Multiplying matrices using n arithmetic operations

It is widely known that the lower bound for the algorithmic complexity of square matrix multiplication resorts to at least $n^2$ arithmetic operations. The justification builds upon the following reasoning: given that there are $2 n^2$ numbers in the input matrices, any algorithm necessarily must operate on each at least once. In this paper, we show that this is not necessarily the case for certain instances of the problem, for instance matrices with natural number entries. We present an algorithm performing a single multiplication and $(n - 1)$ sums, therefore using n arithmetic operations. The ingenuity of the approach relies on encoding the original $2n^2$ elements as two numbers of much greater magnitude. Thus, though processing each of the inputs at least once, it relies on a lower count of arithmetic operations. In the computational model used to analyze this problem, such encoding operation is not available, thus it is not clear this work affects the currently accepted complexity results for matrix multiplication, but the new algorithm complexity (when taking into account the encodings) is $3n^2 + 2n - 1$ operations. In addition, given the exponential increase in multiplication operands magnitude, its practical usage is constrained to certain instances of the problem. Nonetheless, this work presents a novel mathematically inspired algorithm while pointing towards an alternative research path, which opens the possibility of novel algorithms and a taxonomy of matrix multiplications and associated complexities

Matrices as arrows! A biproduct approach to typed linear algebra

Motivated by the need to formalize generation of fast running code for linear algebra applications, we show how an index-free, calculational approach to matrix algebra can be developed by regarding matrices as morphisms of a category with biproducts. This shifts the traditional view of matrices as indexed structures to a type-level perspective analogous to that of the pointfree algebra of programming. The derivation of fusion, cancellation and abide laws from the biproduct equations makes it easy to calculate algorithms implementing matrix multiplication, the kernel operation of matrix algebra, ranging from its divide-and-conquer version to the conventional, iterative one. From errant attempts to learn how particular products and coproducts emerge from biproducts, we not only rediscovered block-wise matrix com- binators but also found a way of addressing other operations calculation- ally such as e.g. Gaussian elimination. A strategy for addressing vector- ization along the same lines is also given.FCT, Mondrian Project funded by contract PTDC/EIA-CCO/108302/2008

Implementation-First Approach of Developing Formal Semantics of a Simulation Language in VDM-SL

Formal specification is a basis for rigorous software implementation. VDM-SL is a formal specification language with an extensive executable subset. Successful cases of VDM-family including VDM-SL have shown that producing a well-tested executable specification can reduce the cost of the implementation phase. This paper introduces and discusses the reversed order of specification and implementation. The development of a multi-agent simulation language called \remobidyc is described and examined as a case study of defining a formal specification after initial implementation and reflecting the specification into the implementation code

Defining Effectiveness Using Finite Sets A Study on Computability

AbstractThis paper studies effectiveness in the domain of computability. In the context of model-theoretical approaches to effectiveness, where a function is considered effective if there is a model containing a representation of such function, our definition relies on a model provided by functions between finite sets and uses category theory as its mathematical foundations. The model relies on the fact that every function between finite sets is computable, and that the finite composition of such functions is also computable. Our approach is an alternative to the traditional model-theoretical based works which rely on (ZFC) set theory as a mathematical foundation, and our approach is also novel when compared to the already existing works using category theory to approach computability results. Moreover, we show how to encode Turing machine computations in the model, thus concluding the model expresses at least the desired computational behavior. We also provide details on what instances of the model would indeed be computable by a Turing machine

Bidirectional UML Visualisation of VDM Models

The VDM-PlantUML Plugin enables translations between the text based UML tool PlantUML and VDM++ and has been released as a part of the VDM VSCode extension. This enhances already extensive feature-set of VDM VSCode with support for UML. The link between VDM and UML is thoroughly described with a set of translation rules that serve as the base of the implementation of the translation plugin. This is however still an early rendition of the plugin with limited usability due to the loss of information between translations and a lack of workflow optimisations, which we plan to solve in the future

Uncertainty Quantification and Runtime Monitoring Using Environment-Aware Digital Twins

A digital twin for a Cyber-Physical System includes a simulation model that predicts how a physical system should behave. We show how to quantify and characterise violation events for a given safety property for the physical system. The analysis uses the digital twin to inform a runtime monitor that checks whether the noise and violations observed fall within expected statistical distributions. The results allow engineers to determine the best system configuration through what-if analysis. We illustrate our approach with a case study of an agricultural vehicle

A Cloud-Based Collaboration Platform for Model-Based Design of Cyber-Physical Systems

Businesses, particularly small and medium-sized enterprises, aiming to start up in Model-Based Design (MBD) face difficult choices from a wide range of methods, notations and tools before making the significant investments in planning, procurement and training necessary to deploy new approaches successfully. In the development of Cyber-Physical Systems (CPSs) this is exacerbated by the diversity of formalisms covering computation, physical and human processes. In this paper, we propose the use of a cloud-enabled and open collaboration platform that allows businesses to offer models, tools and other assets, and permits others to access these on a pay-per-use basis as a means of lowering barriers to the adoption of MBD technology, and to promote experimentation in a sandbox environment

On the physico-mechanical, electrical and dielectric properties of mullite-glass composites

Mullite-glass composites were obtained by solid-state reactive sintering of kaolinite clay and kaolin waste mixtures with waste additions up to 100 wt%. The structural and microstructural analysis of starting powders and sintered samples were evaluated by X-ray diffractometry (XRD) and field-emission scanning electron microscopy (FESEM). The mechanical properties were evaluated by measuring the flexural strength of sintered bodies. Electrical properties of the composites were assessed by impedance spectroscopy (at 30 °C and from 400 to 700 °C) in air. A viscous flux mechanism resulting from the glassy phase filled up the open porosity and increased the mechanical strength. Electrical conductivity, dielectric constant and dielectric loss were strongly dependent on the microstructural features, namely glassy phase and porosity. The activation energies (0.89–0.99 eV) for electrical conduction were lower than typical literature values of mullite-based materials. The results indicated that the herein synthesized mullite-glass composites with up to 53.6 wt% mullite are promising low-cost materials for electronics-related applications.publishe