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 2014
December 24, 2014

Motivation
Ultimate Goal
Conjecutre
S1
2 = S2
where Si
2, i = 1, 2, . . . are Buss’s hierarchies of bounded
arithmetics and S2 = i=1,2,... Si
2

Motivation
Approach using consistency statement
If S2 Con(S1
2 ), we have S1
2 = S2.
Theorem (Pudl´ak, 1990)
S2 Con(S1
2 )
Question
Can we ﬁnd a theory T such that S2 Con(T) but
S1
2 Con(T)?

Motivation
Unprovability
Theorem (Buss and Ignjatovi´c 1995)
S1
2 Con(PV−
)
PV−
: Cook & Urquhart’s equational theory PV minus
induction
Buss and Ignjatovi´c enrich PV−
by propositional logic and
BASICe
-axioms

Motivation
Provability
Theorem (Beckmann 2002)
S1
2 Con(PV−
)
if PV−
is formulated without substitution

Main results
Main result
Theorem
S1
2 Con(PV−
)
Even if PV−
is formulated with substitution
Our PV−
is equational

PV
Cook & Urquhart’s PV : language
We formulate PV as an equational theory of binary digits.
1. Constant :
2. Function symbols for all polynomial time functions
s0, s1, n
, projn
i , . . .

PV
Cook Urquhart’s PV : axioms
1. Recursive deﬁnition of functions
1.1 Composition
1.2 Recursive deﬁnition
2. Equational axioms
3. Substitution
t(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 ↓ v
t : main term (a term of PV)
ρ : complete development (sequence of
substitutions s.t. tρ is closed)
v : value (a binary digit)
Computation sequence A DAG (or sequence with pointers)
which derives computation statements by
inference rules
Conclusion A computation statement which are not used for
assumptions 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 ( , s0, s1)
, ρ ↓ (3)
t, ρ ↓ v
si t, ρ ↓ si v (4)
where i is either 0 or 1.

Big-step semantics
Deﬁnition (Constant function)
n
(t1, . . . , tn), ρ ↓ (5)

Big-step semantics
Deﬁnition (Projection)
ti , ρ ↓ vi
projn
i (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 - )
g (v1, . . . , vn), ρ ↓ v t, ρ ↓ 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 ↓ si v0} t, ρ ↓ v
f (t, t1, . . . , tn), ρ ↓ v
(9)
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

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 are
closed
4. σ : computation s.t. σ t, ρ ↓ v, α
Then,
∃τ, τ u, ρ ↓ v, α (10)

Soundness Theorem
Soundness theorem
Theorem (S1
2 )
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 (S1
2 )
There is no computation which proves , ρ ↓ s1
Proof.
Immediate from the form of inference rules
Corollary
S1
2 Con(PV −
)

Bound of computation
Size of complete development
Lemma (S1
2 )
Assume σ t1, ρ1 ↓ v1, . . . , tm, ρm ↓ vm.
If t, ρ ↓ v ∈ σ then ρ is a subsequence of one of the
ρ1, . . . , ρm.

Size of value
Lemma (S1
2 )
If σ contains t, ρ ↓ v, then ||v|| ≤ |||σ|||.

Upper bound of G¨odel number of main terms
Lemma (S1
2 )
M(σ) : the maximal size of main terms in σ.
M(σ) ≤ max
g∈Φ
{||g|| + 1) · (C + ||g|| + |||σ||| + T)}
Φ = Base(t1, . . . , tm, ρ)
T = ||t1|| + · · · + ||tm|| + ||ρ1|| + · · · + ||ρm||
t1, . . . , tm : main terms of conclusions
ρ1, . . . , ρm : complete developments of conclusions σ.

Proof
Proof
By induction on r

Proof
Equality axioms
Equality axioms - transitivity
.... r1
t = u
.... r2
u = 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
.... 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

Proof
Equality axioms
σ 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
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, 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
We prove that S1
2 Con(PV −
) where PV −
is purely
equational
While S1
2 Con(PV −
) if PV −
is formulated by
propositional logic and BASICe
-axioms

Consistency proof of a feasible arithmetic inside a bounded arithmetic

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