomsโ€ข ๐ธ๐ธ โ†’ ๐‘Ž = ๐‘Žโ€ข ๐ธ๐ธ โƒ— , โƒ— = ๐‘ โ†’ ๐‘“ โƒ— = ๐‘“ ๐‘ ๐‘Ž ๐‘Ž ๐‘Ž
  • 15. Data axiomsโ€ข โ†’ ๐ธ๐ธโ€ข ๐ธ๐ธ โ†’ ๐ธ๐‘ 0 ๐‘Žโ€ข ๐ธ๐ธ โ†’ ๐ธ๐‘ 1 ๐‘Ž
  • 16. Defining axioms ๐‘“ ๐‘ข ๐‘Ž1 , ๐‘Ž2 , โ€ฆ , ๐‘Ž ๐‘› = ๐‘ก(๐‘Ž1 , โ€ฆ , ๐‘Ž ๐‘› ) ๐‘ข ๐‘Ž = 0, ๐‘Ž, ๐‘ 0 ๐‘Ž, ๐‘ 1 ๐‘Ž ๐ธ๐‘Ž1 , โ€ฆ , ๐ธ๐‘Ž ๐‘› , ๐ธ๐ธ ๐‘Ž1 , โ€ฆ , ๐‘Ž ๐‘› โ†’๐‘“ ๐‘ข ๐‘Ž1 , ๐‘Ž2 , โ€ฆ , ๐‘Ž ๐‘› = ๐‘ก(๐‘Ž1 , โ€ฆ , ๐‘Ž ๐‘› )
  • 17. Auxiliary axioms ๐‘Ž = ๐‘ โŠƒ ๐‘Ž#๐‘ = ๐‘#๐‘๐ธ๐ธ#๐‘, ๐ธ๐ธ#๐‘, ๐‘Ž = |๐‘| โ†’ ๐‘Ž#๐‘ = ๐‘#๐‘
  • 18. PIND-rule
  • 19. Bootstrapping ๐‘†2๐‘– ๐ธI. ๐‘†2 ๐ธ โŠข Tot(๐‘“) for any ๐‘“, ๐‘– โ‰ฅ 0 ๐‘–II. ๐‘†2 ๐ธ โŠข BASICโˆ— , equality axioms ๐‘– โˆ—III. ๐‘†2 ๐ธ โŠข predicate logic ๐‘– โˆ—IV. ๐‘†2๐‘– ๐ธโŠข ฮฃ๐‘–๐‘ โˆ’PINDโˆ—
  • 20. CONSISTENCY PROOF OF ๐‘†2 โˆ’1 ๐ธ
  • 21. Valuation treesฯ-valuation tree bounded by 19 ฯ(a)=2, ฯ(b)=3 a=2 a#a=16 b=3 ๐‘ฃ ๐‘Ž#๐‘Ž + ๐‘ , ๐œŒ โ†“19 19 a#a+b=19 ๐‘ฃ ๐‘ก , ๐œŒ โ†“ ๐‘ข ๐‘ is ฮฃ1๐‘
  • 22. Bounded truth definition (1)โ€ข ๐‘‡ ๐‘ข, ๐‘ก1 = ๐‘ก2 , ๐œŒ โ‡”def โˆƒ๐‘ โ‰ค ๐‘ข, ๐‘ฃ ๐‘ก1 , ๐œŒ โ†“ ๐‘ข ๐‘ โˆง ๐‘ฃ ๐‘ก1 , ๐œŒ โ†“ ๐‘ข ๐‘โ€ข ๐‘‡ ๐‘ข, ๐œ™1 โˆง ๐œ™2 , ๐œŒ โ‡”def ๐‘‡ ๐‘ข, ๐œ™1 , ๐œŒ โˆง ๐‘‡ ๐‘ข, ๐œ™2 , ๐œŒโ€ข ๐‘‡ ๐‘ข, ๐œ™1 โˆจ ๐œ™2 , ๐œŒ โ‡”def ๐‘‡ ๐‘ข, ๐œ™1 , ๐œŒ โˆจ ๐‘‡ ๐‘ข, ๐œ™2 , ๐œŒ
  • 23. Bounded truth definition (2)โ€ข ๐‘‡ ๐‘ข, โˆƒ๐‘ฅ โ‰ค ๐‘ก, ๐œ™(๐‘ฅ) , ๐œŒ โ‡”def โˆƒ๐‘ โ‰ค ๐‘ข, ๐‘ฃ ๐‘ก , ๐œŒ โ†“ ๐‘ข ๐‘ โˆง โˆƒ๐‘‘ โ‰ค ๐‘, ๐‘‡ ๐‘ข, ๐œ™ ๐‘ฅ , ๐œŒ ๐‘ฅ โ†ฆ ๐‘‘โ€ข ๐‘‡ ๐‘ข, โˆ€๐‘ฅ โ‰ค ๐‘ก, ๐œ™(๐‘ฅ) , ๐œŒ โ‡”def โˆƒ๐‘ โ‰ค ๐‘ข, ๐‘ฃ ๐‘ก , ๐œŒ โ†“ ๐‘ข ๐‘ โˆง โˆ€๐‘‘ โ‰ค ๐‘, ๐‘‡(๐‘ข, ๐œ™ ๐‘ฅ , ๐œŒ[๐‘ฅ โ†ฆ ๐‘‘]) Remark: If ๐œ™ is ฮฃ ๐‘–๐‘ , ๐‘‡ is ฮฃ ๐‘–+1 ๐‘
  • 24. induction hypothesis ๐‘ข: enough large integer๐‘Ÿ: node of a proof of 0=1ฮ“ ๐‘Ÿ โ†’ ฮ” ๐‘Ÿ : the sequent of node ๐‘Ÿ ๐œŒ: assignment ๐œŒ ๐‘Ž โ‰ค ๐‘ขโˆ€๐‘ขโ€ฒ โ‰ค ๐‘ข โŠ– ๐‘Ÿ, { โˆ€๐ด โˆˆ ฮ“ ๐‘Ÿ ๐‘‡ ๐‘ขโ€ฒ , ๐ด , ๐œŒ โŠƒ [โˆƒ๐ต โˆˆ ฮ”r , ๐‘‡(๐‘ขโ€ฒ โŠ• ๐‘Ÿ, ๐ต , ๐œŒ)]}
  • 25. CONCLUSION
  • 26. Conjectureโ€ข ๐‘†2 ๐ธ is weak enough ๐‘– โ€“ ๐‘†2 can prove ๐‘–-consistency of ๐‘†2 ๐ธ ๐‘–+2 โˆ’1โ€ข While ๐‘†2 ๐ธ is strong enough ๐‘– โ€“ ๐‘†2 ๐ธ can interpret ๐‘†2 ๐‘– ๐‘– ๐‘†2 ๐ธ is a good candidate to separate ๐‘†2 and ๐‘†2 .โ€ข Conjecture โˆ’1 ๐‘– ๐‘–+2
  • 27. Future works ๐‘†2 โŠข ๐‘–โˆ’Con(๐‘†2 ๐ธ)? ๐‘– โˆ’1โ€ข Long-term goal โ€“ Simplify ๐‘†2 ๐ธโ€ข Short-term goal ๐‘–

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