An identification system based on the explicit isomorphism problem

We propose a new identification system based on algorithmic problems related to computing isomorphisms between central simple algebras. We design a statistical zero knowledge protocol which relies on the hardness of computing isomorphisms between orders in division algebras which generalizes a protocol by Hartung and Schnorr, which relies on the hardness of integral equivalence of quadratic forms.

Fast Recognition of Some Parametric Graph Families

Recognizing graphs with high level of symmetries is hard in general, and usually requires additional structural understanding. In this paper we study a particular graph parameter and motivate its usage by devising eÿcient recognition algorithm for the family of I-graphs. For integers m a simple graph is cycle regular if every path of length ` belongs to exactly cycles of length m. We identify all cycle regular I-graphs and, as a conse-quence, describe linear recognition algorithm for the observed family. Similar procedure can be used to devise the recog-nition algorithms for Double generalized Petersen graphs and folded cubes. Besides that, we believe the structural observations and methods used in the paper are of independent interest and could be used for solving other algorithmic problems.

P vs NP: P is Equal to NP: Desired Proof

Computations and computational complexity are fundamental for mathematics and all computer science, including web load time, cryptography (cryptocurrency mining), cybersecurity, artificial intelligence, game theory, multimedia processing, computational physics, biology (for instance, in protein structure prediction), chemistry, and the P vs. NP problem that has been singled out as one of the most challenging open problems in computer science and has great importance as this would essentially solve all the algorithmic problems that we have today if the problem is solved, but the existing complexity is deprecated and does not solve complex computations of tasks that appear in the new digital age as efficiently as it needs. Therefore, we need to realize a new complexity to solve these tasks more rapidly and easily. This paper presents proof of the equality of P and NP complexity classes when the NP problem is not harder to compute than to verify in polynomial time if we forget recursion that takes exponential running time and goes to regress only (every problem in NP can be solved in exponential time, and so it is recursive, this is a key concept that exists, but recursion does not solve the NP problems efficiently). The paper’s goal is to prove the existence of an algorithm solving the NP task in polynomial running time. We get the desired reduction of the exponential problem to the polynomial problem that takes O(log n) complexity.

Modeling of physical phenomena as a methodological means of forming a knowledge structure in physics and programming

To bring the theoretical basis to the improvement of the methodic of carrying out laboratory works in physics and computer sciences, a study of psychological and cognitive processes that accompany the corresponding educational activity was held. The important moment of the work is the research of the mental side of educational activities in the terms of formal logic. The accompanying steps to these mental activities and the talk activity are considered. The content component of the talk activity was considered in the work as an important diagnostic factor that indicates the level of understanding of the nature of the new material or the completion of the corresponding cognitive structure. The original graphical ways of formalization of the cognitive scheme of the object activity and the logic of its formation in the context of mental and materialized education activities are presented. These ideas were used to consider the processes of educational activity on the example of the creation of a simple physical model that is realized through the Python talk with the help of Visual library. Educational activities are considered through the prism of the evolution of the structural organization of mental representation of the objects of this activity. The research demonstrates the cyclical nature of the corresponding cognitive processes, materialized activities, and hierarchical character of the structure of knowledge that is formed in the process of educational activities in physics and programming. The correlation of these processes with the dynamics of the evolution of knowledge is settled. The conceptual ideas that could be useful for modifications of the methodic of realization of laboratory knowledge in the direction of modeling of physical processes and phenomena are formulated. Individual algorithmic problems of modeling of physical processes are considered in the research as components of the holistic system of problems. According to the point of view of the authors, in the course of the preparation of highly professional specialists in the domain of natural sciences, it is necessary to devote a lot of time to the conscious, goal-oriented formation of protocols of abstract logical and causal thinking. The methodology of such work is well formulated, especially in the well-known methods of descending step-by-step detailing and in modular programming

Conjugacy Problem in the Fundamental Groups of High-Dimensional Graph Manifolds

The conjugacy problem for a group G is one of the important algorithmic problems deciding whether or not two elements in G are conjugate to each other. In this paper, we analyze the graph of group structure for the fundamental group of a high-dimensional graph manifold and study the conjugacy problem. We also provide a new proof for the solvable word problem.

Constraint Satisfaction Problems and Their Reduction to Directed Graphs

Constraint satisfaction problems present a general framework for studying a large class of algorithmic problems such as satisfaction of Boolean formulas, solving systems of equations over finite fields, graph colourings, as well as various applied problems in artificial intelligence (scheduling, allocation of cell phone frequencies, among others.) CSP (Constraint Satisfaction Problems) bring together graph theory, complexity theory and universal algebra. It is a well known result, due to Feder and Vardi, that any constraint satisfaction problem over a finite relational structure can be reduced to the homomorphism problem for a finite oriented graph. Until recently, it was not known whether this reduction preserves the type of the algorithm which solves the original constraint satisfaction problem, so that the same algorithm solves the corresponding digraph homomorphism problem. We look at how a recent construction due to Bulin, Deli´c, Jackson, and Niven can be used to show that the polynomial solvability of a constraint satisfaction problem using Datalog, a programming language which is a weaker version of Prolog, translates from arbitrary relational structures to digraphs.

Constraint Satisfaction Problems and Their Reduction to Directed Graphs

Constraint satisfaction problems present a general framework for studying a large class of algorithmic problems such as satisfaction of Boolean formulas, solving systems of equations over finite fields, graph colourings, as well as various applied problems in artificial intelligence (scheduling, allocation of cell phone frequencies, among others.) CSP (Constraint Satisfaction Problems) bring together graph theory, complexity theory and universal algebra. It is a well known result, due to Feder and Vardi, that any constraint satisfaction problem over a finite relational structure can be reduced to the homomorphism problem for a finite oriented graph. Until recently, it was not known whether this reduction preserves the type of the algorithm which solves the original constraint satisfaction problem, so that the same algorithm solves the corresponding digraph homomorphism problem. We look at how a recent construction due to Bulin, Deli´c, Jackson, and Niven can be used to show that the polynomial solvability of a constraint satisfaction problem using Datalog, a programming language which is a weaker version of Prolog, translates from arbitrary relational structures to digraphs.

Chapter 14. Minimal Unsatisfiability and Autarkies

Minimal unsatisfiability describes the reduced kernel of unsatisfiable formulas. The investigation of this property is very helpful in understanding the reasons for unsatisfiability as well as the behaviour of SAT-solvers and proof calculi. Moreover, for propositional formulas and quantified Boolean formulas the computational complexity of various SAT-related problems are strongly related to the complexity of minimal unsatisfiable formulas. While “minimal unsatisfiability” studies the structure of problem instances without redundancies, the study of “autarkies” considers the redundancies themselves, in various guises related to partial assignments which satisfy some part of the problem instance while leaving the rest “untouched”. As it turns out, autarky theory creates many bridges to combinatorics, algebra and logic, and the second part of this chapter provides a solid foundation of the basic ideas and results of autarky theory: the basic algorithmic problems, the algebra involved, and relations to various combinatorial theories (e.g., matching theory, linear programming, graph theory, the theory of permanents). Also the general theory of autarkies as a kind of combinatorial “meta theory” is sketched (regarding its basic notions).

Daisy: Data analysis integrated software system for X-ray experiments

Daisy (Data Analysis Integrated Software System) has been designed for the analysis and visualisation of X-ray experiments. To address the requirements of the Chinese radiation facilities community, spanning an extensive range from purely algorithmic problems to scientific computing infrastructure, Daisy sets up a cloud-native platform to support on-site data analysis services with fast feedback and interaction. Furthermore, the plug-in based application is convenient to process the expected high throughput data flow in parallel at next-generation facilities such as the High Energy Photon Source (HEPS). The objectives, functionality and architecture of Daisy are described in this article.