Abstract Shallit and Wang showed that the automatic complexity $A(x)$ satisfies $A(x)\ge n/13$ for almost all $x\in {\{\mathtt {0},\mathtt {1}\}}^n$ . They also stated Holger Petersen had informed them constant $13$ can be reduced to $7$ Here we show it $2+\epsilon $ any $\epsilon>0$ The result applies nondeterministic $A_N(x)$ In setting is tight inasmuch as $A_N(x)\le n/2+1$ x

We investigate topological, combinatorial, statistical, and enumeration properties of finite graphs with high Kolmogorov complexity (almost all graphs) using the novel incompressibility method. Example results are: (i) the mean and variance of the number of (possibly overlapping) ordered labeled subgraphs of a labeled graph as a function of its randomness deficiency (how far it falls short of t...

We prove the so-called Unitary Hyperbolicity Theorem, a result on hyperbolicity of unitary involutions. The analogous previously known results for the orthogonal and symplectic involutions are formal consequences of the unitary one. While the original proofs in the orthogonal and symplectic cases were based on the incompressibility of generalized Severi-Brauer varieties, the proof in the unitar...

We briefly survey some concepts related to empirical entropy — normal numbers, de Bruijn sequences and Markov processes — and investigate how well it approximates Kolmogorov complexity. Our results suggest lth-order empirical entropy stops being a reasonable complexity metric for almost all strings of length m over alphabets of size n about when nl surpasses m.