On the Kolmogorov-Chaitin Complexity for short sequences

Jean-Paul Delahaye; Hector Zenil

Home/Archive/arXiv

arXiv · 0704.1043

On the Kolmogorov-Chaitin Complexity for short sequences

Jean-Paul Delahaye·Hector Zenil

View original source ↗Read source document ↗Export metadata ↓

Abstract

A drawback of Kolmogorov-Chaitin complexity (K) as a function from s to the shortest program producing s is its noncomputability which limits its range of applicability. Moreover, when strings are short, the dependence of K on a particular universal Turing machine U can be arbitrary. In practice one can approximate it by computable compression methods. However, such compression methods do not always provide meaningful approximations--for strings shorter, for example, than typical compiler lengths. In this paper we suggest an empirical approach to overcome this difficulty and to obtain a stable definition of the Kolmogorov-Chaitin complexity for short sequences. Additionally, a correlation in terms of distribution frequencies was found across the output of two models of abstract machines, namely unidimensional cellular automata and deterministic Turing machine.

Explore related subjects

cs.CC cs.IT math.IT

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Jean-Paul Delahaye, Hector Zenil. 2010-12-17. On the Kolmogorov-Chaitin Complexity for short sequences. https://arxiv.org/abs/0704.1043

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

CVP Is NP-Complete for Principal Cyclotomic Ideals

We prove that exact Euclidean decision-CVP is $\mathsf{NP}$-complete on the coefficient lattices of nonzero principal ideals in the power-of-two cyclotomic rings $R_d:=\mathbb{Z}[y]/(y^d+1)$. Our deterministic reduction from Exact Cover by 3-Sets (X3C) produces a target and a squared threshold $Δ$ such that the closest squared distance is exactly $Δ$ in YES instances and at least $Δ+4$ in NO instances. This also implies $\mathsf{NP}$-hardness of exact search-CVP under polynomial-time Turing reductions. We also transfer the resulting principal-ideal CVP instances to full-rank principal ideals of the cyclic quotient ring $\mathbb{Z}[X]/(X^D-1)$, where $D:=2d$. Their coefficient lattices are invariant under cyclic coordinate shifts. The lift preserves principality and multiplies corresponding squared distances by eight. Thus, on principal cyclic ideal lattices, exact decision-CVP is $\mathsf{NP}$-complete and exact search-CVP is $\mathsf{NP}$-hard. We also obtain uniformly computable fixed cyclotomic and cyclic families in which only the target and threshold depend on the X3C collection. Consequently, a polynomial-time solution to exact decision-CVPP on either family would imply $\mathsf{NP}\subseteq\mathsf{P}/\mathrm{poly}$ and collapse the polynomial hierarchy to $Σ_2^{\mathsf{P}}$. To our knowledge, the cyclic results answer Micciancio's questions of whether exact decision-CVP is $\mathsf{NP}$-hard on cyclic lattices and on a fixed family of cyclic lattices, even when restricted to full-rank principal cyclic ideals. Finally, under the coefficient embedding, we prove that exact decision-module-SIVP is $\mathsf{NP}$-complete on free rank-two modules over the same cyclotomic rings.

cs.CC↗

arXiv2026

Fooling Thresholds of Halfspaces

We initiate the study of constructing explicit pseudorandom generators for thresholds of halfspaces with seed length polylogarithmic in the number of halfspaces. This class of functions lies at the frontier of circuit complexity [CTW26]. We show that the generator designed by O'Donnell, Servedio, and Tan for polytopes [OST22] also fools this broader class. To analyze the generator, we develop a threshold-specific smooth approximation framework based on a Bentkus-type mollifier. We prove derivative bounds for this mollifier and also establish a Boolean anticoncentration theorem for thresholds of halfspaces via a random thinning argument. These ingredients imply that the generator $δ$-fools every $k$-out-of-$m$ threshold of $m$ halfspaces over $\{-1,1\}^n$ with seed length $\widetilde{O}(κ^{6+2\varepsilon}\log^{6+2\varepsilon}\!m\cdotδ^{-(2+2\varepsilon)}\log n)$, for any arbitrarily small constant $\varepsilon>0$, where $κ=\min\{k,m-k+1\}$. The random thinning argument also yields bounds on the noise sensitivity and Gaussian surface area for thresholds of halfspaces, leading to learning algorithms under both the uniform and Gaussian distributions.

cs.CC↗

arXiv2026

An Oracle Separating Conjectures about Incompleteness in the Finite Domain

Pudlák [Pud17] lists several major conjectures from the field of proof complexity and asks for oracles that separate corresponding relativized conjectures. Among these conjectures are: - $\mathsf{DisjNP}$: The class of all disjoint NP-pairs does not have many-one complete elements. - $\mathsf{SAT}$: NP does not contain many-one complete sets that have P-optimal proof systems. - $\mathsf{UP}$: UP does not have many-one complete problems. - $\mathsf{NP}\cap\mathsf{coNP}$: $\text{NP}\cap\text{coNP}$ does not have many-one complete problems. As one answer to this question, we construct an oracle relative to which $\mathsf{DisjNP}$, $\neg \mathsf{SAT}$, $\mathsf{UP}$, and $\mathsf{NP}\cap\mathsf{coNP}$ hold, i.e., there is no relativizable proof for the implication $\mathsf{DisjNP}\wedge \mathsf{UP}\wedge \mathsf{NP}\cap\mathsf{coNP}\Rightarrow\mathsf{SAT}$. In particular, regarding the conjectures by Pudlák this extends a result by Khaniki [Kha19].

cs.CC↗