Bounded arithmetic in free logic

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Bounded arithmetic in free logic

  1. 1. Bounded Arithmetic in Free Logic Yoriyuki Yamagata CTFM, 2013/02/20
  2. 2. Bussโ€™s theories ๐‘†2๐‘– โ€“a#b=2 ๐‘Ž โ‹…|๐‘|โ€ข Language of Peano Arithmetic + โ€œ#โ€โ€ข BASIC axioms ๐‘ฅ ๐ด , ฮ“ โ†’ ฮ”, ๐ด(๐‘ฅ) 2โ€ข PIND ๐ด 0 , ฮ“ โ†’ ฮ”, ๐ด(๐‘ก)where ๐ด ๐‘ฅ โˆˆ ฮฃ ๐‘–๐‘ , i.e. has ๐‘–-alternations ofbounded quantifiers โˆ€๐‘ฅ โ‰ค ๐‘ก, โˆƒ๐‘ฅ โ‰ค ๐‘ก.
  3. 3. PH and Bussโ€™s theories ๐‘†2๐‘– ๐‘†2 = ๐‘†2 = ๐‘†2 = โ€ฆ 1 2 3 ๐‘ƒ = โ–ก(๐‘๐‘) = โ–ก(ฮฃ2 ) = โ€ฆ ๐‘ ImpliesWe can approach (non) collapse of PH from(non) collapse of hierarchy of Bussโ€™s theories (PH = Polynomial Hierarchy)
  4. 4. Our approachโ€ข Separate ๐‘†2๐‘– by Gรถdel incompleteness theoremโ€ข Use analogy of separation of ๐ผฮฃ ๐‘–
  5. 5. Separation of ๐ผฮฃ ๐‘– ๐ผฮฃ3 โŠข Con(Iฮฃ2 ) โ€ฆ ๐ผฮฃ2 โŠข Con Iฮฃ2โŠ† ๐ผฮฃ1โŠ†
  6. 6. Consistency proof inside ๐‘†2๐‘–โ€ข Bounded Arithmetics generally are not โ€“ ๐‘†2 does not prove consistency of Q (Paris, Wilkie) capable to prove consistency. โ€“ ๐‘†2 does not prove bounded consistency of ๐‘†2 (Pudlรกk) 1 โ€“ ๐‘†2 does not prove consistency the ๐ต ๐‘– ๐‘ fragement ๐‘– of ๐‘†2 (Buss and Ignjatoviฤ‡) โˆ’1
  7. 7. Buss and Ignjatoviฤ‡(1995) โ€ฆ๐‘†2 โŠข ๐ต3 โˆ’ Con(๐‘†2 ) 3 b โˆ’1 ๐‘†2 2 โŠข ๐ต2 b โˆ’ Con(๐‘†2 ) โˆ’1โŠ† ๐‘†2 1 โŠข ๐ต1 b โˆ’ Con(๐‘†2 ) โˆ’1โŠ†
  8. 8. Whereโ€ฆโ€ข ๐ต ๐‘– ๐‘ โˆ’ ๐ถ๐ถ๐ถ ๐‘‡ โ€“ consistency of ๐ต ๐‘– ๐‘ โˆ’proofs โ€“ ๐ต ๐‘– ๐‘ โˆ’proofs : the proofs by ๐ต ๐‘– ๐‘ -formule โ€“ ๐ต ๐‘– ๐‘ :ฮฃ0๐‘ (ฮฃ ๐‘–๐‘ )โ€ฆ Formulas generated from ฮฃ ๐‘–๐‘ by Boolean connectives and sharply boundedโ€ข ๐‘†2 โˆ’1 quantifiers. โ€“ Induction free fragment of ๐‘†2๐‘–
  9. 9. Ifโ€ฆ ๐‘†2 โŠข ๐ตi โˆ’ Con ๐‘†2 ,j > i ๐‘— b โˆ’1Then, Bussโ€™s hierarchy does not collapse.
  10. 10. Consistency proof of ๐‘†2 inside ๐‘†2๐‘– โˆ’1Problemโ€ข No truth definition, becauseโ€ข No valuation of terms, because โ€ข The values of terms increase exponentiallyIn ๐‘†2 world, terms do not have values a priori. ๐‘– โ€ข E.g. 2#2#2#2#2#...#2โ€ข We introduce the predicate ๐ธ which signifies existence ofโ€ข Thus, we must prove the existence of values in proofs. values.
  11. 11. Our result(2012)๐‘†2โ€ฆ5 โŠข3โˆ’ Con(๐‘†2 โˆ’1 ๐ธ)๐‘†2 4 โŠข2โˆ’ Con(๐‘†2 โˆ’1 ๐ธ)โŠ†๐‘†2 3 โŠข1โˆ’ Con(๐‘†2 โˆ’1 ๐ธ)โŠ†
  12. 12. Whereโ€ฆโ€ข ๐‘– โˆ’ ๐ถ๐ถ๐ถ ๐‘‡ โ€“ consistency of ๐‘–-normal proofs โ€“ ๐‘–-normal proofs : the proofs by ๐‘–-normal formulas โ€“ ๐‘–-normal formulas: Formulas in the form: โˆƒ๐‘ฅ1 โ‰ค ๐‘ก1 โˆ€๐‘ฅ2 โ‰ค ๐‘ก2 โ€ฆ ๐‘„๐‘ฅ ๐‘– โ‰ค ๐‘ก ๐‘– ๐‘„๐‘ฅ ๐‘–+1 โ‰ค ๐‘ก ๐‘–+1 . ๐ด(โ€ฆ ) Where ๐ด is quantifier free
  13. 13. Whereโ€ฆโ€ข ๐‘†2 ๐ธ โˆ’1 โ€“ Induction free fragment of ๐‘†2 ๐ธ ๐‘– โ€“ have predicate ๐ธ which signifies existence of values โ€ข Such logic is called Free logic
  14. 14. ๐‘†2๐‘– ๐ธ(Language)โ€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
  15. 15. ๐‘†2๐‘– ๐ธ(Axioms)โ€ข ๐ธ-axiomsโ€ข Equality axiomsโ€ข Data axiomsโ€ข Defining axiomsโ€ข Auxiliary axioms
  16. 16. Idea behind axiomsโ€ฆ โ†’ ๐‘Ž= ๐‘ŽBecause there is no guarantee of ๐ธ๐ธThus, we add ๐ธ๐ธ in the antecedent ๐ธ๐ธ โ†’ ๐‘Ž = ๐‘Ž
  17. 17. 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‘—
  18. 18. 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
  19. 19. 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
  20. 20. 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‘› )
  21. 21. 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‘
  22. 22. PIND-rulewhere ๐ด is an ฮฃ ๐‘–๐‘ -formulas
  23. 23. 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—
  24. 24. Theorem (Consistency) ๐‘†2๐‘–+2 โŠขiโˆ’ Con(๐‘†2 โˆ’1 ๐ธ)
  25. 25. 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‘
  26. 26. 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
  27. 27. 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‘‡ ๐‘ข, ๐œ™ is ฮฃ ๐‘–+1 ๐‘
  28. 28. 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)]}
  29. 29. Conjecture ๐‘†2 โˆ’1 ๐ธ is weak enough โ€“ ๐‘†2 can prove ๐‘–-consistency of ๐‘†2 ๐ธโ€ข ๐‘–+2 โˆ’1โ€ข While ๐‘†2 ๐ธ is strong enough โˆ’1 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
  30. 30. 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‘–
  31. 31. Publicationsโ€ข Bounded Arithmetic in Free Logic Logical Methods in Computer Science Volume 8, Issue 3, Aug. 10, 2012

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