Some Applications of Coding Theory in Computational Complexity

Two-Dimensional Kolmogorov Complexity and Validation of the Coding Theorem Method by Compressibility

We propose a measure based upon the fundamental theoretical concept in algorithmic information theory that provides a natural approach to the problem of evaluating $n$-dimensional complexity by using an $n$-dimensional deterministic Turing machine. The technique is interesting because it provides a natural algorithmic process for symmetry breaking generating complex $n$-dimensional structures from perfectly symmetric and fully deterministic computational rules producing a distribution of patterns as described by algorithmic probability. Algorithmic probability also elegantly connects the frequ…

随机Kolmogorov复杂度的最优编码及其应用

Optimal Coding for Randomized Kolmogorov Complexity and Its Applications

本文在无单向函数假设下，为随机Kolmogorov复杂度建立了高效编码定理，解决了先前仅适用于概率版本的瓶颈，并由此证明了多个平均情况复杂度结果及解决了两个开放问题。

Optimal Coding Theorems in Time-Bounded Kolmogorov Complexity

The classical coding theorem in Kolmogorov complexity states that if an $n$-bit string $x$ is sampled with probability $δ$ by an algorithm with prefix-free domain then K$(x) \leq \log(1/δ) + O(1)$. In a recent work, Lu and Oliveira [LO21] established an unconditional time-bounded version of this result, by showing that if $x$ can be efficiently sampled with probability $δ$ then rKt$(x) = O(\log(1/δ)) + O(\log n)$, where rKt denotes the randomized analogue of Levin's Kt complexity. Unfortunately, this result is often insufficient when transferring applications of the classical coding theorem to…

On the Descriptive Complexity of Color Coding

Color coding is an algorithmic technique used in parameterized complexity theory to detect "small" structures inside graphs. The idea is to derandomize algorithms that first randomly color a graph and then search for an easily-detectable, small color pattern. We transfer color coding to the world of descriptive complexity theory by characterizing -- purely in terms of the syntactic structure of describing formulas -- when the powerful second-order quantifiers representing a random coloring can be replaced by equivalent, simple first-order formulas. Building on this result, we identify syntacti…

Algorithmic complexity for psychology: A user-friendly implementation of the coding theorem method

Kolmogorov-Chaitin complexity has long been believed to be impossible to approximate when it comes to short sequences (e.g. of length 5-50). However, with the newly developed \emph{coding theorem method} the complexity of strings of length 2-11 can now be numerically estimated. We present the theoretical basis of algorithmic complexity for short strings (ACSS) and describe an R-package providing functions based on ACSS that will cover psychologists' needs and improve upon previous methods in three ways: (1) ACSS is now available not only for binary strings, but for strings based on up to 9 dif…

Interpreting CNN for Low Complexity Learned Sub-pixel Motion Compensation in Video Coding

Deep learning has shown great potential in image and video compression tasks. However, it brings bit savings at the cost of significant increases in coding complexity, which limits its potential for implementation within practical applications. In this paper, a novel neural network-based tool is presented which improves the interpolation of reference samples needed for fractional precision motion compensation. Contrary to previous efforts, the proposed approach focuses on complexity reduction achieved by interpreting the interpolation filters learned by the networks. When the approach is imple…