Eric Allender Rutgers University Curiouser and Curiouser: The Link between Incompressibility and Complexity CiE Special Session, June 19, 2012.

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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, 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 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]. BUT The same P/poly bound holds, even if we consider P R K t instead of P tt R K t. and recall PSPACE is contained in P R. 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 P tt R is in P/poly. We give a proof of a statement of the form: A n A j ( n,j ) such that: if for each n and j there is a proof in PA of (n,j) then the conjecture holds. Eric Allender: Curiouser & Curiouser: The Link between Incompressibility and Complexity Basic Proof Theory Recall that Peano Arithmetic cannot prove the statement PA is consistent. Let PA 1 be PA + PA is consistent. Similarly, one can define PA 2, PA 3, 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 Approach Conjecture: Every decidable set in P tt R is in P/poly. We give a proof (in PA 1 ) of a statement of the form: A n A j ( n,j ) such that: if for each n and j there is a proof in PA of (n,j) then the conjecture holds. Eric Allender: Curiouser & Curiouser: The Link between Incompressibility and Complexity The Central Conjecture: The Earlier Approach Fails The connections to proof theory were unexpected and intriguing, and seemed promising But unfortunately, it turns out that many of the statements (n,j) are independent of PA (and a related approach yields statements (n,j,k) that are independent of each system PA r ). Eric Allender: Curiouser & Curiouser: The Link between Incompressibility and Complexity A High-Level View of the Earlier Approach Let A be decidable, and let M be a poly-time machine computing a tt -reduction from A to R. Let Q(x) be the set of queries that M asks on input x. Let the size of Q(x) be at most f(|x|). Then there is a d such that for all x, there is a V containing only strings of length at most d+log f(|x|), such that M V (x) = A(x). Note: V says long queries are non-random. Eric Allender: Curiouser & Curiouser: The Link between Incompressibility and Complexity A Warm-Up Let A be decidable, and let M be a poly-time machine computing a tt -reduction from A to R. Let Q(x) be the set of queries that M asks on input x. Let the size of Q(x) be at most f(|x|). Then there is a d such that for all x, there is a V containing only strings of length at most d+log f(|x|), such that M V (x) = A(x). Note: If some V works for all x of length n, then A is in P/poly. Eric Allender: Curiouser & Curiouser: The Link between Incompressibility and Complexity Proof Assume that for each d there is some x such that, for all V containing strings of length at most d+log f(|x|), M V (x)A(x). Consider the machine that takes input (d,r) and finds x (as above) and outputs the r th element of Q(x). This shows that each element y of Q(x) has C(y) log d + log f(|x|) + O(1) Eric Allender: Curiouser & Curiouser: The Link between Incompressibility and Complexity Proof Assume that for each d there is some x such that, for all V containing strings of length at most d+log f(|x|), M V (x)A(x). Consider the machine that takes input (d,r) and finds x (as above) and outputs the r th element of Q(x). This shows that each element y of Q(x) has C(y) log d + log f(|x|) + O(1) < d + log f(|x|). Thus if we pick V* to be R{0,1} d+log f(|x|), we see that M V* (x) = M R (x) Eric Allender: Curiouser & Curiouser: The Link between Incompressibility and Complexity Proof Assume that for each d there is some x such that, for all V containing strings of length at most d+log f(|x|), M V (x)A(x). Consider the machine that takes input (d,r) and finds x (as above) and outputs the r th element of Q(x). This shows that each element y of Q(x) has C(y) log d + log f(|x|) + O(1) < d + log f(|x|). Thus if we pick V* to be R{0,1} d+log f(|x|), we see that M V* (x) = M R (x) = A(x). Contradiction! Eric Allender: Curiouser & Curiouser: The Link between Incompressibility and Complexity Cleaning Things Up Let A be decidable, and let M be a poly-time machine computing a tt -reduction from A to R. Let Q(x) be the set of queries that M asks on input x. Let the size of Q(x) be at most f(|x|). Then there is a d such that for all x, there is a V containing only strings of length at most d+log f(|x|) g A (|x|), such that M V (x) = A(x). Eric Allender: Curiouser & Curiouser: The Link between Incompressibility and Complexity Cleaning Things Up Let A be decidable, and let M be a poly-time machine computing a tt -reduction from A to R. Let Q(x) be the set of queries that M asks on input x. Let the size of Q(x) be at most f(|x|). Then there is a d such that for all x, there is a V containing only strings of length at most d+log f(|x|) g A (|x|), such that M V (x) = A(x). Eric Allender: Curiouser & Curiouser: The Link between Incompressibility and Complexity Cleaning Things Up Let A be decidable, and let M be a poly-time machine computing a tt -reduction from A to R. Let Q(x) be the set of queries that M asks on input x. Let the size of Q(x) be at most f(|x|). Then for all x, there is a V containing only strings in R of length at most g A (|x|) such that M V (x) = A(x). Eric Allender: Curiouser & Curiouser: The Link between Incompressibility and Complexity A Refinement Let A be decidable, and let M be a poly-time machine computing a tt -reduction from A to R. Let Q(x) be the set of queries that M asks on input x. Let the size of Q(x) be at most f(|x|). Then for all x, there is a V containing only strings in R of length at most g A (|x|) such that M V (x) = A(x). Eric Allender: Curiouser & Curiouser: The Link between Incompressibility and Complexity Approximating R We can obtain a series of approximations to R (up to length g A (n)) as follows: R n,0 = all strings of length at most g A (n). R n,i+1 = R n,i minus the i+1 st string of length at most g A (n) that is found, in an enumeration of non-random strings. R n,0, R n,1, R n,2, R n,i* = R{0,1} g A (n) Eric Allender: Curiouser & Curiouser: The Link between Incompressibility and Complexity A Refinement Let A be decidable, and let M be a poly-time machine computing a tt -reduction from A to R. Let Q(x) be the set of queries that M asks on input x. Let the size of Q(x) be at most f(|x|). Then for all x{0,1} n, for all i, there is a V containing only strings in R n,i such that M V (x) = A(x). Eric Allender: Curiouser & Curiouser: The Link between Incompressibility and Complexity Proof Assume that for each d there is some x,i such that, for all V containing strings in R n,i of length at most d+log f(|x|), M V (x)A(x). Consider the machine that takes input (d,r) and finds x,i (as above) and outputs the r th element of Q(x). This shows that each element y of Q(x) has C(y) log d + log f(|x|) + O(1) < d + log f(|x|). Thus if we pick V* to be R{0,1} d+log f(|x|), we see that M V* (x) = M R (x) = A(x). Contradiction! Eric Allender: Curiouser & Curiouser: The Link between Incompressibility and Complexity Where does PA come in?? Let A be decidable, and let M be a poly-time machine computing a tt -reduction from A to R. Let Q(x) be the set of queries that M asks on input x. Let the size of Q(x) be at most f(|x|). Then for all x{0,1} n, for all i, there is a V containing only strings in R n,i such that M V (x) = A(x) and there is not a length-k proof that for all i, V is not equal to R n,i . Eric Allender: Curiouser & Curiouser: The Link between Incompressibility and Complexity What went wrong with the earlier approach. We have shown: For all x{0,1} n, for all i, there is a V containing only short strings in R n,i such that M V (x) = A(x). We were aiming at showing that one can swap the quantifiers, so that for all n, there is a V containing only short strings in R n,i such that, for all x of length n, M V (x) = A(x). But there is a (useless) reduction M for which this is false. (M already knows the outcome of its queries, assuming that the oracle is R.) Eric Allender: Curiouser & Curiouser: The Link between Incompressibility and Complexity Open Questions: Decrease the gap (NEXP vs EXPSPACE) between the lower and upper bounds on the complexity of the problems that are in NP R K for every U. Some of our proofs rely on using R K. Do similar results hold also for R C ? Disprove: The halting problem is in P tt R C. Can the PSPACE P/poly bound (in the time- bounded setting) be improved to BPP? Is this approach relevant at all to the P=BPP question? Eric Allender: Curiouser & Curiouser: The Link between Incompressibility and Complexity P vs BPP Our main intuition for P=BPP comes from [Impagliazzo, Wigderson]. Circuit lower bounds imply derandomization. Note that this provides much more than merely P=BPP; it gives a recipe for simulating any probabilistic algorithm. Goldreich has argued that any proof of P=BPP actually yields pseudorandom generators (and hence a recipe as above) but this has only been proved for the promise problem formulation of P=BPP. Eric Allender: Curiouser & Curiouser: The Link between Incompressibility and Complexity P vs BPP Recall that TTRT sits between BPP and PSPACE P/poly. A proof that TTRT = P would show that BPP = P but it is not clear that this would yield any sort of recipe for constructing useful pseudorandom generators. Although it would be a less useful approach, perhaps it might be an easier approach? Eric Allender: Curiouser & Curiouser: The Link between Incompressibility and Complexity Thank you!

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