A Tight Karp-Lipton Collapse Result in Bounded Arithmetic

Transcription

1 A Tight Karp-Lipton Collapse Result in Bounded Arithmetic Olaf Beyersdorff 1 Sebastian Müller 2 1 Institut für Theoretische Informatik, Leibniz-Universität Hannover 2 Institut für Informatik, Humboldt-Universität zu Berlin 17th Workshop on Computer Science Logic Bertinoro 2009

2 Outline 1 Karp-Lipton Collapse Results 2 Arithmetic Theories Defining Complexity Classes Provable Inclusions 3 Ingredients of the Proof Conclusion

3 Karp-Lipton Collapse Results Theorem (Karp, Lipton 82) If NP P/poly, then PH Σ p 2.

4 Karp-Lipton Collapse Results Theorem (Karp, Lipton 82) If NP P/poly, then PH Σ p 2. Theorem (Köbler, Watanabe 98) If NP P/poly, then PH ZPP NP.

5 Karp-Lipton Collapse Results Theorem (Karp, Lipton 82) If NP P/poly, then PH Σ p 2. Theorem (Köbler, Watanabe 98) If NP P/poly, then PH ZPP NP. Theorem (Sengupta (Cai 01)) If NP P/poly, then PH S p 2.

6 Karp-Lipton Collapse Results in Bounded Arithmetic A stronger assumption Assume provability of NP P/poly in some weak arithmetic theory, i.e. NP P/poly holds and we have an easy proof for it.

7 Karp-Lipton Collapse Results in Bounded Arithmetic A stronger assumption Assume provability of NP P/poly in some weak arithmetic theory, i.e. NP P/poly holds and we have an easy proof for it. Theorem (Cook, Krajíček 07) If PV proves NP P/poly, then PH BH, and this collapse is provable in PV.

8 Karp-Lipton Collapse Results in Bounded Arithmetic Theorem (Cook, Krajíček 07) If PV NP P/poly, then PV PH BH. If S2 1 NP P/poly, then S1 2 PH PNP[O(log)]. If S2 2 NP P/poly, then S2 2 PH PNP.

9 Karp-Lipton Collapse Results in Bounded Arithmetic Theorem (Cook, Krajíček 07) If PV NP P/poly, then PV PH BH. If S2 1 NP P/poly, then S1 2 PH PNP[O(log)]. If S2 2 NP P/poly, then S2 2 PH PNP. Stronger assumptions yield stronger collapses PH collapses to BH P NP[O(log)] P NP S p 2 in the theory PV S 1 2 S 2 2 ZFC

10 A Tight Karp-Lipton Collapse Result Open Problem (Cook, Krajíček 07) Do the converse implications also hold? I.e., do these collapse consequences characterize the assertion NP P/poly in the respective theories?

11 A Tight Karp-Lipton Collapse Result Open Problem (Cook, Krajíček 07) Do the converse implications also hold? I.e., do these collapse consequences characterize the 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.

12 Arithmetic Formulas Arithmetic Theories Defining Complexity Classes Provable Inclusions Definition The language of arithmetic uses the symbols 0, S, +,,, #... Σ b 1-formulas are formulas in prenex normal form with only bounded -quantifiers, i.e. ( x t(y))ψ(x, y). Π b 1-formulas: ( x t(y))ψ(x, y) alternating blocks of bounded and -quantifiers define Σ b i and Π b i -formulas for i 0.

13 Arithmetic Theories Arithmetic Theories Defining Complexity Classes Provable Inclusions Bounded arithmetic = weak fragments of PA Axiomatized by: basic axioms + some induction Definition (Cook 75) For the theory PV, the language additionally contains function symbols 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.

14 Arithmetic Theories Arithmetic Theories Defining Complexity Classes Provable Inclusions 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 ) S2 i = BASIC + Σb i LIND

15 Arithmetic Theories Defining Complexity Classes Provable Inclusions Complexity Classes in Arithmetic Theories Definition A class of arithmetic formulas F represents a complexity class C if for each A Σ we have A C if and only if A is definable by an F-formula ϕ(x).

16 Arithmetic Theories Defining Complexity Classes Provable Inclusions Complexity Classes in Arithmetic Theories Definition A class of arithmetic formulas F represents a complexity class C if for each A Σ we have A C if and only if A is definable by an F-formula ϕ(x). Theorem (Wrathall 78) NP is represented by Σ b 1 -formulas. conp is represented by Π b 1 -formulas. PH is represented by all bounded formulas.

17 The Boolean Hierarchy Arithmetic Theories Defining Complexity Classes Provable Inclusions Definition The levels of BH are denoted BH k and are inductively defined by BH 1 = NP and Proposition BH k+1 = {L 1 \ L 2 L 1 NP and L 2 BH k }. BH k is represented by formulas of the type ϕ 1 (X) (ϕ 2 (X)... (ϕ k 1 (X) ϕ k (X))... ) with Σ b 1 -formulas ϕ 1,...,ϕ k.

18 Arithmetic Theories Defining Complexity Classes Provable Inclusions Provable Inclusions between Complexity Classes Definition Let A and B are complexity classes represented by the formula classes A and B, respectively. For an arithmetic theory T T A B abbreviates that for every formula ϕ A A there exists a formula ϕ B B, such that T ϕ A (X) ϕ B (X).

19 Arithmetic Theories Defining Complexity Classes Provable Inclusions Provable Inclusions between Complexity Classes Definition Let A and B are complexity classes represented by the formula classes A and B, respectively. For an arithmetic theory T T A B abbreviates that for every formula ϕ A A there exists a formula ϕ B B, such that T ϕ A (X) ϕ B (X). Thus we can write statements like PV PH BH.

20 Complexity Classes with Advice Arithmetic Theories Defining Complexity Classes Provable Inclusions Definition Let k be a constant. Then T conp NP/k abbreviates that, for every ϕ Π b 1 there exist Σb 1 -formulas ϕ 1,...,ϕ 2 k, such that T ( n) 1 i 2 k ( X) X = n (ϕ(x) ϕ i (X)). Similarly, we formalize T NP P/poly as: for all ϕ Σ b 1 there exists ψ Σb 0 such that T ( n)( C t(n))( X) X = n (ϕ(x) ψ(x, C)).

21 Arithmetic Theories Defining Complexity Classes Provable Inclusions The Strength of Bounded Arithmetic Fundamental Question Which inclusions between complexity classes are provable in bounded arithmetic?

22 Arithmetic Theories Defining Complexity Classes Provable Inclusions The Strength of Bounded Arithmetic Fundamental Question Which inclusions between complexity classes are provable in bounded arithmetic? Which arguments formalize in bounded arithmetic? easy: combinatorial arguments difficult: counting arguments or inductive proofs a strong tool: witnessing theorems

23 Ingredients of the Proof Conclusion Back to the Karp-Lipton Collapse Theorem (Cook, Krajíček 07) If PV NP P/poly, then PV PH BH.

24 Ingredients of the Proof Conclusion Back to the Karp-Lipton Collapse Theorem (Cook, Krajíček 07) If PV NP P/poly, then PV PH BH. The main ingredient in the proof of this collapse result: Theorem (Cook, Krajíček 07) PV NP P/poly if and only if PV conp NP/O(1).

25 Ingredients of the Proof Conclusion Back to the Karp-Lipton Collapse Theorem (Cook, Krajíček 07) If PV NP P/poly, then PV PH BH. The main ingredient in the proof of this collapse result: Theorem (Cook, Krajíček 07) PV NP P/poly if and only if PV conp NP/O(1). Proof. uses strong witnessing arguments (KPT witnessing for -formulas).

26 The Converse Implication Ingredients of the Proof Conclusion Theorem (Buhrman, Chang, Fortnow 03) For every constant k we have conp NP/k if and only if PH BH 2 k.

27 The Converse Implication Ingredients of the Proof Conclusion Theorem (Buhrman, Chang, Fortnow 03) For every constant k we have conp NP/k if and only if PH BH 2 k. Proof. : relatively easy : intricate hard/easy argument

28 The Converse Implication Ingredients of the Proof Conclusion Theorem (Buhrman, Chang, Fortnow 03) For every constant k we have conp NP/k if and only if PH BH 2 k. Proof. : relatively easy : intricate hard/easy argument We formalize this argument in PV and obtain: Theorem If PV PH BH 2 k, then PV conp NP/k.

29 The Optimality of the Collapse Ingredients of the Proof Conclusion Corollary PV NP P/poly if and only if PV PH BH. Proof. : Cook, Krajíček 07 : follows by PV PH BH PV conp NP/O(1) PV conp NP/O(1) PV NP P/poly (Cook, Krajíček 07)

30 Ingredients of the Proof Conclusion Conditions of the form conp NP/O(1) naturally lead to propositional proof systems with advice (Cook, Krajíček 07).

31 Ingredients of the Proof Conclusion Conditions of the form conp NP/O(1) naturally lead to propositional proof systems with advice (Cook, Krajíček 07). Question Do there exist optimal proof systems with advice?

32 Ingredients of the Proof Conclusion Conditions of the form conp NP/O(1) naturally lead to propositional proof systems with advice (Cook, Krajíček 07). Question Do there exist optimal proof systems with advice? Answer Without advice optimal propositional proof systems probably do not exist (Köbler, Messner, Torán 03). For proof systems with advice we obtain optimal and p-optimal proof systems for several measures of advice.

33 Conclusion Karp-Lipton Collapse Results Ingredients of the Proof Conclusion Are the other collapse consequences also optimal? Is PH P NP[O(log)] equivalent to NP P/poly in S 1 2? Is PH P NP equivalent to NP P/poly in S 2 2?

34 Conclusion Karp-Lipton Collapse Results Ingredients of the Proof Conclusion Are the other collapse consequences also optimal? Is PH P NP[O(log)] equivalent to NP P/poly in S 1 2? Is PH P NP equivalent to NP P/poly in S 2 2? Theorem (Cook, Krajíček 07) S2 1 NP P/poly if and only if S1 2 conp NP/O(log).

35 Two Conjectures Ingredients of the Proof Conclusion Theorem (Buhrman, Chang, Fortnow 03) conp NP/O(1) if and only if PH BH = P NP[O(1)].

36 Two Conjectures Ingredients of the Proof Conclusion Theorem (Buhrman, Chang, Fortnow 03) conp NP/O(1) if and only if PH BH = P NP[O(1)]. Do the following equivalences hold? conp NP/O(log) if and only if PH P NP[O(log)]. conp NP/poly if and only if PH P NP.

37 Two Conjectures Ingredients of the Proof Conclusion Theorem (Buhrman, Chang, Fortnow 03) conp NP/O(1) if and only if PH BH = P NP[O(1)]. Do the following equivalences hold? conp NP/O(log) if and only if PH P NP[O(log)]. conp NP/poly if and only if PH P NP. Known results conp NP/O(log) PH P NP[O(log)] [Cook, Krajíček 07]. conp NP/poly PH S NP 2 [Cai et al. 05].