Related papers: Limit Complexities, Minimal Descriptions, and $n$-…
The dimension of random simplicial complexes (defined as the maximal dimension among all faces) is a natural extreme value associated with the complex, and is closely related to other functionals defined by a maximum, such as the clique…
We prove the formula C(a,b) = K(a|C(a,b)) + C(b|a,C(a,b)) + O(1) that expresses the plain complexity of a pair in terms of prefix and plain conditional complexities of its components.
We study the notion of an asymptotically automatic sequence, which generalises the notion of an automatic sequence. While $k$-automatic sequences are characterised by finiteness of $k$-kernels, the $k$-kernels of asymptotically…
In this paper, we introduce the n-th discrete topological complexity and study its properties such as its relation with simplicial Lusternik-Schnirelmann category and how the higher dimensions of discrete topological complexity relate with…
We study the classification performance of Kronecker-structured models in two asymptotic regimes and developed an algorithm for separable, fast and compact K-S dictionary learning for better classification and representation of…
The computational complexity of a Delta 2 set will be calibrated by the amount of changes needed for any of its computable approximations. Firstly, we study Martin-Loef random sets, where we quantify the changes of initial segments.…
We initiate the theory of communication complexity of individual inputs held by the agents, rather than worst-case or average-case. We consider total, partial, and partially correct protocols, one-way versus two-way, with and without help…
TThe problem is to identify a probability associated with a set of natural numbers, given an infinite data sequence of elements from the set. If the given sequence is drawn i.i.d. and the probability mass function involved (the target)…
We consider representing of natural numbers by expressions using 1's, addition, multiplication and parentheses. $\left\| n \right\|$ denotes the minimum number of 1's in the expressions representing $n$. The logarithmic complexity $\left\|…
We demonstrate some novel links between entropy and description complexity, a notion referring to the minimal formula length for specifying given properties. Let MLU be the logic obtained by extending propositional logic with the universal…
Nies and Scholz defined quantum Martin-L\"of randomness (q-MLR) for states (infinite qubitstrings). We define a notion of quantum Solovay randomness and show it to be equivalent to q-MLR using purely linear algebraic methods. Quantum…
We present several philosophical ideas emerging from the studies of complex systems. We make a brief introduction to the basic concepts of complex systems, for then defining "abstraction levels". These are useful for representing…
Define $\|n\|$ to be the complexity of $n$, the smallest number of ones needed to write $n$ using an arbitrary combination of addition and multiplication. John Selfridge showed that $\|n\|\ge 3\log_3 n$ for all $n$. Based on this, this…
The theory of asymptotic complexity provides an approach to characterizing the behavior of programs in terms of bounds on the number of computational steps executed or use of computational resources. We describe work using ACL2 to prove…
We study definable sets $D$ of SU-rank 1 in $M^{eq}$, where $M$ is a countable homogeneous and simple structure in a language with finite relational vocabulary. Each such $D$ can be seen as a `canonically embedded structure', which inherits…
We determine, up to the equivalence of first-order interdefinability, all structures which are first-order definable in the random partial order. It turns out that these structures fall into precisely five equivalence classes. We achieve…
Non-compact symmetries cannot be fully broken by randomness since non-compact groups have no invariant probability distributions. In particular, this makes trickier the "Copernican" random choice of the place of the observer in infinite…
Let $\Omega$ be a non-singular syplectic form on some vector space $V$. Denote by $S^{n}_{k}(\Omega)$ the set of all $k$-dimensional planes $s$ in $V$ such that the restriction of $\Omega$ onto $s$ is singular. For the cases when $k=2,n-2$…
The rate of randomness (or dimension) of a string $\sigma$ is the ratio $C(\sigma)/|\sigma|$ where $C(\sigma)$ is the Kolmogorov complexity of $\sigma$. While it is known that a single computable transformation cannot increase the rate of…
Kolmogorov Complexity constitutes an integral part of computability theory, information theory, and computational complexity theory -- in the discrete setting of bits and Turing machines. Over real numbers, on the other hand, the…