a tight karp-lipton collapse result in bounded arithmetic filekarp-lipton collapse results...

Karp-Lipton Collapse ResultsComplexity Classes in Bounded Arithmetic

The Optimal Karp-Lipton Collapse in PV

A Tight Karp-Lipton Collapse Result inBounded Arithmetic

Olaf Beyersdorff1 Sebastian Müller2

1Institut für Theoretische Informatik, Leibniz-Universität Hannover

2Institut für Informatik, Humboldt-Universität zu Berlin

17th Workshop on Computer Science LogicBertinoro 2009

Olaf Beyersdorff, Sebastian Müller A Tight Karp-Lipton Collapse Result in Bounded Arithmetic



Outline

1 Karp-Lipton Collapse Results

2 Complexity Classes in Bounded ArithmeticArithmetic TheoriesDefining Complexity ClassesProvable Inclusions

3 The Optimal Karp-Lipton Collapse in PVIngredients of the ProofFurther ResultsConclusion




Karp-Lipton Collapse Results

Theorem (Karp, Lipton 82)

If NP ⊆ P/poly, then PH ⊆ Σp2.







Theorem (Köbler, Watanabe 98)

If NP ⊆ P/poly, then PH ⊆ ZPPNP.







Theorem (Köbler, Watanabe 98)

If NP ⊆ P/poly, then PH ⊆ ZPPNP.

Theorem (Sengupta (Cai 01))

If NP ⊆ P/poly, then PH ⊆ Sp2.




Karp-Lipton Collapse Results in Bounded Arithmetic

A stronger assumption

Assume provability of NP ⊆ P/poly in some weakarithmetic theory, i.e.

NP ⊆ P/poly holds and we have an easy proof for it.





A stronger assumption

Assume provability of NP ⊆ P/poly in some weakarithmetic theory, i.e.

NP ⊆ P/poly holds and we have an easy proof for it.

Theorem (Cook, Krajícek 07)

If PV proves NP ⊆ P/poly, then PH ⊆ BH, and this collapse isprovable in PV.






If PV ⊢ NP ⊆ P/poly, then PV ⊢ PH ⊆ BH.

If S12 ⊢ NP ⊆ P/poly, then S1

2 ⊢ PH ⊆ PNP[O(log)].


2 ⊢ PH ⊆ PNP.








2 ⊢ PH ⊆ PNP[O(log)].


2 ⊢ PH ⊆ PNP.

Stronger assumptions yield stronger collapses

PH collapses to BH ⊆ PNP[O(log)] ⊆ PNP ⊆ Sp2

in the theory PV S12 S2

2 ZFC




A Tight Karp-Lipton Collapse Result

Open Problem (Cook, Krajícek 07)

Do the converse implications also hold?I.e., do these collapse consequences characterizethe assertion NP ⊆ P/poly in the respective theories?




A Tight Karp-Lipton Collapse Result

Open Problem (Cook, Krajícek 07)

Do the converse implications also hold?I.e., do these collapse consequences characterizethe assertion NP ⊆ P/poly in the respective theories?

Our Contribution

BH is the optimal Karp-Lipton collapse in PV , i.e.,PV proves NP ⊆ P/poly if and only if PV proves PH ⊆ BH.




Arithmetic TheoriesDefining Complexity ClassesProvable Inclusions

Arithmetic Formulas

Definition

The language of arithmetic uses the symbols

0, S, +, ∗, ≤, # . . .

Σb1-formulas are formulas in prenex normal form with only

bounded ∃-quantifiers, i.e. (∃x ≤ t(y))ψ(x , y).

Πb1-formulas: (∀x ≤ t(y))ψ(x , y)

alternating blocks of bounded ∃ and ∀-quantifiers define Σbi

and Πbi -formulas for i ≥ 0.





Arithmetic Theories

Bounded arithmetic = weak fragments of PA

Axiomatized by: basic axioms + some induction

Definition (Cook 75)

For the theory PV , the language additionally contains functionsymbols for all functions from FP.PV is axiomatized by

defining axioms for all these functions(Cobham’s recursive definition of FP)

induction for open formulas.





Arithmetic Theories

Bounded arithmetic = weak fragments of PA

Axiomatized by: basic axioms + some induction

Definition (Buss 86)

Length induction scheme LIND for ϕ

ϕ(0) ∧ (∀x)(ϕ(x) → ϕ(x + 1)) → (∀x)ϕ(|x |)

Si2 = BASIC + Σb

i − LIND





Complexity Classes in Arithmetic Theories

Definition

A class of arithmetic formulas F represents a complexity classC if for each A ⊆ Σ∗ we have A ∈ C if and only if A is definableby an F-formula ϕ(X ).





Complexity Classes in Arithmetic Theories

Definition

A class of arithmetic formulas F represents a complexity classC if for each A ⊆ Σ∗ we have A ∈ C if and only if A is definableby an F-formula ϕ(X ).

Theorem (Wrathall 78)

NP is represented by Σb1-formulas.

coNP is represented by Πb1-formulas.

PH is represented by all bounded formulas.





The Boolean Hierarchy

Definition

The levels of BH are denoted BHk and are inductively definedby BH1 = NP and

BHk+1 = {L1 \ L2 | L1 ∈ NP and L2 ∈ BHk} .

Proposition

BHk is represented by formulas of the type

ϕ1(X ) ∧ ¬(ϕ2(X ) ∧ . . .¬(ϕk−1(X ) ∧ ¬ϕk (X )) . . . )

with Σb1-formulas ϕ1, . . . , ϕk .





Provable Inclusions between Complexity Classes

Definition

Let A and B are complexity classes represented by the formulaclasses A and B, respectively.For an arithmetic theory T

T ⊢ A ⊆ B

abbreviates that for every formula ϕA ∈ A there exists aformula ϕB ∈ B, such that

T ⊢ ϕA(X ) ↔ ϕB(X ) .





Provable Inclusions between Complexity Classes

Definition

Let A and B are complexity classes represented by the formulaclasses A and B, respectively.For an arithmetic theory T

T ⊢ A ⊆ B

abbreviates that for every formula ϕA ∈ A there exists aformula ϕB ∈ B, such that

T ⊢ ϕA(X ) ↔ ϕB(X ) .

Thus we can write statements like PV ⊢ PH ⊆ BH.





Complexity Classes with Advice

Definition

Let k be a constant. Then T ⊢ coNP ⊆ NP/k abbreviates that,for every ϕ ∈ Πb

1 there exist Σb1-formulas ϕ1, . . . , ϕ2k , such that

T ⊢ (∀n)∨

1≤i≤2k

(∀X ) |X | = n → (ϕ(X ) ↔ ϕi(X )) .

Similarly, we formalize T ⊢ NP ⊆ P/poly as:for all ϕ ∈ Σb

1 there exists ψ ∈ Σb0 such that

T ⊢ (∀n)(∃C ≤ t(n))(∀X ) |X | = n → (ϕ(X ) ↔ ψ(X ,C)) .





The Strength of Bounded Arithmetic

Fundamental QuestionWhich inclusions between complexity classes are provable inbounded arithmetic?





The Strength of Bounded Arithmetic

Fundamental QuestionWhich inclusions between complexity classes are provable inbounded arithmetic?

Which arguments formalize in bounded arithmetic?

easy: combinatorial arguments

difficult: counting arguments or inductive proofs

a strong tool: witnessing theorems




Ingredients of the ProofFurther ResultsConclusion

Back to the Karp-Lipton Collapse










The main ingredient in the proof of this collapse result:


PV ⊢ NP ⊆ P/poly if and only if PV ⊢ coNP ⊆ NP/O(1).








The main ingredient in the proof of this collapse result:


PV ⊢ NP ⊆ P/poly if and only if PV ⊢ coNP ⊆ NP/O(1).

Proof.

uses strong witnessing arguments(KPT witnessing for ∀∃∀-formulas).





The Converse Implication

Theorem (Buhrman, Chang, Fortnow 03)

For every constant k we havecoNP ⊆ NP/k if and only if PH ⊆ BH2k .








Proof.⇒: relatively easy⇐: intricate hard/easy argument








Proof.⇒: relatively easy⇐: intricate hard/easy argument

We formalize this argument in PV and obtain:

Theorem

If PV ⊢ PH ⊆ BH2k , then PV ⊢ coNP ⊆ NP/k.





The Optimality of the Collapse

Corollary

PV ⊢ NP ⊆ P/poly if and only if PV ⊢ PH ⊆ BH.

Proof.

⇒: Cook, Krajícek 07⇐: follows by

PV ⊢ PH ⊆ BH ⇒ PV ⊢ coNP ⊆ NP/O(1)

PV ⊢ coNP ⊆ NP/O(1) ⇒ PV ⊢ NP ⊆ P/poly(Cook, Krajícek 07)





Further Results

Conditions of the form coNP ⊆ NP/O(1) naturally lead topropositional proof systems with advice (Cook, Krajícek 07).





Further Results


Question

Do there exist optimal proof systems with advice?





Further Results


Question

Do there exist optimal proof systems with advice?

Answer

Without advice optimal propositional proof systemsprobably do not exist (Köbler, Messner, Torán 03).

For proof systems with advice we obtain optimal andp-optimal proof systems for several measures of advice.





Conclusion

Are the other collapse consequences also optimal?

Is PH ⊆ PNP[O(log)] equivalent to NP ⊆ P/poly in S12?

Is PH ⊆ PNP equivalent to NP ⊆ P/poly in S22?





Conclusion

Are the other collapse consequences also optimal?

Is PH ⊆ PNP[O(log)] equivalent to NP ⊆ P/poly in S12?

Is PH ⊆ PNP equivalent to NP ⊆ P/poly in S22?


S12 ⊢ NP ⊆ P/poly if and only if S1

2 ⊢ coNP ⊆ NP/O(log).





Two Conjectures


coNP ⊆ NP/O(1) if and only if PH ⊆ BH = PNP[O(1)].





Two Conjectures



Do the following equivalences hold?

coNP ⊆ NP/O(log) if and only if PH ⊆ PNP[O(log)].

coNP ⊆ NP/poly if and only if PH ⊆ PNP.





Two Conjectures



Do the following equivalences hold?

coNP ⊆ NP/O(log) if and only if PH ⊆ PNP[O(log)].

coNP ⊆ NP/poly if and only if PH ⊆ PNP.

Known results

coNP ⊆ NP/O(log) ⇒ PH ⊆ PNP[O(log)]

[Cook, Krajícek 07].

coNP ⊆ NP/poly ⇒ PH ⊆ SNP2 [Cai et al. 05].


a tight karp-lipton collapse result in bounded arithmetic filekarp-lipton collapse results...

Documents