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

Proof 2014December 24, 2014


Motivation

Ultimate Goal

Conjecutre

S12 6= S2

where S i2, i = 1, 2, . . . are Buss’s hierarchies of bounded

arithmetics and S2 =⋃

i=1,2,... Si2


Motivation

Approach using consistency statement

If S2 ` Con(S12 ), we have S1

2 6= S2.

Theorem (Pudlak, 1990)S2 6` Con(S1

2 )

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

2 6` Con(T )?


Motivation

Unprovability

Theorem (Buss and Ignjatovic 1995)

S12 6` Con(PV−)

PV− : Cook & Urquhart’s equational theory PV minusinductionBuss and Ignjatovic enrich PV− by propositional logic andBASIC e-axioms


Motivation

Provability

Theorem (Beckmann 2002)

S12 ` Con(PV−)

if PV− is formulated without substitution


Main results

Main result

Theorem

S12 ` Con(PV−)

Even if PV− is formulated with substitutionOur PV− is equational


PV

Cook & Urquhart’s PV : language

We formulate PV as an equational theory of binary digits.

1. Constant : ε

2. Function symbols for all polynomial time functionss0, s1, ε

n, projni , . . .


PV

Cook Urquhart’s PV : axioms

1. Recursive definition of functions

1.1 Composition1.2 Recursive definition

2. Equational axioms

3. Substitutiont(x) = u(x)

t(s) = u(s) (1)

4. PIND for all formulas


Big-step semantics

Computation (sequence)

Definition

Computation statement A form 〈t, ρ〉 ↓ vt : main term (a term of PV)ρ : complete development (sequence ofsubstitutions s.t. tρ is closed)v : value (a binary digit)

Computation sequence A DAG (or sequence with pointers)which derives computation statements byinference rules

Conclusion A computation statement which are not used forassumptions of inferences


Big-step semantics

Inference rules for 〈t, ρ〉 ↓ v

Definition (Substitution)

〈t, ρ2〉 ↓ v〈x , ρ1[t/x ]ρ2〉 ↓ v (2)

where ρ1 does not contain a substitution to x .


Big-step semantics


Definition (ε, s0, s1)

〈ε, ρ〉 ↓ ε (3)

〈t, ρ〉 ↓ v〈si t, ρ〉 ↓ siv (4)

where i is either 0 or 1.


Big-step semantics


Definition (Constant function)

〈εn(t1, . . . , tn), ρ〉 ↓ ε (5)


Big-step semantics


Definition (Projection)

〈ti , ρ〉 ↓ vi〈projni (t1, . . . , tn), ρ〉 ↓ vi (6)

for i = 1, · · · , n.


Big-step semantics


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 (7)

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


Big-step semantics


Definition (Recursion - ε)

〈gε(v1, . . . , vn), ρ〉 ↓ v 〈t, ρ〉 ↓ ε 〈t, ρ〉 ↓ v〈f (t, t1, . . . , tn), ρ〉 ↓ v (8)

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


Big-step semantics


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

(9)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


Soundness Theorem

Soundness theorem

Theorem (Unbounded)Let

1. π `PV− 0 = 1

2. r ` t = u be a sub-proof of π

3. ρ : sequence of substitution such that tρ and uρ areclosed

4. σ : computation s.t. σ ` 〈t, ρ〉 ↓ v , αThen,

∃τ, τ ` 〈u, ρ〉 ↓ v , α (10)


Soundness Theorem

Soundness theorem

Theorem (S12 )

Let

1. π `PV− 0 = 1, U > ||π||2. r ` t = u be a sub-proof of π

3. ρ : sequence of substitution such that tρ is closed and||ρ|| ≤ U − ||r ||

4. σ ` 〈t, ρ〉 ↓ v , α and |||σ|||,Σα∈αM(α) ≤ U − ||r ||Then,

∃τ, τ ` 〈u, ρ〉 ↓ v , α (11)

s.t. |||τ ||| ≤ |||σ|||+ ||r ||


Soundness Theorem

Consistency of big-step semantics

Lemma (S12 )

There is no computation which proves 〈ε, ρ〉 ↓ s1ε

Proof.Immediate from the form of inference rules

Corollary

S12 ` Con(PV−)


Bound of computation

Size of complete development

Lemma (S12 )

Assume σ ` 〈t1, ρ1〉 ↓ v1, . . . , 〈tm, ρm〉 ↓ vm.If 〈t, ρ〉 ↓ v ∈ σ then ρ is a subsequence of one of theρ1, . . . , ρm.



Size of value

Lemma (S12 )

If σ contains 〈t, ρ〉 ↓ v , then ||v || ≤ |||σ|||.



Upper bound of Godel number of main terms

Lemma (S12 )

M(σ) : the maximal size of main terms in σ.

M(σ) ≤ maxg∈Φ{||g ||+ 1) · (C + ||g ||+ |||σ|||+ T )}

Φ = Base(t1, . . . , tm, ρ)

T = ||t1||+ · · ·+ ||tm||+ ||ρ1||+ · · ·+ ||ρm||

t1, . . . , tm : main terms of conclusionsρ1, . . . , ρm : complete developments of conclusions σ.


Proof

Proof

By induction on r


Proof

Equality axioms

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 (12)

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

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

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 , α

|||σ1||| ≤ |||σ|||+ 1 (13)

≤ U − ||r ||+ 1 (14)

≤ U − ||r1|| (15)

and

||f (t, t2, . . . , tk)||+ Σα∈αM(α) ≤ U − ||r ||+ ||f (t, t2, . . . , tk)||≤ U − ||r1|| (16)


Proof

Equality axioms

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 ||


Conclusion and future works

Conclusion

I We prove that S12 ` Con(PV−) where PV− is purely

equational

I While S12 6` Con(PV−) if PV− is formulated by

propositional logic and BASIC e-axioms

Consistency proof of a feasible arithmetic inside a bounded arithmetic

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