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

Sendai Logic Seminar, Oct. 31, 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,... S i2

Motivation

Approach using consistency statement

If S2 ` Con(S12 ), we have S1

2 6= S2.

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

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

Consistency of PV-

Soundness theorem

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 of PV-

Soundness theorem

Cook Urquhart’s PV : axioms

1. Recursive definition of functions

1.1 Composition1.2 Recursion

2. Equational axioms

3. Substitutiont(x) = u(x)

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

4. PIND for all formulas

Consistency of PV-

Soundness theorem

Consistency proof of PV− inside S12

Theorem (Consistency theorem)

S12 ` Con(PV−) (2)

Consistency of PV-

Soundness theorem

TheoremLet

1. π `PV− 0 = 1

2. r ` t = u be a sub-proof of π

3. ρ : sequence of substitution such that tρ is closed

4. σ ` 〈t, ρ〉 ↓ v , α

Then,∃τ, τ ` 〈u, ρ〉 ↓ v , α (3)

Consistency of PV-

Soundness theorem

Theorem (S12 )

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 (4)

s.t. |||τ ||| ≤ |||σ|||+ ||r ||

Consistency of PV-

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

Consistency of PV-

Big-step semantics

Inference rules for 〈t, ρ〉 ↓ v

Definition (Substitution)

〈t, ρ2〉 ↓ v

〈x , ρ1[t/x ]ρ2〉 ↓ v (5)

where ρ1 does not contain a substitution to x .

Consistency of PV-

Big-step semantics

Definition (ε, s0, s1)

〈ε, ρ〉 ↓ ε (6)

〈t, ρ〉 ↓ v

〈si t, ρ〉 ↓ siv (7)

where i is either 0 or 1.

Consistency of PV-

Big-step semantics

Definition (Constant function)

〈εn(t1, . . . , tn), ρ〉 ↓ ε (8)

Consistency of PV-

Big-step semantics

Definition (Projection)

〈ti , ρ〉 ↓ vi〈projni (t1, . . . , tn), ρ〉 ↓ vi (9)

for i = 1, · · · , n.

Consistency of PV-

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 (10)

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

Consistency of PV-

Big-step semantics

Definition (Recursion - ε)

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

〈f (t, t1, . . . , tn), ρ〉 ↓ v (11)

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

Consistency of PV-

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(12)

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 of PV-

Big-step semantics

Computation (sequence)

Definition

Computation sequence A DAG (or sequence with pointers)which is built by these inference rules

Computation statement A form 〈t, ρ〉 ↓ vt : main termρ : environmentv : value

Conclusion A computation statement which are not used forassumptions of inferences

Consistency of PV-

Bounding Godel numbers by number of nodes

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, ρ1, . . . , ρm)

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

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

Consistency of PV-

Size of environments

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 of PV-

Size of value

Lemma (S12 )

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

Consistency of PV-

Size of arguments

Lemma (S12 )

σ contains 〈t, ρ〉 ↓ v and u is a subterm of ts.t. u is a numeralThen,

||u|| ≤ max{|||σ|||,T} (13)

Consistency of PV-

Proof of bound for Godel numbers of main terms

σ = σ1, . . . σnM(σi) be the size of the main term of σi .Prove

M(σi) ≤ maxg∈Φ{(||g ||+ 1) · (C + ||g ||+ |||σ|||+ T )} (14)

by induction on n − iIf σi is a conclusion, M(σi) ≤ TAssume σi is an assumption of σj , j > i .

Consistency of PV-

Assume that M(σj) satisfies induction hypothesisWe analyze different cases of the inference which derives σj

〈t, ρ〉 ↓ v

〈x , ρ′[t/x ]ρ〉 ↓ v (15)

Since ||t|| ≤ ||ρ′[t/x ]ρ|| ≤ T , we are done

Consistency of PV-

〈f1(v1), ρ〉 ↓ w1, . . . , 〈fl1(v1,w), ρ〉 ↓ wl1 , 〈tk1 , ρ〉 ↓ v21 , . . . , 〈tkl2 , ρ〉 ↓ v

〈f (t), ρ〉 ↓ v(16)

v 1, v 2,w , v are numeralstk1 , . . . , tkl2 are a subsequence t.f1, . . . , fl1 ∈ Φthe elements of v 1 come from either v 2 or t that are numerals.

Consistency of PV-

||tk1||, . . . , ||tkl2 || ≤ ||f (t)|| (17)

Consistency of PV-

||fk(v 1)|| ≤ ||fk ||+ ar(fk) · (C + |||σ|||+ T ) (18)

≤ maxg∈Φ{||g ||+ ar(g) · (C + |||σ|||+ T )} (19)

≤ maxg∈Φ{(||g ||+ 1) · (C + |||σ|||+ T )} (20)

where t1 are terms ti s.t. tiρ is a numeral

Consistency of PV-

Polynomial bound of number of symbols in

computation sequence

CorollaryThere is a polynomial p such that

||σ|| ≤ p(|||σ|||, ||α||) (21)

where ||α|| are conclusions of σ

Consistency of PV-

Proof of Corollary

Proof.

M(σ) ≤ maxg∈Φ{(||g ||+ 1) · (C + ||g ||+ |||σ|||+ T )} (22)

≤ (||α||+ 1) · (C + ||α||+ |||σ|||+ T ) (23)

Consistency of PV-

Polynomial bound of Godel number of

computation sequence

Definition

`B α⇔def ∃σ, |||σ||| ≤ B ∧ σ ` α (24)

Corollary`B α is a Σb

1-formula

Consistency of PV-

Soundness of defining axioms

Backward lemma

Lemma (S12 )

If f has a defining axiom

f (c(t), t) = tf ,c(t, f (t, t), t) (25)

c(t) : ε, s0t, s1t, t

`B 〈f (c(t), t), ρ〉 ↓ v , α

⇒`B+||(25)|| 〈tf ,c(t, f (t, t), t), ρ〉 ↓ v , α (26)

Consistency of PV-

Forward lemma

Lemma (S12 )

`B 〈tf ,c(t, f (t, t), t), ρ〉 ↓ v , α

⇒`B+||(25)|| 〈f (c(t), t), ρ〉 ↓B v , α (27)

Consistency of PV-

Fusion lemma

Lemma (S12 )

σ `n α (28)

〈f (v), ρ〉 ↓ v , 〈t, ρ〉 ↓ v ∈ σ (29)

v are numeralsThen ∃τ such that

τ ` 〈f (t), ρ〉 ↓ v , α (30)

τ |n = σ (31)

|||τ ||| ≤ |||σ|||+ 1 (32)

Consistency of PV-

Constructor lemma

Lemma (S12 )

〈ε, ρ〉 ↓ v ∈ σ ⇒ v ≡ ε

〈si t, ρ〉 ↓ v ∈ σ ⇒ v ≡ siv0, 〈t, ρ〉 ↓ v0 ∈ σ

Consistency of PV-

Proof of Backward lemmaFor the case that f = g(h1(x), . . . , hn(x))

〈g(w), ρ〉 ↓ v 〈h(v), ρ〉 ↓ w 〈t, ρ〉 ↓ v

〈f (t1, . . . , tn), ρ〉 ↓ v

By Fusion lemma, τ1 ` 〈g(w), ρ〉 ↓ v , 〈h(t), ρ〉 ↓ w where|||τ1||| ≤ |||σ|||+ #hBy Fusion lemma τ ` 〈g(h(t)), ρ〉 ↓ v

|||τ ||| ≤ |||τ1|||+ 1

≤ |||σ|||+ #h + 1

≤ |||σ|||+ ||g(h(t))||

Consistency of PV-

Proof of Forward lemma

β1, 〈t0, ρ〉 ↓ v

〈h1(t), ρ〉 ↓ w1 , . . .

〈g(h(t)), ρ〉 ↓ v

is transformed to

αg(w),

〈h1(v), ρ〉 ↓ w1 , . . . , 〈t0, ρ〉 ↓ v

〈f (t), ρ〉 ↓ v .

Consistency of PV-

Proof of soundness theorem - defining axioms

f (c(t), t) = tf ,c(t, f (t, t), t)

By Backward and Forward lemma,

`B 〈f (c(t), t), ρ〉 ↓ v , α⇒`B+||r || 〈tf ,c(t, f (t, t), t), ρ〉 ↓ v , α

`B 〈tf ,c(t, f (t, t), t), ρ〉 ↓ v , α⇒`B+||r || 〈f (c(t), t), ρ〉 ↓ v , α

Consistency of PV-

Substitution lemma

Substitution I

Lemma (S12 )

U ,V : large integers

σ ` 〈t1[u/x ], ρ〉 ↓ v1, . . . , 〈tm[u/x ], ρ〉 ↓ vm, α (33)

|||σ||| ≤ U − ||t1[u/x ]|| − · · · − ||tm[u/x ]|| (34)

M(σ) ≤ V (35)

∃τ, τ ` 〈t1, [u/x ]ρ〉 ↓ v1 . . . , 〈tm, [u/x ]ρ〉 ↓ vm, α (36)

|||τ ||| ≤ |||σ|||+ ||t1[u/x ]||+ · · ·+ ||tm[u/x ]|| (37)

M(τ) ≤ M(σ) (38)

Consistency of PV-

Substitution lemma

Substitution IILemma (S1

2 )U ,V : large integers

σ ` 〈t1, [u/x ]ρ〉 ↓ v1, . . . , 〈tm, [u/x ]ρ〉 ↓ vm, α (39)

|||σ||| ≤ U − ||t1[u/x ]|| − · · · − ||tm[u/x ]|| (40)

M(σ) ≤ V (41)

∃τ, τ ` 〈t1[u/x ], ρ〉 ↓ v1 . . . , 〈tm[u/x ], ρ〉 ↓ vm, α (42)

|||τ ||| ≤ |||σ|||+ ||t1[u/x ]||+ · · ·+ ||tm[u/x ]|| (43)

M(τ) ≤ M(σ) · ||u|| (44)

Consistency of PV-

Substitution lemma

Proof of Substitution I

By induction on ||t1[u/x ]||+ · · ·+ ||tm[u/x ]||Sub-induction on the length of σCase analysis of the last inference of σ

Consistency of PV-

Substitution lemma

〈s, ρ1〉 ↓ v

〈y , ρ′[s/y ]ρ1〉 ↓ vR

Two possibility:

1. t1 ≡ x and u ≡ y

2. t1 ≡ y

2 is impossible

Consistency of PV-

Substitution lemma

.... σ1

〈s, ρ1〉 ↓ v

〈x [y/x ], ρ′[s/y ]ρ1〉 ↓ vR

M(σ1) ≤ M(σ), we can apply IH to σ1

Consistency of PV-

Substitution lemma

Let τ be .... τ1

〈s, ρ1〉 ↓ v

〈y , [s/y ]ρ1〉 ↓ v

〈x , [y/x ][s/y ]ρ1〉 ↓ v

|||τ ||| ≤ |||τ1|||+ 2

≤ |||σ1|||+ ||t2[u/x ]||+ · · ·+ ||tm[u/x ]||+ 2

≤ |||σ1|||+ ||t1[u/x ]||+ · · ·+ ||tm[u/x ]||+ 1

≤ |||σ|||+ ||t1[u/x ]||+ · · ·+ ||tm[u/x ]||

Consistency of PV-

Substitution lemma

The case that R is not a substitutionWe consider two cases

1. t1[u/x ] ≡ f (u1[u/x ], . . . , ul [u/x ])

2. t1[u/x ] ≡ u

We only consider the first caseAssume u1[u/x ], . . . , ul [u/x ] is partitioned intou1

1 [u/x ], . . . , u1l1[u/x ] which are not numeral, and numerals

u21 [u/x ], . . . , u2

l2[u/x ]

Consistency of PV-

Substitution lemma

β, 〈u11 [u/x ], ρ〉 ↓ v 1

1 , . . . , 〈u1l1[u/x ], ρ〉 ↓ v 1

〈t1[u/x ], ρ〉 ↓ vR

|||σ0||| ≤ |||σ||| − 1 + l1

≤ U − ||t1[u/x ]|| − · · · − ||tm[u/x ]|| − 1 + l1

≤ U − ||u11 [u/x ]|| − · · · − ||u1

l1[u/x ]|| − · · ·− ||t2[u/x ]|| − · · · − ||tm[u/x ]||

σ0 : The computation which is obtained by removing the lastelement of σ and make all 〈uj [u/x ], ρ〉 ↓ v 1

j conclusions

Consistency of PV-

Substitution lemma

τ0 ` 〈u11 , [u/x ]ρ〉 ↓ v 1

1 , . . . , 〈u1l1 , [u/x ]ρ〉 ↓ v 1

l1 , . . . (45)

|||τ0||| ≤ |||σ0|||+ ||u11 [u/x ]||+ · · ·+ ||u1

l1[u/x ]||+||t2[u/x ]||+ · · ·+ ||tm[u/x ]|| (46)

Consistency of PV-

Substitution lemma

We add the proof of 〈u2k2 , [u/x ]ρ〉 ↓ v 1

k2 for all k2 = 1, . . . , l2

whose numbers of nodes are bounded by ||u2k2[u/x ]||+ 1

τ1 ` 〈u1, [u/x ]ρ〉 ↓ v1, . . . , 〈ul , [u/x ]ρ〉 ↓ vl , . . . (47)

|||τ1||| ≤ |||σ0|||+ ||u1[u/x ]||+ · · ·+ ||ul [u/x ]||+ l2 + ||t2[u/x ]||+ · · ·+ ||tm[u/x ]|| (48)

Consistency of PV-

Substitution lemma

τ ` 〈t1, [u/x ]ρ〉 ↓ v1, . . . , 〈tm, [u/x ]ρ〉 ↓ vm, α (49)

|||τ ||| ≤ |||τ1|||+ 1

≤ |||σ0|||+ ||u1[u/x ]||+ · · ·+ ||uk [u/x ]||+l2 + ||t2[u/x ]||+ · · ·+ ||tm[u/x ]||+ 1

≤ |||σ|||+ l1 + ||u1[u/x ]||+ · · ·+ ||ul [u/x ]||+l2 + ||t2[u/x ]||+ · · ·+ ||tm[u/x ]||+ 1

≤ |||σ|||+ ||t1[u/x ]||+ · · ·+ ||tm[u/x ]||

Consistency of PV-

Substitution lemma

Proof of Substitution II

Similar to the proof of Substitution I

Consistency of PV-

Substitution lemma

Proof of soundness theorem - substitution.... r1

t = ut[t0/x ] = u[t0/x ]

〈t[t0/x ], ρ〉 ↓B v (Assumption)

〈t, [t0/x ]ρ〉 ↓B+||t[t0/x]|| v (Substitution I)

Since ||[t0/x ]ρ|| ≤ U − ||r ||+ ||[t0/x ]|| ≤ U − ||r1||

〈u, [t0/x ]ρ〉 ↓B+||t[t0/x]||+||r1|| v (IH)

〈u[t0/x ], ρ〉 ↓B+||t[t0/x]||+||r1||+||u[t0/x]|| v (Substitution II)

Since B + ||t[t0/x ]||+ ||r1||+ ||u[t0/x ]|| ≤ B + ||r ||

Consistency of PV-

Equality axioms

Equality axioms - transitivity.... r1

.... r2u = w

By induction hypothesis,

σ `B 〈t, ρ〉 ↓ v ⇒ ∃τ, τ `B+||r1|| 〈u, ρ〉 ↓ v (50)

||ρ|| ≤ U − ||r || ≤ U − ||r2|||||τ ||| ≤ U − ||r ||+ ||r1||

≤ U − ||r2||

By induction hypothesis, ∃δ, δ `B+||r1||+||r2|| 〈w , ρ〉 ↓ v

Consistency of PV-

Equality axioms

Equality axioms - substitution

.... r1

t = uf (t) = f (u)

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

Consistency of PV-

Equality axioms

Proof of soundness of equality axiomsσ 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 (51)

≤ U − ||r ||+ 1 (52)

≤ U − ||r1|| (53)

||f (t, t2, . . . , tk)||+ Σα∈αM(α) ≤ U − ||r ||+ ||f (t, t2, . . . , tk)||≤ U − ||r1|| (54)

Consistency of PV-

Equality axioms

Proof of soundness of 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 , . . . , 〈t, ρ〉 ↓ v0, 〈tk1 , ρ〉 ↓ v1, . . .

〈f (t, t2, . . . , tk), ρ〉 ↓ v

Consistency of PV-

Equality axioms

Proof of soundness of equality axioms

〈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

consistency statementif

feasible arithmeticinside

cons12 questioncan

cont buts126

s2where si2

theorem pudlak

ignjatovic 1995s126

theory t

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