Related papers: On the Complexity of Computing G\"odel Numbers
To explore the limitation of a class of quantum algorithms originally proposed for the Hilbert's tenth problem, we consider two further classes of mathematically non-decidable problems, those of a modified version of the Hilbert's tenth…
When designing an algorithm, one cares about arithmetic/computational complexity, but data movement (I/O) complexity plays an increasingly important role that highly impacts performance and energy consumption. For a given algorithm and a…
We argue that parameterized complexity is a useful tool with which to study global constraints. In particular, we show that many global constraints which are intractable to propagate completely have natural parameters which make them…
The halting problem is undecidable --- but can it be solved for "most" inputs? This natural question was considered in a number of papers, in different settings. We revisit their results and show that most of them can be easily proven in a…
We study the algorithmic complexity of the problem of deciding whether a Linear Time Invariant dynamical system with rational coefficients has bounded trajectories. Despite its ubiquitous and elementary nature in Systems and Control, it…
Let T be Goedel's system of primitive recursive functionals of finite type in the lambda formulation. We define by constructive means using recursion on nested multisets a multivalued function I from the set of terms of T into the set of…
Genetic programming (GP) is an evolutionary computation technique to solve problems in an automated, domain-independent way. Rather than identifying the optimum of a function as in more traditional evolutionary optimization, the aim of GP…
We study the classical and parameterized complexity of computing the positive non-clashing teaching dimension of a set of concepts, that is, the smallest number of examples per concept required to successfully teach an intelligent learner…
We develop a theory of complexity for numerical computations that takes into account the condition of the input data and allows for roundoff in the computations. We follow the lines of the theory developed by Blum, Shub, and Smale for…
Recursive Neural Networks are non-linear adaptive models that are able to learn deep structured information. However, these models have not yet been broadly accepted. This fact is mainly due to its inherent complexity. In particular, not…
Robin Hirsch posed in 1996 the 'Really Big Complexity Problem': classify the computational complexity of the network satisfaction problem for all finite relation algebras A. We provide a complete classification for the case that A is…
The Weisfeiler-Leman dimension of a graph $G$ is the least number $k$ such that the $k$-dimensional Weisfeiler-Leman algorithm distinguishes $G$ from every other non-isomorphic graph. The dimension is a standard measure of the descriptive…
In the paper we present results to develop an irreducible theory of complex systems in terms of self-organization processes of prime integer relations. Based on the integers and controlled by arithmetic only the self-organization processes…
Neural operator architectures employ neural networks to approximate operators mapping between Banach spaces of functions; they may be used to accelerate model evaluations via emulation, or to discover models from data. Consequently, the…
We call an $\alpha \in \mathbb{R}$ regainingly approximable if there exists a computable nondecreasing sequence $(a_n)_n$ of rational numbers converging to $\alpha$ with $\alpha - a_n < 2^{-n}$ for infinitely many $n \in \mathbb{N}$. We…
A generic computation of a subset $A$ of $\mathbb{N}$ is a computation which correctly computes most of the bits of $A$, but which potentially does not halt on all inputs. The motivation for this concept is derived from complexity theory,…
We show that every countable group H with solvable word problem (=computable group) can be subnormally embedded into a 2-generated group G which also has solvable word problem. Moreover, the membership problem for H < G is also solvable. We…
Linear differential equations and recurrences reveal many properties about their solutions. Therefore, these equations are well-suited for representing solutions and computing with special functions. We identify a large class of existing…
Recent years have seen significant activity on the problem of using data for the purpose of learning properties of quantum systems or of processing classical or quantum data via quantum computing. As in classical learning, quantum learning…
This document contains notes based on lectures given by Hendrik Lenstra at the PCMI summer school 2022. There are many problems in algebraic number theory which one would like to solve algorithmically, for example computation of the maximal…