Related papers: The complexity of computing in continuous time: sp…
Quantum computation with quantum data that can traverse closed timelike curves represents a new physical model of computation. We argue that a model of quantum computation in the presence of closed timelike curves can be formulated which…
Continuing the study of complexity theory of Koepke's Ordinal Turing Machines (OTMs) that was started by Rin, L\"owe and the author, we prove the following results: (1) An analogue of Ladner's theorem for OTMs holds: That is, there are…
We study the complexity of computing the projection of an arbitrary $d$-polytope along $k$ orthogonal vectors for various input and output forms. We show that if $d$ and $k$ are part of the input (i.e. not a constant) and we are interested…
It is natural to expect the following loosely stated approximation principle to hold: a numerical approximation solution should be in some sense as smooth as its target exact solution in order to have optimal convergence. For piecewise…
There is an important and interesting open question in computational complexity on the relation between the complexity classes $\mathcal{NP}$ and $\mathcal{PSPACE}$. It is a widespread belief that $\mathcal{NP}\ne\mathcal{PSPACE}$. In this…
We recall from previous work a model-independent framework of computational complexity theory. Notably for the present paper, the framework allows formalization of the issues of precision that present themselves when one considers physical,…
We study the weak call-by-value $\lambda$-calculus as a model for computational complexity theory and establish the natural measures for time and space -- the number of beta-reductions and the size of the largest term in a computation -- as…
The connection of Taylor maps and polynomial neural networks (PNN) to solve ordinary differential equations (ODEs) numerically is considered. Having the system of ODEs, it is possible to calculate weights of PNN that simulates the dynamics…
The often-asked question whether space-time is discrete or continuous may not be the right question to ask: Mathematically, it is possible that space-time possesses the differentiability properties of manifolds as well as the ultraviolet…
Lasso problems arise in many areas, including signal processing, machine learning, and control, and are closely connected to sparse coding mechanisms observed in neuroscience. A continuous-time ordinary differential equation (ODE)…
Robustness is a correctness notion for concurrent programs running under relaxed consistency models. The task is to check that the relaxed behavior coincides (up to traces) with sequential consistency (SC). Although computationally simple…
I study the class of problems efficiently solvable by a quantum computer, given the ability to "postselect" on the outcomes of measurements. I prove that this class coincides with a classical complexity class called PP, or Probabilistic…
Neural ordinary differential equations (Neural ODEs) are an effective framework for learning dynamical systems from irregularly sampled time series data. These models provide a continuous-time latent representation of the underlying…
Typestate systems ensure many desirable properties of imperative programs, including initialization of object fields and correct use of stateful library interfaces. Abstract sets with cardinality constraints naturally generalize typestate…
Polynomial chaos expansion (PCE) is a classical and widely used surrogate modeling technique in physical simulation and uncertainty quantification. By taking a linear combination of a set of basis polynomials - orthonormal with respect to…
We show that \emph{efficient Turing computability} at any fixed input/output precision implies the existence of \emph{compositionally sparse} (bounded-fan-in, polynomial-size) DAG representations and of corresponding neural approximants…
Intuitively, if we can prove that a program terminates, we expect some conclusion regarding its complexity. But the passage from termination proofs to complexity bounds is not always clear. In this work we consider Monotonicity Constraint…
This paper studies the computational difficulty of clustering problems that are defined directly on a continuous probability density. Rather than working with finite samples, we assume the density is given as a polynomial and ask whether it…
In this paper, the space complexity of nonuniform quantum computations is investigated. The model chosen for this are quantum branching programs, which provide a graphic description of sequential quantum algorithms. In the first part of the…
We survey the complexity class $\exists \mathbb{R}$, which captures the complexity of deciding the existential theory of the reals. The class $\exists \mathbb{R}$ has roots in two different traditions, one based on the Blum-Shub-Smale model…