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Eric Allender: Curiouser & Curiouser: The Link between Incompressibility and Complexity < 3 >< 3 > Complexity Classes P NP BPP PSPACE NEXP EXPSPACE P/poly

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Eric Allender Rutgers University Curiouser and Curiouser: The Link between Incompressibility and Complexity CiE Special Session, June 19, 2012 Eric Allender: Curiouser & Curiouser: The Link between Incompressibility and Complexity < 2 >< 2 > Todays Goal: To present new developments in a line of research dating back to 2002, presenting some unexpected connections between Kolmogorov Complexity (the theory of randomness), and Computational Complexity Theory Which ought to have nothing to do with each other! Eric Allender: Curiouser & Curiouser: The Link between Incompressibility and Complexity < 3 >< 3 > Complexity Classes P NP BPP PSPACE NEXP EXPSPACE P/poly Eric Allender: Curiouser & Curiouser: The Link between Incompressibility and Complexity < 4 >< 4 > A Jewel of Derandomization [Impagliazzo, Wigderson, 1997]: If there is a problem computable in time 2 n that requires circuits of size 2 n, then P = BPP. Eric Allender: Curiouser & Curiouser: The Link between Incompressibility and Complexity < 5 >< 5 > Kolmogorov Complexity C(x) = min{|d| : U(d) = x} U is a universal Turing machine K(x) = min{|d| : U(d) = x} U is a universal prefix-free Turing machine Important property Invariance: 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 < 6 >< 6 > Kolmogorov Complexity C(x) = min{|d| : U(d) = x} U is a universal Turing machine K(x) = min{|d| : U(d) = x} U is a universal prefix-free Turing machine Important property Invariance: 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 < 7 >< 7 > K, C, and Randomness K(x) and C(x) are close: C(x) K(x) C(x) + 2 log |x| Two notions of randomness: R C = {x : C(x) |x|} R K = {x : K(x) |x|} actually, infinitely many notions of randomness: R C U = {x : C U (x) |x|} Eric Allender: Curiouser & Curiouser: The Link between Incompressibility and Complexity < 8 >< 8 > K, C, and Randomness K(x) and C(x) are close: C(x) K(x) C(x) + 2 log |x| Two notions of randomness: R C = {x : C(x) |x|} R K = {x : K(x) |x|} actually, infinitely many notions of randomness: R C U = {x : C U (x) |x|}, R K U = {x : K U (x) |x|} Eric Allender: Curiouser & Curiouser: The Link between Incompressibility and Complexity < 9 >< 9 > K, C, and Randomness When it makes no difference, well write R instead of R C or R K. Basic facts: R is undecidable but 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 Inclusions NEXP is contained in NP R. [ABK06] PSPACE is contained in P R. [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 Inclusions NEXP is contained in NP R. [ABK06] PSPACE is contained in P R. [ABKMR06] BPP is contained in P tt R. [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 Inclusions NEXP is contained in NP R. [ABK06] PSPACE is contained in P R. [ABKMR06] BPP is contained in P tt R. [BFKL10] Bizarre, because a non-computable upper bound is presented on complexity classes! If we restrict attention to R K, then we can do better Eric Allender: Curiouser & Curiouser: The Link between Incompressibility and Complexity Three Bizarre Inclusions NEXP is contained in NP R K. The decidable sets that are in NP R K for every U are in EXPSPACE. [AFG11] PSPACE is contained in P R K. BPP is contained in P tt R K. The decidable sets that are in P tt R K for every U are in PSPACE. [AFG11] Eric Allender: Curiouser & Curiouser: The Link between Incompressibility and Complexity Three Bizarre Inclusions NEXP is contained in NP R K (for every U). The decidable sets that are in NP R K for every U are in EXPSPACE. [AFG11] PSPACE is contained in P R K (for every U). BPP is contained in P tt R K (for every U). The decidable sets that are in P tt R K for every U are in PSPACE. [AFG11] [CELM] The sets that are in P tt R K for every U are decidable. Eric Allender: Curiouser & Curiouser: The Link between Incompressibility and Complexity Three Bizarre Inclusions NEXP is contained in NP R K (for every U). The decidable sets that are in NP R K for every U are in EXPSPACE. [AFG11] PSPACE is contained in P R K (for every U). BPP is contained in P tt R K (for every U). The sets that are in P tt R K for every U are in PSPACE. [AFG11] Eric Allender: Curiouser & Curiouser: The Link between Incompressibility and Complexity Three Bizarre Inclusions NEXP is contained in NP R K (for every U). The sets that are in NP R K for every U are in EXPSPACE. [AFG11] PSPACE is contained in P R K (for every U). BPP is contained in P tt R K (for every U). The sets that are in P tt R K for every U are in PSPACE. [AFG11] Eric Allender: Curiouser & Curiouser: The Link between Incompressibility and Complexity Three Bizarre Inclusions NEXP is contained in NP R K (for every U). The sets that are in NP R K for every U are in EXPSPACE. [AFG11] Conjecture: This should hold for R C, too. BPP is contained in P tt R K (for every U). The sets that are in P tt R K for every U are in PSPACE. [AFG11] Eric Allender: Curiouser & Curiouser: The Link between Incompressibility and Complexity Three Bizarre Inclusions NEXP is contained in NP R K (for every U). The sets that are in NP R K for every U are in EXPSPACE. [AFG11] This holds even for sets in EXP tt R K for all U! BPP is contained in P tt R K (for every U). The sets that are in P tt R K for every U are in PSPACE. [AFG11] Eric Allender: Curiouser & Curiouser: The Link between Incompressibility and Complexity Three Bizarre Inclusions NEXP is contained in NP R K (for every U). The sets that are in NP R K for every U are in EXPSPACE. [AFG11] Conjecture: This class is exactly NEXP. BPP is contained in P tt R K (for every U). The sets that are in P tt R K 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 Inclusions NEXP is contained in NP R K (for every U). The sets that are in NP R K for every U are in EXPSPACE. [AFG11] Conjecture: This class is exactly NEXP. BPP is contained in P tt R K (for every U). The sets that are in P tt R K 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 P BPP is contained in P tt R K (for every U). The sets that are in P tt R K for every U are in PSPACE. Conjecture: This class is exactly P. Some support for this conjecture [ABK06]: The decidable sets that are in P dtt R C for every U are in P. The decidable sets that are in P parity-tt R C for every U are in P. Eric Allender: Curiouser & Curiouser: The Link between Incompressibility and Complexity K-Complexity and BPP vs P BPP is contained in P tt R K (for every U). The sets that are in P tt R K 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 P tt R is in P/poly. Eric Allender: Curiouser & Curiouser: The Link between Incompressibility and Complexity K-Complexity and BPP vs P BPP is contained in P tt R K (for every U). The sets that are in P tt R K 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 P tt R 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 Conjecture Conjecture: Every decidable set in P tt R is in P/poly. What can we show? Eric Allender: Curiouser & Curiouser: The Link between Incompressibility and Complexity The Central Conjecture Conjecture: Every decidable set in P tt R 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-complexity Let t be a time bound. (Think of t as being large, such as Ackermanns function.) Define K t (x) to be min{|d| : U(d) = x in at most t(|x|) steps}. Define R K t to be {x : K t (x) |x|}. Define TTRT = {A : A is in P tt R K t for all large enough time bounds t}. Vague intuition: Poly-time reductions should not be able to distinguish between R K t and R K, for large t. Eric Allender: Curiouser & Curiouser: The Link between Incompressibility and Complexity The Central Conjecture Conjecture: Every decidable set in P tt R 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) = 2 2 n, then R K t is NOT in P/poly. which supports our vague intuition, becaus

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