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# 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 โ|๐|โข Language of Peano Arithmetic + โ#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 โ ฮฃ ๐๐ , i.e. has ๐-alternations ofbounded quantifiers โ๐ฅ โค ๐ก, โ๐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 = 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 โข Con(Iฮฃ2 ) โฆ ๐ผฮฃ2 โข Con Iฮฃ2โ ๐ผฮฃ1โ
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) โ1
7. 7. Buss and Ignjatoviฤ(1995) โฆ๐2 โข ๐ต3 โ Con(๐2 ) 3 b โ1 ๐2 2 โข ๐ต2 b โ Con(๐2 ) โ1โ ๐2 1 โข ๐ต1 b โ Con(๐2 ) โ1โ
8. 8. Whereโฆโข ๐ต ๐ ๐ โ ๐ถ๐ถ๐ถ ๐ โ consistency of ๐ต ๐ ๐ โproofs โ ๐ต ๐ ๐ โ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 โ1 quantifiers. โ Induction free fragment of ๐2๐
9. 9. Ifโฆ ๐2 โข ๐ตi โ Con ๐2 ,j > i ๐ b โ1Then, Bussโs hierarchy does not collapse.
10. 10. Consistency proof of ๐2 inside ๐2๐ โ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 โข3โ Con(๐2 โ1 ๐ธ)๐2 4 โข2โ Con(๐2 โ1 ๐ธ)โ๐2 3 โข1โ Con(๐2 โ1 ๐ธ)โ
12. 12. Whereโฆโข ๐ โ 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: โ๐ฅ1 โค ๐ก1 โ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 โ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 โจ ๐ต, ๐ด โง 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= ๐Because there is no guarantee of ๐ธ๐ธThus, we add ๐ธ๐ธ in the antecedent ๐ธ๐ธ โ ๐ = ๐
17. 17. E-axiomsโข ๐ธ๐ธ ๐1 , โฆ , ๐ ๐ โ ๐ธ๐ ๐โข ๐1 = ๐2 โ ๐ธ๐ ๐โข ๐1 โ  ๐2 โ ๐ธ๐ ๐โข ๐1 โค ๐2 โ ๐ธ๐ ๐โข ยฌ๐1 โค ๐2 โ ๐ธ๐ ๐
18. 18. Equality axiomsโข ๐ธ๐ธ โ ๐ = ๐โข ๐ธ๐ธ โ , โ = ๐ โ ๐ โ = ๐ ๐ ๐ ๐ ๐
19. 19. Data axiomsโข โ ๐ธ๐ธโข ๐ธ๐ธ โ ๐ธ๐ 0 ๐โข ๐ธ๐ธ โ ๐ธ๐ 1 ๐
20. 20. Defining axioms ๐ ๐ข ๐1 , ๐2 , โฆ , ๐ ๐ = ๐ก(๐1 , โฆ , ๐ ๐ ) ๐ข ๐ = 0, ๐, ๐ 0 ๐, ๐ 1 ๐ ๐ธ๐1 , โฆ , ๐ธ๐ ๐ , ๐ธ๐ธ ๐1 , โฆ , ๐ ๐ โ๐ ๐ข ๐1 , ๐2 , โฆ , ๐ ๐ = ๐ก(๐1 , โฆ , ๐ ๐ )
21. 21. Auxiliary axioms ๐ = ๐ โ ๐#๐ = ๐#๐๐ธ๐ธ#๐, ๐ธ๐ธ#๐, ๐ = |๐| โ ๐#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 โข Tot(๐) for any ๐, ๐ โฅ 0 ๐II. ๐2 ๐ธ โข BASICโ , equality axioms ๐ โIII. ๐2 ๐ธ โข predicate logic ๐ โIV. ๐2๐ ๐ธโข ฮฃ๐๐ โPINDโ
24. 24. Theorem (Consistency) ๐2๐+2 โขiโ Con(๐2 โ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#๐ + ๐ , ๐ โ19 19 a#a+b=19 ๐ฃ ๐ก , ๐ โ ๐ข ๐ is ฮฃ1๐
26. 26. Bounded truth definition (1)โข ๐ ๐ข, ๐ก1 = ๐ก2 , ๐ โdef โ๐ โค ๐ข, ๐ฃ ๐ก1 , ๐ โ ๐ข ๐ โง ๐ฃ ๐ก1 , ๐ โ ๐ข ๐โข ๐ ๐ข, ๐1 โง ๐2 , ๐ โdef ๐ ๐ข, ๐1 , ๐ โง ๐ ๐ข, ๐2 , ๐โข ๐ ๐ข, ๐1 โจ ๐2 , ๐ โdef ๐ ๐ข, ๐1 , ๐ โจ ๐ ๐ข, ๐2 , ๐
27. 27. Bounded truth definition (2)โข ๐ ๐ข, โ๐ฅ โค ๐ก, ๐(๐ฅ) , ๐ โdef โ๐ โค ๐ข, ๐ฃ ๐ก , ๐ โ ๐ข ๐ โง โ๐ โค ๐, ๐ ๐ข, ๐ ๐ฅ , ๐ ๐ฅ โฆ ๐โข ๐ ๐ข, โ๐ฅ โค ๐ก, ๐(๐ฅ) , ๐ โdef โ๐ โค ๐ข, ๐ฃ ๐ก , ๐ โ ๐ข ๐ โง โ๐ โค ๐, ๐(๐ข, ๐ ๐ฅ , ๐[๐ฅ โฆ ๐]) Remark: If ๐ is ฮฃ ๐๐ , ๐ ๐ข, ๐ is ฮฃ ๐+1 ๐
28. 28. induction hypothesis ๐ข: enough large integer๐: node of a proof of 0=1ฮ ๐ โ ฮ ๐ : the sequent of node ๐ ๐: assignment ๐ ๐ โค ๐ขโ๐ขโฒ โค ๐ข โ ๐, { โ๐ด โ ฮ ๐ ๐ ๐ขโฒ , ๐ด , ๐ โ [โ๐ต โ ฮr , ๐(๐ขโฒ โ ๐, ๐ต , ๐)]}
29. 29. Conjecture ๐2 โ1 ๐ธ is weak enough โ ๐2 can prove ๐-consistency of ๐2 ๐ธโข ๐+2 โ1โข While ๐2 ๐ธ is strong enough โ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 โ1 ๐ ๐+2
30. 30. Future works ๐2 โข ๐โCon(๐2 ๐ธ)? ๐ โ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