Formal methods in mathematics

The research aims to analyze geodetic methods for determining the location of natural reserve areas and standardizing the obtained data. The object of the research is the standardization of geodetic data for determining the boundaries of natural reserve areas. Normative and legal documents related to the classification and determination of territories of natural reserve areas and the main geodetic methods used to determine the boundaries of such areas were analyzed in the article. The structure and description of the main geodetic methods for the practical implementation of work to determine the location of natural reserve areas, including geodetic measurements, global navigation, and remote sensing methods, were also discussed. In addition, a number of discrepancies were identified in the process of obtaining, storing, and reproducing geodetic data. Overall, the article contains important information about methods for determining the boundaries of natural reserve areas and how mathematical methods can be used to standardize geodetic data by formalizing it. The research showed that formalization of geodetic data can be done using mathematical methods such as mathematical logic, Boolean algebra, and category theory. The article proposes the use of mathematical methods of data formalization, such as mathematical logic, Boolean algebra, and category theory, to structure and standardize geodetic data. Analytical and graphical models for standardizing geodetic data for determining the boundaries of natural reserve areas were developed using mathematical methods of formalization such as mathematical logic, Boolean algebra, and category theory.

The complexity of living organisms surpasses our unaided habilities of analysis. Hence, computational and mathematical methods are necessary for increasing our understanding of biological systems. At the same time, there has been a phenomenal recent progress allowing the application of novel formal methods to new domains. This progress has spurred a conspicuous optimism in computational biology. This optimism, in turn, has promoted a rapid increase in collaboration between specialists of biology with specialists of computer science. Through sheer complexity, however, many important biological problems are at present intractable, and it is not clear whether we will ever be able to solve such problems. We are in the process of learning what kind of model and what kind of analysis and synthesis techniques to use for a particular problem. Some existing formalisms have been readily used in biological problems, others have been adapted to biological needs, and still others have been especially developed for biological systems. This Research Topic has examples of cases (1) employing existing methods, (2) adapting methods to biology, and (3) developing new methods. We can also see discrete and Boolean models, and the use of both simulators and model checkers. Synthesis is exemplified by manual and by machine-learning methods. We hope that the articles collected in this Research Topic will stimulate new research.

On the Use of Multiple Models and Formal Control Synthesis in Batch Control

Hybrid systems air he/erogenous dynamical systems cliaracterized by interacting continuous and discrete dynamics. Such mathematical mode's have proved fruitful in a great d:ersiry of engineering applications. Including air-traffic control, automated manufacturing, and cliemical process control. The high-profile and safely-critical nature of the application areas /'.as fostered a large and growing body of work on formal methods for hybrid systems: mathematical logics, computational models and methods, and computer-aided reasoning tools supporting the formal specification and verification of performance requirements for hybrid systems, and the design and synthesis of control programs for hybrid systems that are provably correct with respect to formal specifications. This paper offers a synthetic overview of, am! original contributions to, the use of logics and formal methods in the analysis of hybrid systems. Keywords-Automata, computer-aided analysis, computer-aided software engineering, design automation, formal languages, hybrid control systems, logic, software verification, temporal logic.

This article presents a formalization approach for the requirements of object-oriented programs with state machines, using a spacecraft control system as a case study. It proposes a state pattern implementation, where each state is represented as a class with clearly defined responsibilities, and the transitions between states are controlled by the state objects themselves. Additionally, the application of model checking, theorem proving, and code generation techniques are discussed. The effectiveness of the proposed approach in ensuring compliance with the specified requirements is demonstrated, while also identifying potential drawbacks and limitations of the approach. The implementation is validated using a range of formal verification techniques, including model checking and theorem proving. The article also discusses how the approach can be extended and applied to other complex systems. Overall, the valuable insights into the formalization of requirements for object-oriented programs with state machines are provided, offering a practical and effective approach for verifying the correctness and completeness of such implementations. The results of this work have important implications for the development of safety-critical systems and can potentially improve the quality and reliability of software systems in various domains. By using mathematical models and rigorous formal methods, it is possible to detect and eliminate errors early in the development process, leading to higher confidence in the correctness of the final product. Future research in this area could explore the use of more advanced techniques, such as model-driven development and automatic code synthesis, to further streamline the software development process. Additionally, the development of more efficient and user-friendly tools could make these techniques more accessible to a wider range of developers and organizations. Altogether, the combination of formal methods and software engineering has the potential to revolutionize the way software systems are designed, developed, and verified, leading to safer and more reliable software for critical applications.

The article underlines the basic problems that usually evolve at a time of assessment of IT projects feasibility and economic justification. The article also defines basic approaches of the ongoing scientific research consisted in the integration within a single methodology a number of known methods of assessment of IT projects feasibility and economic justification with mathematical models and formal methods developed in the methodology IDEF0.

Although the search for new drugs with the desired activity spectrum is among the most important problems in pharmaceutical chemistry, the useful properties of a new drug are usually predicted either intuitively or proceeding from the existing notions about the mechanisms of interaction of this compound with physiologically important biological targets (receptors and active or regulatory centers of enzymes). Verification of such a prognosis is a long and expensive procedure. An alternative approach to establishing quantitative structure – activity relationships (QSARs) is offered by mathematical methods based on the construction of formal models of such relations. Unfortunately, the mathematical hypotheses obtained using such formal methods cannot always be interpreted; the accuracy of these methods is not always sufficient for reliably predicting the activity of new compounds. The most promising way out consists in using a combined approach, whereby a mathematical QSAR model is formulated using additional data on the structure (conformation) of molecules at the active center of an enzyme. Below we describe the application of such a combined approach to establishing QSARs and predicting the activity of some nucleoside inhibitors of reverse transcriptase of type 1 HIV.

An overview of the geomagnetic observations made in the northern part of Russia is presented from a historical perspective. Several stations were deployed on the territory of the former Soviet Union during the International Geophysical Year, 1957–1958, with the active participation and guidance of the Interagency Geophysical Committee which is inherited by the Geophysical Center of the Russian Academy of Sciences (GC RAS). In the 1990s, the majority of these stations, especially those in the remoter regions, were closed. Nowadays, the geomagnetic network, including the observatories of the INTERMAGNET program, has been restored. Examples of high-latitude geomagnetic variations in the Russian longitudinal sector are shown, and maps and trends of the secular variation over the territory of Russia presented. Particular attention is paid to the automated processing of data and to the analysis methods used. To process the growing amount of high-resolution geomagnetic data, sophisticated mathematical methods based on the fuzzy logic approach and new discrete mathematical analysis algorithms have been developed. The formal methods and algorithms for recognizing both artificial and natural disturbances in the magnetograms are described.

Requirements for cost-effective high-quality software are becoming more and more stringent both in Safety-Critical and in Complex Software Development Environments. Stimulated in part by recent issues from the UK Ministry of Defence, the use of Formal Mathematical Methods is going to be taken into greater consideration also in industrial contexts due to their descriptive and analytical capabilities. Nevertheless, there is a little guidance as to the manner in which formal techniques should be effectively used. In the area of Software Engineering several methodologies can be found which effectively support the software development process in terms of costs and productivity, but, unfortunately, not in terms of quality and reliability. In this paper we propose to integrate the Z formal specification notation with the HOOD object-oriented design methodology (i.e. the standard European Space Agency design method). A twofold benefit is produced: on the one hand the structuring principles of HOOD compensate a class of problems which are peculiar to formal specifications in general and, on the other hand, the behaviour of a HOOD object can be formally described allowing us to rigorously verify some properties. Also, like a “Troian” horse, this could be a promising approach for promoting industrial use of formal methods.KeywordsOperation SchemaSoftware Development ProcessParent ObjectPredicate PartMaster CopyThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Fine-Grained Parallel and Distributed Spatial Stochastic Simulation of Biological Reactions

ΩMEGA [7], a mathematical assistant environment comprising an interactive proof assistant, a proof planner, a structured knowledge base, a graphical user interface, access to external reasoners, etc., is being developed since the early 90’s at Saarland University. Similar to HOL4, Isabelle/HOL, Coq, or Mizar, the overall goal of the project is to develop a system platform for formal methods (not only) in mathematics and computer science. In ΩMEGA, user and system interact in order to produce verifiable and trusted proofs. By continously improving (not only) automation and interaction support in the system we want to ease the usually very tedious formalization and proving task for the user.KeywordsSuccessor StateNatural DeductionAssessment ModuleInteractive ProofProof StepThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In 1970, Michael O. Finkelstein (with William B. Fairley) proposed that under some circumstances a jury in a criminal trial might be invited to use Bayes' Theorem to address the issue of the identity of the criminal perpetrator. In 1971, Laurence Tribe responded with a rhetorically powerful and wide-ranging attack on what he called 'trial by mathematics'. Finkelstein responded to Tribe's attack by further explaining, refining and defending his proposal. Although Tribe soon fell silent on the use of mathematical and formal methods to dissect or regulate uncertain factual proof in legal proceedings, the Finkelstein-Tribe exchange precipitated a decades-long debate about trial by mathematics. But that debate, which continues to this day, became generally unproductive and sterile years ago. This happened in part because two misunderstandings plagued much of the debate almost from the start. The first misunderstanding was a widespread failure to appreciate that mathematics is part of a broader family of rigorous methods of reasoning, a family of methods that is often called 'formal'. The second misunderstanding was a widespread failure to appreciate that mathematical and formal analyses (including analyses that use numbers) can have a large variety of purposes. Before any further major research project on trial by mathematics is begun, interested researchers in mathematics, probability, logic and related fields, on the one hand, and interested legal professionals, on the other hand, should try to reach agreement about the possible distinct purposes that any given mathematical or formal analysis of inconclusive argument about uncertain factual hypotheses might serve. The article lists some of those possible purposes. Language: en

In the first draft of my list, more than half the papers I had chosen had already been mentioned in the three lists that preceded mine (in particular, [5, 6, 7, 15, 16, 12, 10]). Therefore, I decided to narrow the scope of my list around a theme, following David Wetherall’s example. If the idea of the CCR recommended reading list continues (which I hope), I would assume that at least some of the future lists will likewise be focused on particular subtopics or on particular viewpoints. My list focuses on network models. Networking is an engineering discipline, and many of the key advances in networking came out of efforts with an applied flavor, favoring systems, architecture, and experimentation over mathematical modeling and formal methods of analysis. Often, major new applications did not even emerge from academic projects, but were developed by creative young people outside the research community. This is in contrast to other disciplines, e.g., signal processing and communications, that live and breathe through their models. I believe that one of the reasons for this is the sheer complexity of the task at hand: building a global, scalable, robust, secure, evolvable communication infrastructure, satisfying a broad panoply of needs and interests, resisting a wide range of threats, absorbing new technologies and supporting new applications, is much more likely to fail because of an inadequate architecture than because of inadequate underlying models. The former can doom the whole undertaking (and we all know the acronyms lying dead by the wayside), while the latter is often only a matter of efficiency, of throwing more resources at the problem. Furthermore, Moore’s law and related exponential trends move the goalposts so quickly that clean, relevant modeling questions are often hard to find. I do believe, however, that models have something to say. First, models allow us to reason about scale before we are able or willing to build a large-scale artifact. Second, models can reduce the complexity of a question, by forcing us to try to filter out the salient features of a problem at hand. Third, new and practically relevant ways of approaching and solving a problem can come out of modeling efforts. Fourth, models are a useful didactic tool for the study of the systems we are building, in analogy to the natural sciences. Here, then, are some papers on network models that inspired me, and that I believe made a contribution beyond theory, by providing fresh insights or by showing the way to new solutions to old problems. The usual disclaimer applies: the list is heavily biased by my own interests. Enjoy!

Formal specification of Multilingual Alphanumeric Ordering Systems

The first and necessary step in systems approach to study biological phenomena is building a formal model. One of the possibilities is to construct a model based on Petri nets. They have an intuitive graphical representation on one hand, and on the other, can be analyzed using formal mathematical methods. Finding homologies or conserved processes playing important roles in various biological systems can be done by comparing models. The ones expressed as Petri nets are especially well-suited for such a comparison, but there is a lack of software tools for this task. To resolve this problem, a new analytical tool has been implemented in Holmes application and described in this article. It offers four different comparison methods, i.e. the ones based on t-invariants, decomposition, graphlets and branching vertices. Available at http://www.cs.put.poznan.pl/mradom/Holmes/holmes.html. Supplementary data are available at Bioinformatics online.