Related papers: O-notation in algorithm analysis
Developing new ways to estimate probabilities can be valuable for science, statistics, and engineering. By considering the information content of different output patterns, recent work invoking algorithmic information theory has shown that…
Inductive proofs can be represented as proof schemata, i.e. as parameterized sequences of proofs defined in a primitive recursive way. Applications of proof schemata can be found in the area of automated proof analysis where the schemata…
Classic algorithms and machine learning systems like neural networks are both abundant in everyday life. While classic computer science algorithms are suitable for precise execution of exactly defined tasks such as finding the shortest path…
The basic distinction between already known algorithmic characterizations of matroids and antimatroids is in the fact that for antimatroids the ordering of elements is of great importance. While antimatroids can also be characterized as set…
The paper discusses the limitations of deep learning models in identifying and utilizing features that remain invariant under a bijective transformation on the data entries, which we refer to as combinatorial patterns. We argue that the…
A nonnegative matrix $A$ is called primitive if $A^k$ is positive for some integer $k>0$. A generalization of this concept to finite sets of matrices is as follows: a set of matrices $\mathcal M = \{A_1, A_2, \ldots, A_m \}$ is primitive if…
We introduce a taxonomy of objects for EO programming language. This taxonomy is designed with a few principles in mind: non-redundancy, simplicity, and so on. The taxonomy is supposed to be used as a navigation map by EO programmers. It…
Termination analysis of linear loops plays a key r\^{o}le in several areas of computer science, including program verification and abstract interpretation. Already for the simplest variants of linear loops the question of termination…
First-order iterative optimization methods play a fundamental role in large scale optimization and machine learning. This paper presents control interpretations for such optimization methods. First, we give loop-shaping interpretations for…
We analyze the algorithm in [Holub, 2009], which decides whether a given word is a fixed point of a nontrivial morphism. We show that it can be implemented to have complexity in O(mn), where n is the length of the word and m the size of the…
A nonnegative matrix A is said to be primitive if there exists a positive integer m such that entries in A^m are positive and smallest such m is called the exponent of A: Primitive matrices are useful in the study of finite Markov chains…
The exact parameter values of mathematical models are often uncertain or even unknown. Nevertheless, we may have access to crude information about the parameters, e.g., that some of them are nonzero. Such information can be captured by…
Using tools from topology and functional analysis, we provide a framework where artificial neural networks, and their architectures, can be formally described. We define the notion of machine in a general topological context and show how…
Neural Algorithmic Reasoning (NAR) trains neural networks to simulate classical algorithms, enabling structured and interpretable reasoning over complex data. While prior research has predominantly focused on learning exact algorithms for…
Two matrices are said to be principal minor equivalent if they have equal corresponding principal minors of all orders. We give a characterization of principal minor equivalence and a deterministic polynomial time algorithm to check if two…
We introduce the notions of triviality and order-triviality for global invariant types in an arbitrary first-order theory and show that they are well behaved in the NIP context. We show that these two notions agree for invariant global…
Despite the widespread use of machine learning algorithms to solve problems of technological, economic, and social relevance, provable guarantees on the performance of these data-driven algorithms are critically lacking, especially when the…
We derive formulas for characterizing bounded orthogonally additive polynomials in two ways. Firstly, we prove that certain formulas for orthogonally additive polynomials derived in \cite{Kusa} actually characterize them. Secondly, by…
Graph games with {\omega}-regular winning conditions provide a mathematical framework to analyze a wide range of problems in the analysis of reactive systems and programs (such as the synthesis of reactive systems, program repair, and the…
Mechanistic interpretability aims to explain what a neural network has learned at a nuts-and-bolts level. What are the fundamental primitives of neural network representations? Previous mechanistic descriptions have used individual neurons…