Related papers: Logical Zonotopes: A Set Representation for the Fo…
We prove that each bounded polytope can be represented as a polynomial zonotope, which we refer to as the Z-representation of polytopes. Previous representations are the vertex representation (V-representation) and the halfspace…
Using a binary representation for basis elements of an algebra combined with a framework of multiplier and index functions, a connection has been established between the structure of a large class of algebras and the XOR componentwise…
We demonstrate how a genetic ring oscillator network with quorum sensing feedback can operate as a robust logic gate. Specifically we show how a range of logic functions, namely AND/NAND, OR/NOR and XOR/XNOR, can be realized by the system,…
A bitmap is a data structure designed to compactly represent sets of integers; it provides very fast operations for querying and manipulating such sets, exploiting bit-level parallelism. In this paper, we describe a technique for the…
Representing meaning in the form of high dimensional vectors is a common and powerful tool in biologically inspired architectures. While the meaning of a set of concepts can be summarized by taking a (possibly weighted) sum of their…
A wealth of geometric and combinatorial properties of a given linear endomorphism $X$ of $\R^N$ is captured in the study of its associated zonotope $Z(X)$, and, by duality, its associated hyperplane arrangement ${\cal H}(X)$. This…
We propose a binary representation of categorical values using a linear map. This linear representation preserves the neighborhood structure of categorical values. In the context of evolutionary algorithms, it means that every categorical…
Non-linear neuron models overcomes the limitations of linear binary models of neurons that have the inability to compute linearly non-separable functions such as XOR. While several biologically plausible models based on dendrite thresholds…
Activation-based steering provides control of LLM behavior at inference time, but the dominant paradigm reduces each concept to a single direction whose geometry is left largely unexamined. Rather than selecting a single steering direction,…
Polynomial threshold gates are basic processing units of an artificial neural network. When the input vectors are binary vectors, these gates correspond to Boolean functions and can be analyzed via their polynomial representations. In…
Mixed integer set representations, and specifically hybrid zonotopes, have enabled new techniques for reachability and verification of nonlinear and hybrid systems. Mixed-integer sets which have the property that their convex relaxation is…
This paper considers the problem of set-based state estimation for linear time-invariant (LTI) systems under time-varying sensor attacks. Provided that the LTI system is stable and observable via every single sensor and that at least one…
We present a design and protocol to add binary numbers using discrete solitons in waveguide arrays. We show that the nonlinear interaction between discrete solitons in waveguide arrays can be exploited to design half and full adders. By…
The linearly inseparable XOR problem and the related problem of representing binary logical gates is revisited from the point of view of temporal encoding and its solvability by spiking neural networks with minimal configurations of leaky…
To represent real $m$-dimensional vectors, a positional vector system given by a non-singular matrix $M \in \mathbb{Z}^{m \times m}$ and a digit set $\mathcal{D} \subset \mathbb{Z}^m$ is used. If $m = 1$, the system coincides with the well…
We introduce constrained polynomial zonotopes, a novel non-convex set representation that is closed under linear map, Minkowski sum, Cartesian product, convex hull, intersection, union, and quadratic as well as higher-order maps. We show…
Stable dissipative solitons are perfect carries of optical information due to remarkable stability of their waveforms that allows the signal transmission with extremely dense soliton packing without loosing the encoded information. Apart of…
Many security- and performance-critical domains, such as cryptography, rely on low-level verification to minimize the trusted computing surface and allow code to be written directly in assembly. However, verifying assembly code against a…
In formal safety verification, many proposed algorithms use parametric set representations and convert the computation of the relevant sets into an optimization problem; consequently, the choice of parameterization and objective function…
How do artificial neural networks bind concepts to form complex semantic structures? Here, we propose a simple neural code, whereby the existence and the type of relations between entities are represented by the distance and the direction…