Slides for LCC'15

1. Consistency proof of a feasible arithmetic inside a bounded arithmetic
Consistency proof of a feasible arithmetic
inside a bounded arithmetic
Yoriyuki Yamagata
National Intsisute of Advanced Industrial Science and Technology (AIST)
LCC’15
July 4, 2015

2. Consistency proof of a feasible arithmetic inside a bounded arithmetic
Background and Motivation
Buss’s hierarchy Si
2 of bounded arithmetics
Language : 0, S, +, ·, ·
2
, | · |, #, =, ≤ where
a#b = 2|a|·|b|
BASIC-axioms
Σb
i -PIND
A(0) ⊃ ∀x{A(
x
2
) ⊃ A(x)} ⊃ ∀xA(x)
for Σb
i -formula A(x).

3. Consistency proof of a feasible arithmetic inside a bounded arithmetic
Background and Motivation
Significance of Si
2
Provably total functions in Si
2
Functions in the i-th polynomial-time hierarchy
In particular, S1
2 corresponds P

4. Consistency proof of a feasible arithmetic inside a bounded arithmetic
Background and Motivation
Fundamental problem in bounded arithmetic
Conjecutre
S1
2 = S2
where S2 = i=1,2,... Si
2
Otherwise, the polynomial-time hierarchy collapses to P

5. Consistency proof of a feasible arithmetic inside a bounded arithmetic
Background and Motivation
Approach using consistency statement
If S2 Con(S1
2 ), we have S1
2 = S2.
S2 Con(Q) (Wilkie and Paris, 1987)
S2 BDCon(S1
2 ) (Pudl´ak, 1990)
Question
Can we find a theory T such that S2 BDCon(T) but
S1
2 BDCon(T)?
T must be weaker than S1
2

6. Consistency proof of a feasible arithmetic inside a bounded arithmetic
Background and Motivation
Strategy to obtain T
Weaker theories : Q, S−∞
2 , S−1
2 , PV−
, . . .
Weaker notions of provability : i-normal proof...

7. Consistency proof of a feasible arithmetic inside a bounded arithmetic
Background and Motivation
Strengthen Pudl´ak’s result
S1
2 BDCon(S−1
2 ) (Buss 1986, Takeuti 1990, Buss and
Ignjatovi´c 1995)
S1
2 Con(PV−
p ) (Buss and Ignjatovi´c 1995)
PV−
: Cook & Urquhart’s equational theory PV minus
induction
PV−
p :PV−
enriched by propositional logic and
BASIC, BASICe
-axioms

8. Consistency proof of a feasible arithmetic inside a bounded arithmetic
Background and Motivation
Lower bound of T
Theorem (Beckmann 2002)
S1
2 Con(S−∞
2 )
S1
2 Con(PV−−
)
PV−−
: PV−
minus substitution
While Takeuti, 1991 conjectured S1
2 Con(S−∞
2 )

9. Consistency proof of a feasible arithmetic inside a bounded arithmetic
Main result
Main result
Theorem
S1
2 Con(PV−
)
Even if PV−
is formulated with substitution

10. Consistency proof of a feasible arithmetic inside a bounded arithmetic
Proof
The system C
The DAGs of which nodes are forms t, ρ ↓ v
t : main term (a term of PV)
ρ : complete development (sequence of
substitutions s.t. tρ is closed)
v : value (a binary digit)
If σ is a DAG of which terminal nodes are α, σ is said to be an
inference of σ in C and written as σ α
If |||σ||| ≤ B, we write σ B σ

11. Consistency proof of a feasible arithmetic inside a bounded arithmetic
Proof
Notations
||r|| : Number of symbols in r
|||σ||| : Number of nodes in σ if σ is a graph
M(α) : Size of main term of α

12. Consistency proof of a feasible arithmetic inside a bounded arithmetic
Proof
Inference rules of C
Definition (Substitution)
t, ρ2 ↓ v
x, ρ1[t/x]ρ2 ↓ v
where ρ1 does not contain a substitution to x.

13. Consistency proof of a feasible arithmetic inside a bounded arithmetic
Proof
Inference rules of C
Definition ( , s0, s1)
, ρ ↓
t, ρ ↓ v
si t, ρ ↓ si v
where i is either 0 or 1.

14. Consistency proof of a feasible arithmetic inside a bounded arithmetic
Proof
Inference rules of C
Definition (Constant function)
n
(t1, . . . , tn), ρ ↓

15. Consistency proof of a feasible arithmetic inside a bounded arithmetic
Proof
Inference rules of C
Definition (Projection)
ti , ρ ↓ vi
projn
i (t1, . . . , tn), ρ ↓ vi
for i = 1, · · · , n.

16. Consistency proof of a feasible arithmetic inside a bounded arithmetic
Proof
Inference rules of C
Definition (Composition)
If f is defined by g(h(t)),
g(w), ρ ↓ v h(v), ρ ↓ w t, ρ ↓ v
f (t1, . . . , tn), ρ ↓ v
t : members of t1, . . . , tn such that t are not numerals

17. Consistency proof of a feasible arithmetic inside a bounded arithmetic
Proof
Inference rules of C
Definition (Recursion - )
g (v1, . . . , vn), ρ ↓ v t, ρ ↓ t, ρ ↓ v
f (t, t1, . . . , tn), ρ ↓ v
t : members ti of t1, . . . , tn such that ti are not numerals

18. Consistency proof of a feasible arithmetic inside a bounded arithmetic
Proof
Inference rules of C
Definition (Recursion - s0, s1)
gi (v0, w, v), ρ ↓ v f (v0, v), ρ ↓ w { t, ρ ↓ si v0} t, ρ ↓ v
f (t, t1, . . . , tn), ρ ↓ v
i = 0, 1
t : members ti of t1, . . . , tn such that ti are not numerals
The clause { t, ρ ↓ si v0} is there if t is not a numeral

19. Consistency proof of a feasible arithmetic inside a bounded arithmetic
Proof
Soundness theorem
Theorem (Unbounded)
Let
1. π PV − 0 = 1
2. r t = u be a sub-proof of π
3. ρ : complete development of t and u
4. σ : inference of C s.t. σ t, ρ ↓ v, α
Then,
∃τ, τ u, ρ ↓ v, α

20. Consistency proof of a feasible arithmetic inside a bounded arithmetic
Proof
Soundness theorem
Theorem (S1
2 )
Let
1. π PV − 0 = 1, U > ||π||
2. r t = u be a sub-proof of π
3. ρ : complete development of t and u such that
||ρ|| ≤ U − ||r||
4. σ : inference of C s.t. σ t, ρ ↓ v, α and
|||σ|||, Σα∈αM(α) ≤ U − ||r||
Then, there is an inference τ of C s.t.
τ u, ρ ↓ v, α
and |||τ||| ≤ |||σ||| + ||r||

21. Consistency proof of a feasible arithmetic inside a bounded arithmetic
Proof
Soundness theorem : remark
Remark
It is enough to bound |||τ|||
||τ|| can be polynomially bounded by |||τ||| and the size of
its conclusions (non-trivial but tedious to prove)

22. Consistency proof of a feasible arithmetic inside a bounded arithmetic
Proof
Proof of soundness theorem
By induction on r

23. Consistency proof of a feasible arithmetic inside a bounded arithmetic
Proof
Equality axioms - transitivity
.... r1
t = u
.... r2
u = w
t = w
By induction hypothesis,
σ B t, ρ ↓ v ⇒ ∃τ, τ B+||r1|| u, ρ ↓ v (1)
Since
||ρ|| ≤ U − ||r|| ≤ U − ||r2||
|||τ||| ≤ U − ||r|| + ||r1||
≤ U − ||r2||
By induction hypothesis, ∃δ, δ B+||r1||+||r2|| w, ρ ↓ v

24. Consistency proof of a feasible arithmetic inside a bounded arithmetic
Proof
Equality axioms - substitution
.... r1
t = u
f (t) = f (u)
σ : computation of f (t), ρ ↓ v.
We only consider that the case t is not a numeral

25. Consistency proof of a feasible arithmetic inside a bounded arithmetic
Proof
Equality axioms - substitution
σ has a form
f1(v0, v), ρ ↓ w1 , . . . , t, ρ ↓ v0, tk1 , ρ ↓ v1, . . .
f (t, t2, . . . , tk), ρ ↓ v
∃σ1, s.t. σ1 t, ρ ↓ v0, f (t, t2, . . . , tk), ρ ↓ v, α
IH can be applied to r1 because
|||σ1||| ≤ |||σ||| + 1 (2)
≤ U − ||r|| + 1 (3)
≤ U − ||r1|| (4)
||f (t, t2, . . . , tk)|| + Σα∈αM(α) ≤ U − ||r|| + ||f (t, t2, . . . , tk)||
≤ U − ||r1|| (5)

26. Consistency proof of a feasible arithmetic inside a bounded arithmetic
Proof
Equality axioms - substitution
By IH, ∃τ1 s.t. τ1 u, ρ ↓ v0, f (t, t2, . . . , tk), ρ ↓ v, α and
|||τ1||| ≤ |||σ1||| + ||r1||
f1(v0, v), ρ ↓ w1 , . . . , u, ρ ↓ v0, tk1 , ρ ↓ v1, . . .
f (u, t2, . . . , tk), ρ ↓ v
Let this derivation be τ
|||τ||| ≤ |||σ1||| + ||r1|| + 1
≤ |||σ||| + ||r1|| + 2
≤ |||σ||| + ||r||

27. Consistency proof of a feasible arithmetic inside a bounded arithmetic
Proof
The proof of main result
Since C never prove 0, ρ ↓ 1, the consistency of PV−
is
immediate from soundness theorem

28. Consistency proof of a feasible arithmetic inside a bounded arithmetic
Conclusion and future works
Main Contribution
If we want to separate S2 from S1
2 by consistency of a theory
T, the lower bound of T is now PV−

29. Consistency proof of a feasible arithmetic inside a bounded arithmetic
Conclusion and future works
Future works
S2 ? Con(PV−
p )
S2 ? Con(PV−
p (d)) constant depth PV−
p

30. Consistency proof of a feasible arithmetic inside a bounded arithmetic
Reference
Full paper
Yoriyuki Yamagata
Consistency proof of a feasible arithmetic inside a bounded
arithmetic.
eprint arXiv:1411.7087
http://arxiv.org/abs/1411.7087

31. Consistency proof of a feasible arithmetic inside a bounded arithmetic
Reference
Reference
Arnold Beckmann.
Proving consistency of equational theories in bounded arithmetic.
Journal of Symbolic Logic, 67(1):279–296, March 2002.
Samuel R. Buss and Aleksandar Ignjatovi´c.
Unprovability of consistency statements in fragments of bounded arithmetic.
Annals of pure and applied logic, 74:221–244, 1995.
P. Pudl´ak.
A note on bounded arithmetic.
Fundamenta mathematicae, 136:85–89, 1990.
Gaisi Takeuti.
Some relations among systems for bounded arithmetic.
In PetiocPetrov Petkov, editor, Mathematical Logic, pages 139–154. Springer US, 1990.
A. Wilkie and J. Paris.
On the scheme of induction for bounded arithmetic formulas.
Annals of pure and applied logic, 35:261–302, 1987.