Impossibility of independence amplification in Kolmogorov complexity theory

The paper studies randomness extraction from sources with bounded independence and the issue of independence amplification of sources, using the framework of Kolmogorov complexity. The dependency of strings $x$ and $y$ is ${\rm dep}(x,y) = \max\{C(x) - C(x \mid y), C(y) - C(y\mid x)\}$, where $C(\cdot)$ denotes the Kolmogorov complexity. It is shown that there exists a computable Kolmogorov extractor $f$ such that, for any two $n$-bit strings with complexity $s(n)$ and dependency $\alpha(n)$, it outputs a string of length $s(n)$ with complexity $s(n)- \alpha(n)$ conditioned by any one of the input strings. It is proven that the above are the optimal parameters a Kolmogorov extractor can achieve. It is shown that independence amplification cannot be effectively realized. Specifically, if (after excluding a trivial case) there exist computable functions $f_1$ and $f_2$ such that ${\rm dep}(f_1(x,y), f_2(x,y)) \leq \beta(n)$ for all $n$-bit strings $x$ and $y$ with ${\rm dep}(x,y) \leq \alpha(n)$, then $\beta(n) \geq \alpha(n) - O(\log n)$

Torts—Negligence—Standard of Care

Caldwell v. Village of Island Park, 304 N. Y. 268, 107 N. E. 2d 441 (1952y, rev\u27g 279 App. Div. 746, 108 N. Y. S. 2d 334 (2d Dep\u27t 1951)

Decedent\u27s Estate—Right of Spouse to Elect Against Will

In re Shupack\u27s Will, 1 N. Y. 2d 482, 136 N. E. 2d 513 (1956), modifying 1 A. D. 2d 841, 149 N. Y. S. 2d 20 (2d Dep\u27t 1956)

Torts—Negligence—Duty

Woods v. Lancet, 303 N. Y. 349, 102 N. E. 2d 691 (1951), rev\u27g 278 App. Div. 913, 105 N. Y. S. 2d 417 (Ist Dep\u27t 1951)

Torts—Negligence—Causation

Dunham v. Village of Canisteo, 303 N. Y. 498, 104 N. E. 2d 872 (1952), rev\u27g 278 App. Div. 743, 103 N. Y. S. 2d 519 (4th Dep\u27t 1951)

Domestic Relations—Annulment: Fraud

Woronzoff-Daschkoff v. Woronzoff-Daschkoff, 303 W: Y. 506, 104 N. E. 2d 877 (1952), rev\u27g 278 App. Div. 924, 105 N. Y. S. 2d 910 (1st Dep\u27t 1951)

Domestic Relations—Annulment: Fraud

Woronzoff-Daschkoff v. Woronzoff-Daschkoff, 303 W: Y. 506, 104 N. E. 2d 877 (1952), rev\u27g 278 App. Div. 924, 105 N. Y. S. 2d 910 (1st Dep\u27t 1951)

Conflict of Laws—Foreign Marriage Incestuous by Lex Domicilii Held Valid

In Re May\u27s Estate, __ App. Div. __, 117 N. Y. S. 2d 345 (3d Dep\u27t 1952)

Wills—Effect of Taxes on Elective Share

In Re Wolf\u27s Will, 282 App. Div. 1018, 126 N. Y. S. 2d 302 (1st. Dep\u27t 1953), affirming per curiam, 204 Misc. 356, 121 N. Y. S. 2d 412 (Surr. Ct. 1953)

Administrative Law—A Widening of Judicial Review

Holland v. Edwards, 282 App. Div. 353, 122 N. Y. S. 2d 721 (1st Dep\u27t 1953)