Bounded arithmetic in free logic

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Bounded arithmetic in free logic

  1. 1. Bounded Arithmetic in Free Logic Yoriyuki Yamagata RIMS, 2012/09/12
  2. 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. 3. Publicationsโ€ข Bounded Arithmetic in Free Logic Logical Methods in Computer Science Volume 8, Issue 3, Aug. 10, 2012
  4. 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. 5. BOUNDED ARITHMETIC ANDCOMPTATIONAL COMPLEXITY
  6. 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. 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. 8. Separation of ๐ผฮฃ ๐‘– ๐ผฮฃ3 โ€ฆ ๐ผฮฃ2โŠ† ๐ผฮฃ1โŠ†
  9. 9. Separation of ๐ผฮฃ ๐‘– ๐ผฮฃ3 โŠข Con(Iฮฃ2 ) โ€ฆ ๐ผฮฃ2 โŠข Con Iฮฃ2โŠ† ๐ผฮฃ1โŠ†
  10. 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. 11. SYSTEM ๐‘†2๐‘– ๐ธ
  12. 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. 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. 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. 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. 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. 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. 18. PIND-rule
  19. 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. 20. CONSISTENCY PROOF OF ๐‘†2 โˆ’1 ๐ธ
  21. 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. 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. 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. 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. 25. CONCLUSION
  26. 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. 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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