Consistency proof of a feasible arithmetic inside a bounded arithmetic

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’15July 4, 2015


Background and Motivation

Buss’s hierarchy S i2 of bounded arithmetics

I Language : 0, S ,+, ·, b ·2c, | · |,#,=,≤ where

a#b = 2|a|·|b|

I BASIC -axioms

I Σbi -PIND

A(0) ⊃ ∀x{A(bx2c) ⊃ A(x)} ⊃ ∀xA(x)

for Σbi -formula A(x).



Significance of S i2

Provably total functions in S i2

qFunctions in the i -th polynomial-time hierarchy

In particular, S12 corresponds P



Fundamental problem in bounded arithmetic

Conjecutre

S12 6= S2

where S2 =⋃

i=1,2,... Si2

Otherwise, the polynomial-time hierarchy collapses to P



Approach using consistency statement

If S2 ` Con(S12 ), we have S1

2 6= S2.

I S2 6` Con(Q) (Wilkie and Paris, 1987)

I S2 6` BDCon(S12 ) (Pudlak, 1990)

QuestionCan we find a theory T such that S2 ` BDCon(T ) butS1

2 6` BDCon(T )?

T must be weaker than S12



Strategy to obtain T

I Weaker theories : Q, S−∞2 , S−12 ,PV−, . . .

I Weaker notions of provability : i -normal proof...



Strengthen Pudlak’s result

I S12 6` BDCon(S−1

2 ) (Buss 1986, Takeuti 1990, Buss andIgnjatovic 1995)

I S12 6` Con(PV−p ) (Buss and Ignjatovic 1995)

PV− : Cook & Urquhart’s equational theory PV minusinductionPV−p :PV− enriched by propositional logic andBASIC ,BASIC e-axioms



Lower bound of T

Theorem (Beckmann 2002)

S12 ` Con(S−∞2 )

S12 ` Con(PV−−)

PV−− : PV− minus substitutionWhile Takeuti, 1991 conjectured S1

2 6` Con(S−∞2 )


Main result

Main result

Theorem

S12 ` Con(PV−)

Even if PV− is formulated with substitution


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 ofsubstitutions 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 aninference of σ in C and written as σ ` αIf |||σ||| ≤ B , we write σ `B σ


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


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 .


Proof


Definition (ε, s0, s1)

〈ε, ρ〉 ↓ ε

〈t, ρ〉 ↓ v〈si t, ρ〉 ↓ siv

where i is either 0 or 1.


Proof


Definition (Constant function)

〈εn(t1, . . . , tn), ρ〉 ↓ ε


Proof


Definition (Projection)

〈ti , ρ〉 ↓ vi〈projni (t1, . . . , tn), ρ〉 ↓ vi

for i = 1, · · · , n.


Proof


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


Proof


Definition (Recursion - ε)

〈gε(v1, . . . , vn), ρ〉 ↓ v 〈t, ρ〉 ↓ ε 〈t, ρ〉 ↓ v〈f (t, t1, . . . , tn), ρ〉 ↓ v

t : members ti of t1, . . . , tn such that ti are not numerals


Proof


Definition (Recursion - s0, s1)

〈gi(v0,w , v), ρ〉 ↓ v 〈f (v0, v), ρ〉 ↓ w {〈t, ρ〉 ↓ siv0} 〈t, ρ〉 ↓ v〈f (t, t1, . . . , tn), ρ〉 ↓ v

i = 0, 1t : members ti of t1, . . . , tn such that ti are not numeralsThe clause {〈t, ρ〉 ↓ siv0} is there if t is not a numeral


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 , αThen,

∃τ, τ ` 〈u, ρ〉 ↓ v , α


Proof

Soundness theoremTheorem (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 ||


Proof

Soundness theorem : remark

RemarkIt is enough to bound |||τ |||∵ ||τ || can be polynomially bounded by |||τ ||| and the size ofits conclusions (non-trivial but tedious to prove)


Proof

Proof of soundness theorem

∵ By induction on r


Proof

Equality axioms - transitivity.... r1

t = u

.... r2u = 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


Proof

Equality axioms - substitution

.... r1t = u

f (t) = f (u)

σ : computation of 〈f (t), ρ〉 ↓ v .We only consider that the case t is not a numeral


Proof

Equality axioms - substitutionσ 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 , αIH 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)


Proof

Equality axioms - substitutionBy 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 ||


Proof

The proof of main result

Since C never prove 〈0, ρ〉 ↓ 1, the consistency of PV− isimmediate from soundness theorem


Conclusion and future works

Main Contribution

If we want to separate S2 from S12 by consistency of a theory

T , the lower bound of T is now PV−


Conclusion and future works

Future works

S2 `? Con(PV−p )

S2 `? Con(PV−p (d)) constant depth PV−p


Reference

Full paper

Yoriyuki YamagataConsistency proof of a feasible arithmetic inside a boundedarithmetic.eprint arXiv:1411.7087http://arxiv.org/abs/1411.7087


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 Ignjatovic.

Unprovability of consistency statements in fragments of bounded arithmetic.Annals of pure and applied logic, 74:221–244, 1995.

P. Pudlak.

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.

Consistency proof of a feasible arithmetic inside a bounded arithmetic

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