consistency proof of a feasible arithmetic inside a bounded arithmetic
TRANSCRIPT
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
Consistency proof of a feasible arithmetic inside a bounded arithmetic
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
Consistency proof of a feasible arithmetic inside a bounded arithmetic
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 )?
Consistency proof of a feasible arithmetic inside a bounded arithmetic
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
Consistency proof of a feasible arithmetic inside a bounded arithmetic
Motivation
Provability
Theorem (Beckmann 2002)
S12 ` Con(PV−)
if PV− is formulated without substitution
Consistency proof of a feasible arithmetic inside a bounded arithmetic
Main results
Main result
Theorem
S12 ` Con(PV−)
Even if PV− is formulated with substitutionOur PV− is equational
Consistency proof of a feasible arithmetic inside a bounded arithmetic
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 , . . .
Consistency proof of a feasible arithmetic inside a bounded arithmetic
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
Consistency proof of a feasible arithmetic inside a bounded arithmetic
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
Consistency proof of a feasible arithmetic inside a bounded arithmetic
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 .
Consistency proof of a feasible arithmetic inside a bounded arithmetic
Big-step semantics
Inference rules for 〈t, ρ〉 ↓ v
Definition (ε, s0, s1)
〈ε, ρ〉 ↓ ε (3)
〈t, ρ〉 ↓ v〈si t, ρ〉 ↓ siv (4)
where i is either 0 or 1.
Consistency proof of a feasible arithmetic inside a bounded arithmetic
Big-step semantics
Inference rules for 〈t, ρ〉 ↓ v
Definition (Constant function)
〈εn(t1, . . . , tn), ρ〉 ↓ ε (5)
Consistency proof of a feasible arithmetic inside a bounded arithmetic
Big-step semantics
Inference rules for 〈t, ρ〉 ↓ v
Definition (Projection)
〈ti , ρ〉 ↓ vi〈projni (t1, . . . , tn), ρ〉 ↓ vi (6)
for i = 1, · · · , n.
Consistency proof of a feasible arithmetic inside a bounded arithmetic
Big-step semantics
Inference rules for 〈t, ρ〉 ↓ v
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
Consistency proof of a feasible arithmetic inside a bounded arithmetic
Big-step semantics
Inference rules for 〈t, ρ〉 ↓ v
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
Consistency proof of a feasible arithmetic inside a bounded arithmetic
Big-step semantics
Inference rules for 〈t, ρ〉 ↓ v
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
Consistency proof of a feasible arithmetic inside a bounded arithmetic
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)
Consistency proof of a feasible arithmetic inside a bounded arithmetic
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 ||
Consistency proof of a feasible arithmetic inside a bounded arithmetic
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−)
Consistency proof of a feasible arithmetic inside a bounded arithmetic
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.
Consistency proof of a feasible arithmetic inside a bounded arithmetic
Bound of computation
Size of value
Lemma (S12 )
If σ contains 〈t, ρ〉 ↓ v , then ||v || ≤ |||σ|||.
Consistency proof of a feasible arithmetic inside a bounded arithmetic
Bound of computation
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.
Consistency proof of a feasible arithmetic inside a bounded arithmetic
Proof
Proof
By induction on r
Consistency proof of a feasible arithmetic inside a bounded arithmetic
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
Consistency proof of a feasible arithmetic inside a bounded arithmetic
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
Consistency proof of a feasible arithmetic inside a bounded arithmetic
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)
Consistency proof of a feasible arithmetic inside a bounded arithmetic
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 ||
Consistency proof of a feasible arithmetic inside a bounded arithmetic
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