Short lists for shortest descriptions in short time

Is it possible to find a shortest description for a binary string? The well-known answer is "no, Kolmogorov complexity is not computable." Faced with this barrier, one might instead seek a short list of candidates which includes a laconic description. Remarkably such approximations exist. This paper presents an efficient algorithm which generates a polynomial-size list containing an optimal description for a given input string. Along the way, we employ expander graphs and randomness dispersers to obtain an Explicit Online Matching Theorem for bipartite graphs and a refinement of Muchnik's Conditional Complexity Theorem. Our main result extends recent work by Bauwens, Mahklin, Vereschchagin, and Zimand

Empirical and Strong Coordination via Soft Covering with Polar Codes

We design polar codes for empirical coordination and strong coordination in two-node networks. Our constructions hinge on the fact that polar codes enable explicit low-complexity schemes for soft covering. We leverage this property to propose explicit and low-complexity coding schemes that achieve the capacity regions of both empirical coordination and strong coordination for sequences of actions taking value in an alphabet of prime cardinality. Our results improve previously known polar coding schemes, which (i) were restricted to uniform distributions and to actions obtained via binary symmetric channels for strong coordination, (ii) required a non-negligible amount of common randomness for empirical coordination, and (iii) assumed that the simulation of discrete memoryless channels could be perfectly implemented. As a by-product of our results, we obtain a polar coding scheme that achieves channel resolvability for an arbitrary discrete memoryless channel whose input alphabet has prime cardinality.Comment: 14 pages, two-column, 5 figures, accepted to IEEE Transactions on Information Theor

On Macroscopic Complexity and Perceptual Coding

The theoretical limits of 'lossy' data compression algorithms are considered. The complexity of an object as seen by a macroscopic observer is the size of the perceptual code which discards all information that can be lost without altering the perception of the specified observer. The complexity of this macroscopically observed state is the simplest description of any microstate comprising that macrostate. Inference and pattern recognition based on macrostate rather than microstate complexities will take advantage of the complexity of the macroscopic observer to ignore irrelevant noise

A New Upperbound for the Oblivious Transfer Capacity of Discrete Memoryless Channels

We derive a new upper bound on the string oblivious transfer capacity of discrete memoryless channels. The main tool we use is the tension region of a pair of random variables introduced in Prabhakaran and Prabhakaran (2014) where it was used to derive upper bounds on rates of secure sampling in the source model. In this paper, we consider secure computation of string oblivious transfer in the channel model. Our bound is based on a monotonicity property of the tension region in the channel model. We show that our bound strictly improves upon the upper bound of Ahlswede and Csisz\'ar (2013).Comment: 7 pages, 3 figures, extended version of submission to IEEE Information Theory Workshop, 201