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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 RIMS, 2012/09/12
2. 2. Resultsโข Define ๐2 ๐ธ, bounded arithmetic in free logic ๐โข โBootstrappingโ ๐2๐ ๐ธโข Prove ๐-consistency of ๐2 ๐ธ in ๐2 โ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 โ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 NPโ โ ๐2 1 ๐โ โ
7. 7. PH and Bussโs theories ๐2๐ ๐2 3 โข Tot(๐) ๐โ ๐ ฮฃ2 ๐ โฆ โฆ ๐2 2 ๐ ๐๐โ โ ๐2 1 ๐โ โ
8. 8. Separation of ๐ผฮฃ ๐ ๐ผฮฃ3 โฆ ๐ผฮฃ2โ ๐ผฮฃ1โ
9. 9. Separation of ๐ผฮฃ ๐ ๐ผฮฃ3 โข Con(Iฮฃ2 ) โฆ ๐ผฮฃ2 โข Con Iฮฃ2โ ๐ผฮฃ1โ
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 โจ ๐ต, ๐ด โง ๐ตโข Atomic formula, negated atomic formulaโข Bounded quantifiers
13. 13. 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
14. 14. 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
15. 15. 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
16. 16. 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 )
17. 17. Auxiliary axioms ๐ = ๐ โ ๐#๐ = ๐#๐๐ธ๐ธ#๐, ๐ธ๐ธ#๐, ๐ = |๐| โ ๐#๐ = ๐#๐
18. 18. PIND-rule
19. 19. 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
20. 20. CONSISTENCY PROOF OF ๐2 โ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#๐ + ๐ , ๐ โ19 19 a#a+b=19 ๐ฃ ๐ก , ๐ โ ๐ข ๐ is ฮฃ1๐
22. 22. 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
23. 23. 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 is ฮฃ ๐+1 ๐
24. 24. 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)]}
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 โ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 โ1 ๐ ๐+2
27. 27. 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
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