20,641 research outputs found

    Relating and contrasting plain and prefix Kolmogorov complexity

    Get PDF
    In [3] a short proof is given that some strings have maximal plain Kolmogorov complexity but not maximal prefix-free complexity. The proof uses Levin's symmetry of information, Levin's formula relating plain and prefix complexity and Gacs' theorem that complexity of complexity given the string can be high. We argue that the proof technique and results mentioned above are useful to simplify existing proofs and to solve open questions. We present a short proof of Solovay's result [21] relating plain and prefix complexity: K(x)=C(x)+CC(x)+O(CCC(x))K (x) = C (x) + CC (x) + O(CCC (x)) and C(x)=K(x)KK(x)+O(KKK(x))C (x) = K (x) - KK (x) + O(KKK (x)), (here CC(x)CC(x) denotes C(C(x))C(C(x)), etc.). We show that there exist ω\omega such that lim infC(ω1ωn)C(n)\liminf C(\omega_1\dots \omega_n) - C(n) is infinite and lim infK(ω1ωn)K(n)\liminf K(\omega_1\dots \omega_n) - K(n) is finite, i.e. the infinitely often C-trivial reals are not the same as the infinitely often K-trivial reals (i.e. [1,Question 1]). Solovay showed that for infinitely many xx we have xC(x)O(1)|x| - C (x) \le O(1) and x+K(x)K(x)log(2)xO(log(3)x)|x| + K (|x|) - K (x) \ge \log^{(2)} |x| - O(\log^{(3)} |x|), (here x|x| denotes the length of xx and log(2)=loglog\log^{(2)} = \log\log, etc.). We show that this result holds for prefixes of some 2-random sequences. Finally, we generalize our proof technique and show that no monotone relation exists between expectation and probability bounded randomness deficiency (i.e. [6, Question 1]).Comment: 20 pages, 1 figur

    NP-Hardness of Approximately Solving Linear Equations Over Reals

    Get PDF
    URL lists article on conference siteIn this paper, we consider the problem of approximately solving a system of homogeneous linear equations over reals, where each equation contains at most three variables. Since the all-zero assignment always satisfies all the equations exactly, we restrict the assignments to be “non-trivial”. Here is an informal statement of our result: it is NP-hard to distinguish whether there is a non-trivial assignment that satisfies 1δ1-\delta fraction of the equations or every non-trivial assignment fails to satisfy a constant fraction of the equations with a ``margin" of Ω(δ)\Omega(\sqrt{\delta}). We develop linearity and dictatorship testing procedures for functions f : Rn 7--> R over a Gaussian space, which could be of independent interest. We believe that studying the complexity of linear equations over reals, apart from being a natural pursuit, can lead to progress on the Unique Games Conjecture.National Science Foundation (U.S.) (NSF CAREER grant CCF-0833228)National Science Foundation (U.S.) (Expeditions grant CCF-0832795)U.S.-Israel Binational Science Foundation (BSF grant 2008059
    corecore