Interested in formalizing the generation of fast running code for linear algebra applications, the authors 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 central operation of matrix algebra, ranging from its divide-and-conquer version to its vectorization implementation.From errant attempts to learn how particular products and coproducts emerge from biproducts, not only blocked matrix algebra is rediscovered but also a way of extending other operations (e.g. Gaussian elimination) blockwise, in a calculational style, is found.The prospect of building biproduct-based type checkers for computer algebra systems such as MATLAB™ is also considered.Keywords: Linear algebra, categories of matrices, algebra of programming

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Inspired by the relational algebra of data processing, this paper addresses the foundations of data analytical processing from a linear algebra perspective. The paper investigates, in particular, how aggregation operations such as cross tabulations and data cubes essential to quantitative analysis of data can be expressed solely in terms of matrix multiplication, transposition and the Khatri-Rao variant of the Kronecker product. The approach offers a basis for deriving an algebraic theory of data consolidation, handling the quantitative as well as qualitative sides of data science in a natural, elegant and typed way. It also shows potential for parallel analytical processing (OLAP), as the parallelization theory of such matrix operations is well acknowledged.

In today’s world, critical infrastructure is often controlled by computing systems. This introduces new risks for cyber attacks, which can compromise the security and disrupt the functionality of these systems. It is therefore necessary to build such systems with strong guarantees of resiliency against cyber attacks. One way to achieve this level of assurance is using formal verification, which provides proofs of system compliance with desired cyber security properties. The use of Formal Methods (FM) in aspects of cyber security and safety-critical systems are reviewed in this article. We split FM into the three main classes: theorem proving, model checking and lightweight FM. To allow the different uses of FM to be compared, we define a common set of terms. We further develop categories based on the type of computing system FM are applied in. Solutions in each class and category are presented, discussed, compared and summarised. We describe historical highlights and developments and present a state-of-the-art review in the area of FM in cyber security. This review is presented from the point of view of FM practitioners and researchers, commenting on the trends in each of the classes and categories. This is achieved by considering all types of FM, several types of security and safety-critical systems and by structuring the taxonomy accordingly. The article hence provides a comprehensive overview of FM and techniques available to system designers of security-critical systems, simplifying the process of choosing the right tool for the task. The article concludes by summarising the discussion of the review, focusing on best practices, challenges, general future trends and directions of research within this field.

International audienceAbstract. The number of malicious software (malware) is growing out of control. Syntactic signature based detection cannot cope with such growth and manual construction of malware signature databases needs to be replaced by computer learning based approaches. Currently, a single modern signature capturing the semantics of a malicious behavior can be used to replace an arbitrarily large number of old-fashioned syntactical signatures. However teaching computers to learn such behaviors is a challenge. Existing work relies on dynamic analysis to extract malicious behaviors, but such technique does not guarantee the coverage of all behaviors. To sidestep this limitation we show how to learn malware signatures using static reachability analysis. The idea is to model binary programs using pushdown systems (that can be used to model the stack operations occurring during the binary code execution), use reachability analysis to extract behaviors in the form of trees, and use subtrees that are common among the trees extracted from a training set of malware files as signatures. To detect malware we propose to use a tree automaton to compactly store malicious behavior trees and check if any of the subtrees extracted from the file under analysis is malicious. Experimental data shows that our approach can be used to learn signatures from a training set of malware files and use them to detect a test set of malware that is 5 times the size of the training set