1. Bounded Arithmetic in Free Logic
Yoriyuki Yamagata
RIMS, 2012/09/12

2. Results
• Define 𝑆2 𝐸, bounded arithmetic in free logic
𝑖
• “Bootstrapping” 𝑆2𝑖 𝐸
• Prove 𝑖-consistency of 𝑆2 𝐸 in 𝑆2
−1 𝑖

3. Publications
• Bounded Arithmetic in Free Logic
Logical Methods in Computer Science
Volume 8, Issue 3, Aug. 10, 2012

4. Agenda
• System 𝐸
𝑆2𝑖
• Bounded arithmetic and complexity
• Consistency proof of 𝑆2 𝐸
−1

5. BOUNDED ARITHMETIC AND
COMPTATIONAL COMPLEXITY

6. PH and Buss’s theories 𝑆2𝑖
𝑆2 Σ2
3 𝑝
…
…
𝑆2
2
NP
⊆
⊆
𝑆2
1
𝑃
⊆
⊆

7. PH and Buss’s theories 𝑆2𝑖
𝑆2
3
⊢ Tot(𝑓) 𝑓∈ 𝑃 Σ2
𝑝
…
…
𝑆2
2
𝑃 𝑁𝑁
⊆
⊆
𝑆2
1
𝑃
⊆
⊆

8. Separation of 𝐼Σ 𝑖
𝐼Σ3
…
𝐼Σ2
⊆
𝐼Σ1
⊆

9. Separation of 𝐼Σ 𝑖
𝐼Σ3 ⊢ Con(IΣ2 )
…
𝐼Σ2 ⊢ Con IΣ2
⊆
𝐼Σ1
⊆

10. Separation of 𝑆2𝑖
Problem
• No truth definition
• No valuation of terms
In 𝑆2 world, terms do not have values a priori.
𝑖
• E.g. 2#2#2#2#2#...#2
• the predicate 𝐸 signifies the existence of a value
• We must prove the existence of values in proofs.

11. SYSTEM 𝑆2𝑖
𝐸

12. Language
• =, ≤, 𝐸
Predicates
Function symbols
• Finite number of polynomial functions
Formulas
• 𝐴 ∨ 𝐵, 𝐴 ∧ 𝐵
• Atomic formula, negated atomic formula
• Bounded quantifiers

13. E-axioms
• 𝐸𝐸 𝑎1 , … , 𝑎 𝑛 → 𝐸𝑎 𝑗
• 𝑎1 = 𝑎2 → 𝐸𝑎 𝑗
• 𝑎1 ≠ 𝑎2 → 𝐸𝑎 𝑗
• 𝑎1 ≤ 𝑎2 → 𝐸𝑎 𝑗
• ¬𝑎1 ≤ 𝑎2 → 𝐸𝑎 𝑗

14. Equality axioms
• 𝐸𝐸 → 𝑎 = 𝑎
• 𝐸𝐸 ⃗ , ⃗ = 𝑏 → 𝑓 ⃗ = 𝑓 𝑏
𝑎 𝑎 𝑎

15. Data axioms
• → 𝐸𝐸
• 𝐸𝐸 → 𝐸𝑠0 𝑎
• 𝐸𝐸 → 𝐸𝑠1 𝑎

16. Defining axioms
𝑓 𝑢 𝑎1 , 𝑎2 , … , 𝑎 𝑛 = 𝑡(𝑎1 , … , 𝑎 𝑛 )
𝑢 𝑎 = 0, 𝑎, 𝑠0 𝑎, 𝑠1 𝑎
𝐸𝑎1 , … , 𝐸𝑎 𝑛 , 𝐸𝐸 𝑎1 , … , 𝑎 𝑛 →
𝑓 𝑢 𝑎1 , 𝑎2 , … , 𝑎 𝑛 = 𝑡(𝑎1 , … , 𝑎 𝑛 )

17. Auxiliary axioms
𝑎 = 𝑏 ⊃ 𝑎#𝑐 = 𝑏#𝑐
𝐸𝐸#𝑐, 𝐸𝐸#𝑐, 𝑎 = |𝑏| → 𝑎#𝑐 = 𝑏#𝑐

18. PIND-rule

19. Bootstrapping 𝑆2𝑖
𝐸
I. 𝑆2 𝐸 ⊢ Tot(𝑓) for any 𝑓, 𝑖 ≥ 0
𝑖
II. 𝑆2 𝐸 ⊢ BASIC∗ , equality axioms
𝑖 ∗
III. 𝑆2 𝐸 ⊢ predicate logic
𝑖 ∗
IV. 𝑆2𝑖
𝐸⊢ Σ𝑖𝑏
−PIND∗

20. CONSISTENCY PROOF OF 𝑆2
−1
𝐸

21. Valuation trees
ρ-valuation tree bounded by 19
ρ(a)=2, ρ(b)=3
a=2
a#a=16 b=3
𝑣 𝑎#𝑎 + 𝑏 , 𝜌 ↓19 19
a#a+b=19
𝑣 𝑡 , 𝜌 ↓ 𝑢 𝑐 is Σ1𝑏

22. Bounded truth definition (1)
• 𝑇 𝑢, 𝑡1 = 𝑡2 , 𝜌 ⇔def
∃𝑐 ≤ 𝑢, 𝑣 𝑡1 , 𝜌 ↓ 𝑢 𝑐 ∧ 𝑣 𝑡1 , 𝜌 ↓ 𝑢 𝑐
• 𝑇 𝑢, 𝜙1 ∧ 𝜙2 , 𝜌 ⇔def
𝑇 𝑢, 𝜙1 , 𝜌 ∧ 𝑇 𝑢, 𝜙2 , 𝜌
• 𝑇 𝑢, 𝜙1 ∨ 𝜙2 , 𝜌 ⇔def
𝑇 𝑢, 𝜙1 , 𝜌 ∨ 𝑇 𝑢, 𝜙2 , 𝜌

23. Bounded truth definition (2)
• 𝑇 𝑢, ∃𝑥 ≤ 𝑡, 𝜙(𝑥) , 𝜌 ⇔def
∃𝑐 ≤ 𝑢, 𝑣 𝑡 , 𝜌 ↓ 𝑢 𝑐 ∧
∃𝑑 ≤ 𝑐, 𝑇 𝑢, 𝜙 𝑥 , 𝜌 𝑥 ↦ 𝑑
• 𝑇 𝑢, ∀𝑥 ≤ 𝑡, 𝜙(𝑥) , 𝜌 ⇔def
∃𝑐 ≤ 𝑢, 𝑣 𝑡 , 𝜌 ↓ 𝑢 𝑐 ∧
∀𝑑 ≤ 𝑐, 𝑇(𝑢, 𝜙 𝑥 , 𝜌[𝑥 ↦ 𝑑])
Remark: If 𝜙 is Σ 𝑖𝑏 , 𝑇 is Σ 𝑖+1
𝑏

24. induction hypothesis
𝑢: enough large integer
𝑟: node of a proof of 0=1
Γ 𝑟 → Δ 𝑟 : the sequent of node 𝑟
𝜌: assignment 𝜌 𝑎 ≤ 𝑢
∀𝑢′ ≤ 𝑢 ⊖ 𝑟, { ∀𝐴 ∈ Γ 𝑟 𝑇 𝑢′ , 𝐴 , 𝜌 ⊃
[∃𝐵 ∈ Δr , 𝑇(𝑢′ ⊕ 𝑟, 𝐵 , 𝜌)]}

25. CONCLUSION

26. Conjecture
• 𝑆2 𝐸 is weak enough
𝑖
– 𝑆2 can prove 𝑖-consistency of 𝑆2 𝐸
𝑖+2 −1
• While 𝑆2 𝐸 is strong enough
𝑖
– 𝑆2 𝐸 can interpret 𝑆2
𝑖 𝑖
𝑆2 𝐸 is a good candidate to separate 𝑆2 and 𝑆2 .
• Conjecture
−1 𝑖 𝑖+2

27. Future works
𝑆2 ⊢ 𝑖−Con(𝑆2 𝐸)?
𝑖 −1
• Long-term goal
– Simplify 𝑆2 𝐸
• Short-term goal
𝑖