Algebraic Characteristics of the Matrix Conversions Class and Its Hardware Implementation

This paper derives the algebraic characteristic of the matrix transformations class by the method of isomorphic mappings on the algebraic characteristic of the class of vector transformations using the primitive program algebras. The paper also describes the hardware implementation of the matrix operations accelerator based on the obtained results. The urgency of the work is caused by the fact that today there is a rapid integration of computer technology in all spheres of society and, as a consequence, the amount of data that needs to be processed per unit time is constantly increasing. Many problems involving large amounts of complex computation are solved by methods based on matrix operations. Therefore, the study of matrix calculations and their acceleration is a very important task. In this paper, as a contribution in this direction, we propose a study of the matrix transformations class using signature operations of primitive program algebra such as multi place superposition, branching, cycling, which are refinements of the most common control structures in most high-level programming languages, and also isomorphic mapping. Signature operations of primitive program algebra in combination with basic partial-recursive matrix functions and predicates allow to realize the set of all partial-recursive matrix functions and predicates. Obtained the result on the basis of matrix primitive program algebra. Isomorphism provides the reproduction of partially recursive functions and predicates for matrix transformations as a map of partially recursive vector functions and predicates. The completeness of the algebraic system of matrix transformations is ensured due to the available results on the derivation of the algebraic system completeness for vector transformations. A name model of matrix data has been created and optimized for the development of hardware implementation. The hardware implementation provides support for signature operations of primitive software algebra and for isomorphic mapping. Hardware support for the functions of sum, multiplication and transposition of matrices, as well as the predicate of equality of two matrices is implemented. Support for signature operations of primitive software algebra is provided by the design of the control part of the matrix computer based on the RISC architecture. The hardware support of isomorphism is based on counters, they allow to intuitively implement cycling in the functions of isomorphic mappings. Fast execution of vector operations is provided by the principle of computer calculations SIMD.

Hidden-Markov program algebra with iteration

We use hidden Markov models to motivate a quantitative compositional semantics for noninterference-based security with iteration, including a refinement- or ‘implements’ relation that compares two programs with respect to their information leakage; and we propose a program algebra for source-level reasoning about such programs, in particular as a means of establishing that an ‘implementation’ program leaks no more than its ‘specification’ program.This joins two themes: we extend our earlier work, having iteration but only qualitative (Morgan 2009), by making it quantitative; and we extend our earlier quantitative work (McIver et al. 2010) by including iteration.We advocate stepwise refinement and source-level program algebra – both as conceptual reasoning tools and as targets for automated assistance. A selection of algebraic laws is given to support this view in the case of quantitative noninterference; and it is demonstrated on a simple iterated password-guessing attack.

Algebraic Approach to Algorithmic Logic

Summary We introduce algorithmic logic - an algebraic approach according to [25]. It is done in three stages: propositional calculus, quantifier calculus with equality, and finally proper algorithmic logic. For each stage appropriate signature and theory are defined. Propositional calculus and quantifier calculus with equality are explored according to [24]. A language is introduced with language signature including free variables, substitution, and equality. Algorithmic logic requires a bialgebra structure which is an extension of language signature and program algebra. While-if algebra of generator set and algebraic signature is bialgebra with appropriate properties and is used as basic type of algebraic logic.