Presentation at CTFM (Computability Theory and Foundation of Mathematics)

1. Bounded Arithmetic in Free Logic
Yoriyuki Yamagata
CTFM, 2013/02/20

2. Buss’s theories 𝑆2𝑖
–a#b=2 𝑎 ⋅|𝑏|
• Language of Peano Arithmetic + “#”
• BASIC axioms
𝑥
𝐴 , Γ → Δ, 𝐴(𝑥)
2
• PIND
𝐴 0 , Γ → Δ, 𝐴(𝑡)
where 𝐴 𝑥 ∈ Σ 𝑖𝑏 , i.e. has 𝑖-alternations of
bounded quantifiers ∀𝑥 ≤ 𝑡, ∃𝑥 ≤ 𝑡.

3. PH and Buss’s theories 𝑆2𝑖
𝑆2 = 𝑆2 = 𝑆2 = …
1 2 3
𝑃 = □(𝑁𝑁) = □(Σ2 ) = …
𝑝
Implies
We can approach (non) collapse of PH from
(non) collapse of hierarchy of Buss’s theories
(PH = Polynomial Hierarchy)

4. Our approach
• Separate 𝑆2𝑖 by Gödel incompleteness theorem
• Use analogy of separation of 𝐼Σ 𝑖

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

6. Consistency proof inside 𝑆2𝑖
• Bounded Arithmetics generally are not
– 𝑆2 does not prove consistency of Q (Paris, Wilkie)
capable to prove consistency.
– 𝑆2 does not prove bounded consistency of
𝑆2 (Pudlák)
1
– 𝑆2 does not prove consistency the 𝐵 𝑖 𝑏 fragement
𝑖
of 𝑆2 (Buss and Ignjatović)
−1

7. Buss and Ignjatović(1995)
…𝑆2 ⊢ 𝐵3 − Con(𝑆2 )
3 b −1
𝑆2
2
⊢ 𝐵2
b
− Con(𝑆2 )
−1
⊆
𝑆2
1
⊢ 𝐵1
b
− Con(𝑆2 )
−1
⊆

8. Where…
• 𝐵 𝑖 𝑏 − 𝐶𝐶𝐶 𝑇
– consistency of 𝐵 𝑖 𝑏 −proofs
– 𝐵 𝑖 𝑏 −proofs : the proofs by 𝐵 𝑖 𝑏 -formule
– 𝐵 𝑖 𝑏 :Σ0𝑏 (Σ 𝑖𝑏 )… Formulas generated from Σ 𝑖𝑏 by
Boolean connectives and sharply bounded
• 𝑆2
−1
quantifiers.
– Induction free fragment of 𝑆2𝑖

9. If…
𝑆2 ⊢ 𝐵i − Con 𝑆2 ,j > i
𝑗 b −1
Then, Buss’s hierarchy does not collapse.

10. Consistency proof of 𝑆2 inside 𝑆2𝑖
−1
Problem
• No truth definition, because
• No valuation of terms, because
• The values of terms increase exponentially
In 𝑆2 world, terms do not have values a priori.
𝑖
• E.g. 2#2#2#2#2#...#2
• We introduce the predicate 𝐸 which signifies existence of
• Thus, we must prove the existence of values in proofs.
values.

11. Our result(2012)
𝑆2
…5
⊢3− Con(𝑆2
−1
𝐸)
𝑆2
4
⊢2− Con(𝑆2
−1
𝐸)
⊆
𝑆2
3
⊢1− Con(𝑆2
−1
𝐸)
⊆

12. Where…
• 𝑖 − 𝐶𝐶𝐶 𝑇
– consistency of 𝑖-normal proofs
– 𝑖-normal proofs : the proofs by 𝑖-normal formulas
– 𝑖-normal formulas: Formulas in the form:
∃𝑥1 ≤ 𝑡1 ∀𝑥2 ≤ 𝑡2 … 𝑄𝑥 𝑖 ≤ 𝑡 𝑖 𝑄𝑥 𝑖+1 ≤ 𝑡 𝑖+1 . 𝐴(… )
Where 𝐴 is quantifier free

13. Where…
• 𝑆2 𝐸
−1
– Induction free fragment of 𝑆2 𝐸
𝑖
– have predicate 𝐸 which signifies existence of
values
• Such logic is called Free logic

14. 𝑆2𝑖
𝐸(Language)
• =, ≤, 𝐸
Predicates
Function symbols
• Finite number of polynomial functions
Formulas
• 𝐴 ∨ 𝐵, 𝐴 ∧ 𝐵
• Atomic formula, negated atomic formula
• Bounded quantifiers

15. 𝑆2𝑖
𝐸(Axioms)
• 𝐸-axioms
• Equality axioms
• Data axioms
• Defining axioms
• Auxiliary axioms

16. Idea behind axioms…
→ 𝑎= 𝑎
Because there is no guarantee of 𝐸𝐸
Thus, we add 𝐸𝐸 in the antecedent
𝐸𝐸 → 𝑎 = 𝑎

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

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

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

20. 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 )

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

22. PIND-rule
where 𝐴 is an Σ 𝑖𝑏 -formulas

23. 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∗

24. Theorem (Consistency)
𝑆2𝑖+2
⊢i− Con(𝑆2
−1
𝐸)

25. 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

26. 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

27. 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 𝑢, 𝜙 is Σ 𝑖+1
𝑏

28. 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)]}

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

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

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