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- 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 - 1">Consistency proof of a feasible arithmetic inside a bounded arithmetic

Background and Motivation

Buss’s hierarchy S i of bounded arithmetics

2

Language : 0, S , +, ·, · , | · |, #, =, ≤ where

2

a#b = 2|a|·|b|

BASIC -axioms

Σb-PIND

i

x

A(0) ⊃ ∀x {A(

) ⊃ A(x )} ⊃ ∀xA(x )

2

for Σb-formula A(x ).

i - 1">Consistency proof of a feasible arithmetic inside a bounded arithmetic

Background and Motivation

Significance of S i2

Provably total functions in S i2

Functions in the i -th polynomial-time hierarchy

In particular, S 1 corresponds P

2 - 1">Consistency proof of a feasible arithmetic inside a bounded arithmetic

Background and Motivation

Fundamental problem in bounded arithmetic

Conjecutre

S 1 = S

2

2

where S2 =

S i

i =1,2,...

2

Otherwise, the polynomial-time hierarchy collapses to P - 1">Consistency proof of a feasible arithmetic inside a bounded arithmetic

Background and Motivation

Approach using consistency statement

If S2

Con(S 1), we have S 1 = S

2

2

2.

S2

Con(Q) (Wilkie and Paris, 1987)

S2

BDCon(S 1) (Pudl´

ak, 1990)

2

Question

Can we find a theory T such that S2

BDCon(T ) but

S 1

BDCon(T )?

2

T must be weaker than S 1

2 - 1">Consistency proof of a feasible arithmetic inside a bounded arithmetic

Background and Motivation

Strategy to obtain T

Weaker theories : Q, S −∞, S −1, PV−, . . .

2

2

Weaker notions of provability : i -normal proof... - 1">Consistency proof of a feasible arithmetic inside a bounded arithmetic

Background and Motivation

Strengthen Pudl´

ak’s result

S 1

BDCon(S −1) (Buss 1986, Takeuti 1990, Buss and

2

2

Ignjatovi´

c 1995)

S 1

Con(PV−) (Buss and Ignjatovi´

c 1995)

2

p

PV− : Cook & Urquhart’s equational theory PV minus

induction

PV− :PV− enriched by propositional logic and

p

BASIC , BASIC e-axioms - 1">Consistency proof of a feasible arithmetic inside a bounded arithmetic

Background and Motivation

Lower bound of T

Theorem (Beckmann 2002)

S 1

Con(S −∞)

2

2

S 1

Con(PV−−)

2

PV−− : PV− minus substitution

While Takeuti, 1991 conjectured S 1

Con(S −∞)

2

2 - 1">Consistency proof of a feasible arithmetic inside a bounded arithmetic

Main result

Main result

Theorem

S 1

Con(PV−)

2

Even if PV− is formulated with substitution - 1">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 - 1">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 - 1">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. - 1">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. - 1">Consistency proof of a feasible arithmetic inside a bounded arithmetic

Proof

Inference rules of C

Definition (Constant function)

n(t1, . . . , tn), ρ ↓ - 1">Consistency proof of a feasible arithmetic inside a bounded arithmetic

Proof

Inference rules of C

Definition (Projection)

ti , ρ ↓ vi

projn(t

i

1, . . . , tn), ρ

↓ vi

for i = 1, · · · , n. - 1">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 - 1">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 - 1">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 - 1">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 - 1">Consistency proof of a feasible arithmetic inside a bounded arithmetic

Proof

Soundness theorem

Theorem (S 1)

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 || - 1">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) - 1">Consistency proof of a feasible arithmetic inside a bounded arithmetic

Proof

Proof of soundness theorem

∵ By induction on r - 1">Consistency proof of a feasible arithmetic inside a bounded arithmetic

Proof

Equality axioms - transitivity

..

.

.

.

. r1

.. r2

t = u

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 - 1">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 - 1">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,

tk , ρ ↓ v

1

1,

. . .

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) - 1">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,

tk , ρ ↓ v

1

1,

. . .

f (u, t2, . . . , tk), ρ ↓ v

Let this derivation be τ

|||τ ||| ≤ |||σ1||| + ||r1|| + 1

≤ |||σ||| + ||r1|| + 2

≤ |||σ||| + ||r || - 1">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 - 1">Consistency proof of a feasible arithmetic inside a bounded arithmetic

Conclusion and future works

Main Contribution

If we want to separate S2 from S1 by consistency of a theory

2

T , the lower bound of T is now PV− - 1">Consistency proof of a feasible arithmetic inside a bounded arithmetic

Conclusion and future works

Future works

S2 ? Con(PV−)

p

S2 ? Con(PV−(d ))

constant depth PV−

p

p - 1">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 - 1">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.