Curiouser and Curiouser : The Link between Incompressibility and Complexity

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Curiouser and Curiouser : The Link between Incompressibility and Complexity. CiE Special Session, June 19, 2012. Todays Goal:. To present new developments in a line of research dating back to 2002, presenting some unexpected connections between - PowerPoint PPT Presentation

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Curiouser and Curiouser: The Link between Incompressibility and ComplexityCiE Special Session, June 19, 2012Eric AllenderRutgers UniversityEric Allender: Curiouser & Curiouser: The Link between Incompressibility and Complexity< # >1Todays Goal:To present new developments in a line of research dating back to 2002, presenting some unexpected connections betweenKolmogorov Complexity (the theory of randomness), andComputational Complexity TheoryWhich ought to have nothing to do with each other!Eric Allender: Curiouser & Curiouser: The Link between Incompressibility and Complexity< # >2Complexity ClassesPNPBPPPSPACENEXPEXPSPACEP/polyEric Allender: Curiouser & Curiouser: The Link between Incompressibility and Complexity< # >A Jewel of Derandomization[Impagliazzo, Wigderson, 1997]: If there is a problem computable in time 2n that requires circuits of size 2n, then P = BPP.Eric Allender: Curiouser & Curiouser: The Link between Incompressibility and Complexity< # >4Kolmogorov ComplexityC(x) = min{|d| : U(d) = x}U is a universal Turing machineK(x) = min{|d| : U(d) = x}U is a universal prefix-free Turing machineImportant propertyInvariance: The choice of the universal Turing machine U is unimportant (up to an additive constant). x is random if C(x) |x|.Eric Allender: Curiouser & Curiouser: The Link between Incompressibility and Complexity< # >5Kolmogorov ComplexityC(x) = min{|d| : U(d) = x}U is a universal Turing machineK(x) = min{|d| : U(d) = x}U is a universal prefix-free Turing machineImportant propertyInvariance: The choice of the universal Turing machine U is unimportant (up to an additive constant). x is random if C(x) |x|, or K(x) |x|.Eric Allender: Curiouser & Curiouser: The Link between Incompressibility and Complexity< # >6K, C, and RandomnessK(x) and C(x) are close:C(x) K(x) C(x) + 2 log |x|Two notions of randomness:RC = {x : C(x) |x|}RK = {x : K(x) |x|}actually, infinitely many notions of randomness:RCU = {x : CU(x) |x|}

Eric Allender: Curiouser & Curiouser: The Link between Incompressibility and Complexity< # >K, C, and RandomnessK(x) and C(x) are close:C(x) K(x) C(x) + 2 log |x|Two notions of randomness:RC = {x : C(x) |x|}RK = {x : K(x) |x|}actually, infinitely many notions of randomness:RCU = {x : CU(x) |x|}, RKU = {x : KU(x) |x|}

Eric Allender: Curiouser & Curiouser: The Link between Incompressibility and Complexity< # >K, C, and RandomnessWhen it makes no difference, well write R instead of RC or RK.Basic facts:R is undecidablebut it is not easy to use it as an oracle.R is not NP-hard under poly-time m reductions, unless P=NP.Things get more interesting when we consider more powerful types of reducibility.Eric Allender: Curiouser & Curiouser: The Link between Incompressibility and Complexity< # >Three Bizarre InclusionsNEXP is contained in NPR. [ABK06]PSPACE is contained in PR. [ABKMR06]BPP is contained in {A : A is poly-time tt R}. [BFKL10]A tt reduction is a non-adaptive reduction.On input x, a list of queries is formulated before receiving any answer from the oracle.Eric Allender: Curiouser & Curiouser: The Link between Incompressibility and Complexity< # >Three Bizarre InclusionsNEXP is contained in NPR. [ABK06]PSPACE is contained in PR. [ABKMR06]BPP is contained in PttR. [BFKL10]

Bizarre, because a non-computable upper bound is presented on complexity classes!We have been unable to squeeze larger complexity classes inside. Are these containments optimal?Eric Allender: Curiouser & Curiouser: The Link between Incompressibility and Complexity< # >Three Bizarre InclusionsNEXP is contained in NPR. [ABK06]PSPACE is contained in PR. [ABKMR06]BPP is contained in PttR. [BFKL10]

Bizarre, because a non-computable upper bound is presented on complexity classes!

If we restrict attention to RK, then we can do betterEric Allender: Curiouser & Curiouser: The Link between Incompressibility and Complexity< # >Three Bizarre InclusionsNEXP is contained in NPRK.The decidable sets that are in NPRK for every U are in EXPSPACE. [AFG11]PSPACE is contained in PRK.

BPP is contained in PttRK.The decidable sets that are in PttRK for every U are in PSPACE. [AFG11]

Eric Allender: Curiouser & Curiouser: The Link between Incompressibility and Complexity< # >Three Bizarre InclusionsNEXP is contained in NPRK (for every U).The decidable sets that are in NPRK for every U are in EXPSPACE. [AFG11]PSPACE is contained in PRK (for every U).

BPP is contained in PttRK (for every U).The decidable sets that are in PttRK for every U are in PSPACE. [AFG11][CELM] The sets that are in PttRK for every U are decidable.

Eric Allender: Curiouser & Curiouser: The Link between Incompressibility and Complexity< # >Three Bizarre InclusionsNEXP is contained in NPRK (for every U).The decidable sets that are in NPRK for every U are in EXPSPACE. [AFG11]PSPACE is contained in PRK (for every U).

BPP is contained in PttRK (for every U).The sets that are in PttRK for every U are in PSPACE. [AFG11]

Eric Allender: Curiouser & Curiouser: The Link between Incompressibility and Complexity< # >Three Bizarre InclusionsNEXP is contained in NPRK (for every U).The sets that are in NPRK for every U are in EXPSPACE. [AFG11]PSPACE is contained in PRK (for every U).

BPP is contained in PttRK (for every U).The sets that are in PttRK for every U are in PSPACE. [AFG11]

Eric Allender: Curiouser & Curiouser: The Link between Incompressibility and Complexity< # >Three Bizarre InclusionsNEXP is contained in NPRK (for every U).The sets that are in NPRK for every U are in EXPSPACE. [AFG11]Conjecture: This should hold for RC, too.

BPP is contained in PttRK (for every U).The sets that are in PttRK for every U are in PSPACE. [AFG11]

Eric Allender: Curiouser & Curiouser: The Link between Incompressibility and Complexity< # >Three Bizarre InclusionsNEXP is contained in NPRK (for every U).The sets that are in NPRK for every U are in EXPSPACE. [AFG11]This holds even for sets in EXPttRK for all U!

BPP is contained in PttRK (for every U).The sets that are in PttRK for every U are in PSPACE. [AFG11]Eric Allender: Curiouser & Curiouser: The Link between Incompressibility and Complexity< # >Three Bizarre InclusionsNEXP is contained in NPRK (for every U).The sets that are in NPRK for every U are in EXPSPACE. [AFG11]Conjecture: This class is exactly NEXP.

BPP is contained in PttRK (for every U).The sets that are in PttRK for every U are in PSPACE. [AFG11]Conjecture: This class is exactly BPP.Eric Allender: Curiouser & Curiouser: The Link between Incompressibility and Complexity< # >Three Bizarre InclusionsNEXP is contained in NPRK (for every U).The sets that are in NPRK for every U are in EXPSPACE. [AFG11]Conjecture: This class is exactly NEXP.

BPP is contained in PttRK (for every U).The sets that are in PttRK for every U are in PSPACE. [AFG11]Conjecture: This class is exactly BPP P.Eric Allender: Curiouser & Curiouser: The Link between Incompressibility and Complexity< # >K-Complexity and BPP vs PBPP is contained in PttRK (for every U).The sets that are in PttRK for every U are in PSPACE.Conjecture: This class is exactly P.Some support for this conjecture [ABK06]:The decidable sets that are in PdttRC for every U are in P. The decidable sets that are in Pparity-ttRC for every U are in P. Eric Allender: Curiouser & Curiouser: The Link between Incompressibility and Complexity< # >K-Complexity and BPP vs PBPP is contained in PttRK (for every U).The sets that are in PttRK for every U are in PSPACE.Conjecture: This class is exactly P.New results support a weaker conjecture:Conjecture: This class is contained in PSPACE P/poly.More strongly: Every decidable set in PttR is in P/poly.Eric Allender: Curiouser & Curiouser: The Link between Incompressibility and Complexity< # >K-Complexity and BPP vs PBPP is contained in PttRK (for every U).The sets that are in PttRK for every U are in PSPACE.Conjecture: This class is exactly P.New results support a weaker conjecture :Conjecture: This class is contained in PSPACE P/poly.More strongly: Every decidable set in PttR is in P/poly (i.e., for every U, and for both C and K). Eric Allender: Curiouser & Curiouser: The Link between Incompressibility and Complexity< # >The Central ConjectureConjecture: Every decidable set in PttR is in P/poly.What can we show?Eric Allender: Curiouser & Curiouser: The Link between Incompressibility and Complexity< # >The Central ConjectureConjecture: Every decidable set in PttR is in P/poly.What can we show?We show that a similar statement holds in the context of time-bounded K-complexity.Eric Allender: Curiouser & Curiouser: The Link between Incompressibility and Complexity< # >Time-Bounded K-complexityLet t be a time bound. (Think of t as being large, such as Ackermanns function.)Define Kt(x) to be min{|d| : U(d) = x in at most t(|x|) steps}.Define RKt to be {x : Kt(x) |x|}.Define TTRT = {A : A is in PttRKt for all large enough time bounds t}.Vague intuition: Poly-time reductions should not be able to distinguish between RKt and RK, for large t.Eric Allender: Curiouser & Curiouser: The Link between Incompressibility and Complexity< # >The Central ConjectureConjecture: Every decidable set in PttR is in P/poly.We show that a similar statement holds in the context of time-bounded K-complexity:TTRT is contained in P/poly [ABFL12].If t(n) = 22n, then RKt is NOT in P/poly.which supports our vague intuition, because this set is not reducible to the time-t-random strings for t >> t.Eric Allender: Curiouser & Curiouser: The Link between Incompressibility and Complexity< # >The Central ConjectureConjecture: Every decidable set in PttR is in P/poly.We show that a similar statement holds in the context of time-bounded K-complexity:TTRT is contained in P/poly [ABFL12].BUT The same P/poly bound holds, even if we consider PRKt instead of PttRKt.and recall PSPACE is contained in PR. So the vague intuition is wrong!Eric Allender: Curiouser & Curiouser: The Link between Incompressibility and Complexity< # >The Central Conjecture: An Earlier Approach Conjecture: Every decidable set in PttR is in P/poly.We give a proof of a statement of the form: AnAj(n,j)such that: if for each n and jthere is a proof in PA of (n,j)then the conjecture holds.Eric Allender: Curiouser & Curiouser: The Link between Incompressibility and Complexity< # >Basic Proof TheoryRecall that Peano Arithmetic cannot prove the statement PA is consistent.Let PA1 be PA + PA is consistent.Similarly, one can define PA2, PA3, PA is consistent can be formulated as for all j, there is no length j proof of 0=1.For each j, PA can prove there is no length j proof of 0=1.Eric Allender: Curiouser & Curiouser: The Link between Incompressibility and Complexity< # >The Central Conjecture: An Earlier ApproachConjecture: Every decidable set in PttR is in P/poly.We give a proof (in PA1) of a statement of the form: AnAj(n,j)such that: if for each n and jthere is a proof in PA of (n,j)then the conjecture holds.Eric Allender: Curiouser & Curiouser: The Link between Incomp...

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