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
Consistency proof of a feasible arithmetic inside a bounded arithmetic
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).
Consistency proof of a feasible arithmetic inside a bounded arithmetic
Background and Motivation
Significance of S i2
Provably total functions in S i2
qFunctions in the i -th polynomial-time hierarchy
In particular, S12 corresponds P
Consistency proof of a feasible arithmetic inside a bounded arithmetic
Background and Motivation
Fundamental problem in bounded arithmetic
Conjecutre
S12 6= S2
where S2 =⋃
i=1,2,... Si2
Otherwise, the polynomial-time hierarchy collapses to P
Consistency proof of a feasible arithmetic inside a bounded arithmetic
Background and Motivation
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
Consistency proof of a feasible arithmetic inside a bounded arithmetic
Background and Motivation
Strategy to obtain T
I Weaker theories : Q, S−∞2 , S−12 ,PV−, . . .
I Weaker notions of provability : i -normal proof...
Consistency proof of a feasible arithmetic inside a bounded arithmetic
Background and Motivation
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
Consistency proof of a feasible arithmetic inside a bounded arithmetic
Background and Motivation
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 )
Consistency proof of a feasible arithmetic inside a bounded arithmetic
Main result
Main result
Theorem
S12 ` Con(PV−)
Even if PV− is formulated with substitution
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 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
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
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 .
Consistency proof of a feasible arithmetic inside a bounded arithmetic
Proof
Inference rules of C
Definition (ε, s0, s1)
〈ε, ρ〉 ↓ ε
〈t, ρ〉 ↓ v〈si t, ρ〉 ↓ siv
where i is either 0 or 1.
Consistency proof of a feasible arithmetic inside a bounded arithmetic
Proof
Inference rules of C
Definition (Constant function)
〈εn(t1, . . . , tn), ρ〉 ↓ ε
Consistency proof of a feasible arithmetic inside a bounded arithmetic
Proof
Inference rules of C
Definition (Projection)
〈ti , ρ〉 ↓ vi〈projni (t1, . . . , tn), ρ〉 ↓ vi
for i = 1, · · · , n.
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
Consistency proof of a feasible arithmetic inside a bounded arithmetic
Proof
Inference rules of C
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
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〉 ↓ 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
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 , αThen,
∃τ, τ ` 〈u, ρ〉 ↓ v , α
Consistency proof of a feasible arithmetic inside a bounded arithmetic
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 ||
Consistency proof of a feasible arithmetic inside a bounded arithmetic
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)
Consistency proof of a feasible arithmetic inside a bounded arithmetic
Proof
Proof of soundness theorem
∵ By induction on r
Consistency proof of a feasible arithmetic inside a bounded arithmetic
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
Consistency proof of a feasible arithmetic inside a bounded arithmetic
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
Consistency proof of a feasible arithmetic inside a bounded arithmetic
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)
Consistency proof of a feasible arithmetic inside a bounded arithmetic
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 ||
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− isimmediate from soundness theorem
Consistency proof of a feasible arithmetic inside a bounded arithmetic
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−
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
Consistency proof of a feasible arithmetic inside a bounded arithmetic
Reference
Full paper
Yoriyuki YamagataConsistency proof of a feasible arithmetic inside a boundedarithmetic.eprint arXiv:1411.7087http://arxiv.org/abs/1411.7087
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 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.
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