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
Karp-Lipton Collapse ResultsComplexity Classes in Bounded Arithmetic
The Optimal Karp-Lipton Collapse in PV
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
Olaf Beyersdorff, Sebastian Müller A Tight Karp-Lipton Collapse Result in Bounded Arithmetic
Karp-Lipton Collapse ResultsComplexity Classes in Bounded Arithmetic
The Optimal Karp-Lipton Collapse in PV
Karp-Lipton Collapse Results
Theorem (Karp, Lipton 82)
If NP ⊆ P/poly, then PH ⊆ Σp2.
Olaf Beyersdorff, Sebastian Müller A Tight Karp-Lipton Collapse Result in Bounded Arithmetic
Karp-Lipton Collapse ResultsComplexity Classes in Bounded Arithmetic
The Optimal Karp-Lipton Collapse in PV
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.
Olaf Beyersdorff, Sebastian Müller A Tight Karp-Lipton Collapse Result in Bounded Arithmetic
Karp-Lipton Collapse ResultsComplexity Classes in Bounded Arithmetic
The Optimal Karp-Lipton Collapse in PV
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 (Sengupta (Cai 01))
If NP ⊆ P/poly, then PH ⊆ Sp2.
Olaf Beyersdorff, Sebastian Müller A Tight Karp-Lipton Collapse Result in Bounded Arithmetic
Karp-Lipton Collapse ResultsComplexity Classes in Bounded Arithmetic
The Optimal Karp-Lipton Collapse in PV
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.
Olaf Beyersdorff, Sebastian Müller A Tight Karp-Lipton Collapse Result in Bounded Arithmetic
Karp-Lipton Collapse ResultsComplexity Classes in Bounded Arithmetic
The Optimal Karp-Lipton Collapse in PV
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.
Theorem (Cook, Krajícek 07)
If PV proves NP ⊆ P/poly, then PH ⊆ BH, and this collapse isprovable in PV.
Olaf Beyersdorff, Sebastian Müller A Tight Karp-Lipton Collapse Result in Bounded Arithmetic
Karp-Lipton Collapse ResultsComplexity Classes in Bounded Arithmetic
The Optimal Karp-Lipton Collapse in PV
Karp-Lipton Collapse Results in Bounded Arithmetic
Theorem (Cook, Krajícek 07)
If PV ⊢ NP ⊆ P/poly, then PV ⊢ PH ⊆ BH.
If S12 ⊢ NP ⊆ P/poly, then S1
2 ⊢ PH ⊆ PNP[O(log)].
If S22 ⊢ NP ⊆ P/poly, then S2
2 ⊢ PH ⊆ PNP.
Olaf Beyersdorff, Sebastian Müller A Tight Karp-Lipton Collapse Result in Bounded Arithmetic
Karp-Lipton Collapse ResultsComplexity Classes in Bounded Arithmetic
The Optimal Karp-Lipton Collapse in PV
Karp-Lipton Collapse Results in Bounded Arithmetic
Theorem (Cook, Krajícek 07)
If PV ⊢ NP ⊆ P/poly, then PV ⊢ PH ⊆ BH.
If S12 ⊢ NP ⊆ P/poly, then S1
2 ⊢ PH ⊆ PNP[O(log)].
If S22 ⊢ NP ⊆ P/poly, then S2
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
Olaf Beyersdorff, Sebastian Müller A Tight Karp-Lipton Collapse Result in Bounded Arithmetic
Karp-Lipton Collapse ResultsComplexity Classes in Bounded Arithmetic
The Optimal Karp-Lipton Collapse in PV
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?
Olaf Beyersdorff, Sebastian Müller A Tight Karp-Lipton Collapse Result in Bounded Arithmetic
Karp-Lipton Collapse ResultsComplexity Classes in Bounded Arithmetic
The Optimal Karp-Lipton Collapse in PV
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.
Olaf Beyersdorff, Sebastian Müller A Tight Karp-Lipton Collapse Result in Bounded Arithmetic
Karp-Lipton Collapse ResultsComplexity Classes in Bounded Arithmetic
The Optimal Karp-Lipton Collapse in PV
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.
Olaf Beyersdorff, Sebastian Müller A Tight Karp-Lipton Collapse Result in Bounded Arithmetic
Karp-Lipton Collapse ResultsComplexity Classes in Bounded Arithmetic
The Optimal Karp-Lipton Collapse in PV
Arithmetic TheoriesDefining Complexity ClassesProvable Inclusions
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.
Olaf Beyersdorff, Sebastian Müller A Tight Karp-Lipton Collapse Result in Bounded Arithmetic
Karp-Lipton Collapse ResultsComplexity Classes in Bounded Arithmetic
The Optimal Karp-Lipton Collapse in PV
Arithmetic TheoriesDefining Complexity ClassesProvable Inclusions
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
Olaf Beyersdorff, Sebastian Müller A Tight Karp-Lipton Collapse Result in Bounded Arithmetic
Karp-Lipton Collapse ResultsComplexity Classes in Bounded Arithmetic
The Optimal Karp-Lipton Collapse in PV
Arithmetic TheoriesDefining Complexity ClassesProvable Inclusions
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 ).
Olaf Beyersdorff, Sebastian Müller A Tight Karp-Lipton Collapse Result in Bounded Arithmetic
Karp-Lipton Collapse ResultsComplexity Classes in Bounded Arithmetic
The Optimal Karp-Lipton Collapse in PV
Arithmetic TheoriesDefining Complexity ClassesProvable Inclusions
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.
Olaf Beyersdorff, Sebastian Müller A Tight Karp-Lipton Collapse Result in Bounded Arithmetic
Karp-Lipton Collapse ResultsComplexity Classes in Bounded Arithmetic
The Optimal Karp-Lipton Collapse in PV
Arithmetic TheoriesDefining Complexity ClassesProvable Inclusions
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 .
Olaf Beyersdorff, Sebastian Müller A Tight Karp-Lipton Collapse Result in Bounded Arithmetic
Karp-Lipton Collapse ResultsComplexity Classes in Bounded Arithmetic
The Optimal Karp-Lipton Collapse in PV
Arithmetic TheoriesDefining Complexity ClassesProvable Inclusions
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 ) .
Olaf Beyersdorff, Sebastian Müller A Tight Karp-Lipton Collapse Result in Bounded Arithmetic
Karp-Lipton Collapse ResultsComplexity Classes in Bounded Arithmetic
The Optimal Karp-Lipton Collapse in PV
Arithmetic TheoriesDefining Complexity ClassesProvable Inclusions
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.
Olaf Beyersdorff, Sebastian Müller A Tight Karp-Lipton Collapse Result in Bounded Arithmetic
Karp-Lipton Collapse ResultsComplexity Classes in Bounded Arithmetic
The Optimal Karp-Lipton Collapse in PV
Arithmetic TheoriesDefining Complexity ClassesProvable Inclusions
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)) .
Olaf Beyersdorff, Sebastian Müller A Tight Karp-Lipton Collapse Result in Bounded Arithmetic
Karp-Lipton Collapse ResultsComplexity Classes in Bounded Arithmetic
The Optimal Karp-Lipton Collapse in PV
Arithmetic TheoriesDefining Complexity ClassesProvable Inclusions
The Strength of Bounded Arithmetic
Fundamental QuestionWhich inclusions between complexity classes are provable inbounded arithmetic?
Olaf Beyersdorff, Sebastian Müller A Tight Karp-Lipton Collapse Result in Bounded Arithmetic
Karp-Lipton Collapse ResultsComplexity Classes in Bounded Arithmetic
The Optimal Karp-Lipton Collapse in PV
Arithmetic TheoriesDefining Complexity ClassesProvable Inclusions
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
Olaf Beyersdorff, Sebastian Müller A Tight Karp-Lipton Collapse Result in Bounded Arithmetic
Karp-Lipton Collapse ResultsComplexity Classes in Bounded Arithmetic
The Optimal Karp-Lipton Collapse in PV
Ingredients of the ProofFurther ResultsConclusion
Back to the Karp-Lipton Collapse
Theorem (Cook, Krajícek 07)
If PV ⊢ NP ⊆ P/poly, then PV ⊢ PH ⊆ BH.
Olaf Beyersdorff, Sebastian Müller A Tight Karp-Lipton Collapse Result in Bounded Arithmetic
Karp-Lipton Collapse ResultsComplexity Classes in Bounded Arithmetic
The Optimal Karp-Lipton Collapse in PV
Ingredients of the ProofFurther ResultsConclusion
Back to the Karp-Lipton Collapse
Theorem (Cook, Krajícek 07)
If PV ⊢ NP ⊆ P/poly, then PV ⊢ PH ⊆ BH.
The main ingredient in the proof of this collapse result:
Theorem (Cook, Krajícek 07)
PV ⊢ NP ⊆ P/poly if and only if PV ⊢ coNP ⊆ NP/O(1).
Olaf Beyersdorff, Sebastian Müller A Tight Karp-Lipton Collapse Result in Bounded Arithmetic
Karp-Lipton Collapse ResultsComplexity Classes in Bounded Arithmetic
The Optimal Karp-Lipton Collapse in PV
Ingredients of the ProofFurther ResultsConclusion
Back to the Karp-Lipton Collapse
Theorem (Cook, Krajícek 07)
If PV ⊢ NP ⊆ P/poly, then PV ⊢ PH ⊆ BH.
The main ingredient in the proof of this collapse result:
Theorem (Cook, Krajícek 07)
PV ⊢ NP ⊆ P/poly if and only if PV ⊢ coNP ⊆ NP/O(1).
Proof.
uses strong witnessing arguments(KPT witnessing for ∀∃∀-formulas).
Olaf Beyersdorff, Sebastian Müller A Tight Karp-Lipton Collapse Result in Bounded Arithmetic
Karp-Lipton Collapse ResultsComplexity Classes in Bounded Arithmetic
The Optimal Karp-Lipton Collapse in PV
Ingredients of the ProofFurther ResultsConclusion
The Converse Implication
Theorem (Buhrman, Chang, Fortnow 03)
For every constant k we havecoNP ⊆ NP/k if and only if PH ⊆ BH2k .
Olaf Beyersdorff, Sebastian Müller A Tight Karp-Lipton Collapse Result in Bounded Arithmetic
Karp-Lipton Collapse ResultsComplexity Classes in Bounded Arithmetic
The Optimal Karp-Lipton Collapse in PV
Ingredients of the ProofFurther ResultsConclusion
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
Olaf Beyersdorff, Sebastian Müller A Tight Karp-Lipton Collapse Result in Bounded Arithmetic
Karp-Lipton Collapse ResultsComplexity Classes in Bounded Arithmetic
The Optimal Karp-Lipton Collapse in PV
Ingredients of the ProofFurther ResultsConclusion
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
We formalize this argument in PV and obtain:
Theorem
If PV ⊢ PH ⊆ BH2k , then PV ⊢ coNP ⊆ NP/k.
Olaf Beyersdorff, Sebastian Müller A Tight Karp-Lipton Collapse Result in Bounded Arithmetic
Karp-Lipton Collapse ResultsComplexity Classes in Bounded Arithmetic
The Optimal Karp-Lipton Collapse in PV
Ingredients of the ProofFurther ResultsConclusion
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)
Olaf Beyersdorff, Sebastian Müller A Tight Karp-Lipton Collapse Result in Bounded Arithmetic
Karp-Lipton Collapse ResultsComplexity Classes in Bounded Arithmetic
The Optimal Karp-Lipton Collapse in PV
Ingredients of the ProofFurther ResultsConclusion
Further Results
Conditions of the form coNP ⊆ NP/O(1) naturally lead topropositional proof systems with advice (Cook, Krajícek 07).
Olaf Beyersdorff, Sebastian Müller A Tight Karp-Lipton Collapse Result in Bounded Arithmetic
Karp-Lipton Collapse ResultsComplexity Classes in Bounded Arithmetic
The Optimal Karp-Lipton Collapse in PV
Ingredients of the ProofFurther ResultsConclusion
Further Results
Conditions of the form coNP ⊆ NP/O(1) naturally lead topropositional proof systems with advice (Cook, Krajícek 07).
Question
Do there exist optimal proof systems with advice?
Olaf Beyersdorff, Sebastian Müller A Tight Karp-Lipton Collapse Result in Bounded Arithmetic
Karp-Lipton Collapse ResultsComplexity Classes in Bounded Arithmetic
The Optimal Karp-Lipton Collapse in PV
Ingredients of the ProofFurther ResultsConclusion
Further Results
Conditions of the form coNP ⊆ NP/O(1) naturally lead topropositional proof systems with advice (Cook, Krajícek 07).
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.
Olaf Beyersdorff, Sebastian Müller A Tight Karp-Lipton Collapse Result in Bounded Arithmetic
Karp-Lipton Collapse ResultsComplexity Classes in Bounded Arithmetic
The Optimal Karp-Lipton Collapse in PV
Ingredients of the ProofFurther ResultsConclusion
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?
Olaf Beyersdorff, Sebastian Müller A Tight Karp-Lipton Collapse Result in Bounded Arithmetic
Karp-Lipton Collapse ResultsComplexity Classes in Bounded Arithmetic
The Optimal Karp-Lipton Collapse in PV
Ingredients of the ProofFurther ResultsConclusion
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?
Theorem (Cook, Krajícek 07)
S12 ⊢ NP ⊆ P/poly if and only if S1
2 ⊢ coNP ⊆ NP/O(log).
Olaf Beyersdorff, Sebastian Müller A Tight Karp-Lipton Collapse Result in Bounded Arithmetic
Karp-Lipton Collapse ResultsComplexity Classes in Bounded Arithmetic
The Optimal Karp-Lipton Collapse in PV
Ingredients of the ProofFurther ResultsConclusion
Two Conjectures
Theorem (Buhrman, Chang, Fortnow 03)
coNP ⊆ NP/O(1) if and only if PH ⊆ BH = PNP[O(1)].
Olaf Beyersdorff, Sebastian Müller A Tight Karp-Lipton Collapse Result in Bounded Arithmetic
Karp-Lipton Collapse ResultsComplexity Classes in Bounded Arithmetic
The Optimal Karp-Lipton Collapse in PV
Ingredients of the ProofFurther ResultsConclusion
Two Conjectures
Theorem (Buhrman, Chang, Fortnow 03)
coNP ⊆ NP/O(1) if and only if PH ⊆ BH = PNP[O(1)].
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.
Olaf Beyersdorff, Sebastian Müller A Tight Karp-Lipton Collapse Result in Bounded Arithmetic
Karp-Lipton Collapse ResultsComplexity Classes in Bounded Arithmetic
The Optimal Karp-Lipton Collapse in PV
Ingredients of the ProofFurther ResultsConclusion
Two Conjectures
Theorem (Buhrman, Chang, Fortnow 03)
coNP ⊆ NP/O(1) if and only if PH ⊆ BH = PNP[O(1)].
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].
Olaf Beyersdorff, Sebastian Müller A Tight Karp-Lipton Collapse Result in Bounded Arithmetic