Search Results (14)

Formal verification has achieved remarkable outcomes in both theory advancement and engineering practice, with the formalization of mathematical theories serving as its foundational cornerstone—making this process particularly critical. Axiomatic set theory underpins modern mathematics, providing the rigorous basis for constructing almost all theories. Landau’s Foundations of Analysis starts with pure logical axioms from set theory, does not rely on geometric intuition, strictly constructs number systems, and is a benchmark for axiomatic analysis in modern mathematics. In this paper, we first develop a machine proof system for axiomatic set theory rooted in the Morse–Kelley(MK) system. This system encompasses effective proof automation, scale simplification, and specialized handling of the classification axiom for ordered pairs. We then prove the Transfinite Recursion Theorem, leveraging it to further prove the Recursion Theorem for natural numbers—the key result for defining natural number operations. Finally, we detail the implementation of a machine proof system for analysis, which adopts MK as its description language and adheres to Landau’s Foundations of Analysis. This formalization realized all the contents of the book from natural numbers to complex numbers. All formalization does not need to introduce the standard library and has undergone verification by Rocq(Coq) 8.16 to ensure reliability. Implemented using the Rocq proof assistant, the formalization has undergone verification to ensure reliability. This work holds broader applicability such as the formalization of point-set topology and abstract algebra, while also serving as a valuable resource for teaching axiomatic set theory and mathematical analysis. Full article

This paper establishes a connection between control theory for partially observed discrete-event systems (DESs) and automated theorem proving (ATP) in the calculus of positively constructed formulas (PCFs). The language of PCFs is a complete first-order language providing a powerful tool for qualitative analysis of dynamical systems. Based on ATP in the PCF calculus, a new technique is suggested for checking observability as a property of formal languages, which is necessary for the existence of supervisory control of DESs. In the case of violation of observability, words causing a conflict can also be extracted with the help of a specially designed PCF. With an example of the problem of path planning by a robot in an unknown environment, we show the application of our approach at one of the levels of a robot control system. The prover Bootfrost developed to facilitate PCF refutation is also presented. The tests show positive results and perspectives for the presented approach. Full article

The logico-pluralist LogiKEy knowledge engineering methodology and framework is applied to the modelling of a theory of legal balancing, in which legal knowledge (cases and laws) is encoded by utilising context-dependent value preferences. The theory obtained is then used to formalise, automatically evaluate, and reconstruct illustrative property law cases (involving the appropriation of wild animals) within the Isabelle/HOL proof assistant system, illustrating how LogiKEy can harness interactive and automated theorem-proving technology to provide a testbed for the development and formal verification of legal domain-specific languages and theories. Modelling value-oriented legal reasoning in that framework, we establish novel bridges between the latest research in knowledge representation and reasoning in non-classical logics, automated theorem proving, and applications in legal reasoning. Full article

Creating learning models that can exhibit sophisticated reasoning abilities is one of the greatest challenges in deep learning research, and mathematics is rapidly becoming one of the target domains for assessing scientific progress in this direction. In the past few years there has been an explosion of neural network architectures, datasets, and benchmarks specifically designed to tackle mathematical problems, reporting impressive achievements in disparate fields such as automated theorem proving, numerical integration, and the discovery of new conjectures or matrix multiplication algorithms. However, despite this notable success it is still unclear whether deep learning models possess an elementary understanding of quantities and numbers. This survey critically examines the recent literature, concluding that even state-of-the-art architectures and large language models often fall short when probed with relatively simple tasks designed to test basic numerical and arithmetic knowledge. Full article

The increasing significance of theorem proving-based formalization in mathematics and computer science highlights the necessity for formalizing foundational mathematical theories. In this work, we employ the Coq interactive theorem prover to methodically formalize the language, semantics, and syntax of propositional logic, a fundamental aspect of mathematical reasoning and proof construction. We construct four Hilbert-style axiom systems and a natural deduction system for propositional logic, and establish their equivalences through meticulous proofs. Moreover, we provide formal proofs for essential meta-theorems in propositional logic, including the Deduction Theorem, Soundness Theorem, Completeness Theorem, and Compactness Theorem. Importantly, we present an exhaustive formal proof of the Completeness Theorem in this paper. To bolster the proof of the Completeness Theorem, we also formalize concepts related to mappings and countability, and deliver a formal proof of the Cantor–Bernstein–Schröder theorem. Additionally, we devise automated Coq tactics explicitly designed for the propositional logic inference system delineated in this study, enabling the automatic verification of all tautologies, all internal theorems, and the majority of syntactic and semantic inferences within the system. This research contributes a versatile and reusable Coq library for propositional logic, presenting a solid foundation for numerous applications in mathematics, such as the accurate expression and verification of properties in software programs and digital circuits. This work holds particular importance in the domains of mathematical formalization, verification of software and hardware security, and in enhancing comprehension of the principles of logical reasoning. Full article

On-time delivery of documentation and contracts has been recognized as a crucial requirement for the successful delivery of projects. However, the construction industry still depends on time-consuming traditional contract processes, which negatively affect the overall productivity of projects in the industry. The use of Smart Contracts (SCs) is highlighted as a suitable novel technology to expedite the contract processes and establish a reliable payment environment in the construction industry. Whilst there has been an increase in the debate about the use of SCs in construction in recent years, their use in practice still seems to be in its infancy. As such, the topic will benefit from a thorough review of benefits, drivers, barriers and strategies that can enhance the implementation of SCs in construction. This article presents the key findings from a Systematic Literature Review (SLR) on SCs in the construction industry, critically assessing existing studies on the topic. The study initially involved 171 research papers for the SLR process, and out of that 49 research papers were filtered for further analysis after reading their abstracts. A total of 30 papers were finally filtered after the full-text reading for the SLR. Descriptive and content analysis were used to analyse the full-text findings. The study graphically mapped the bibliographic materials by using the Visualization of Similarities (VoS) Viewer software. As per the findings, the topic has mostly been researched in Asia and the Pacific as a region and China as a country. It was noted that there were more empirical articles than theoretical studies related to SCs, evidencing the industry relevance of the issue. A total of 55% of the articles reviewed have been published in journals with a Q1 ranking. All the articles were written by multiple authors, with 30% of the journal articles having international co-authors and benefitting from the collaboration between authors. Key advantages identified in the literature go beyond contract and payment provisions and include aspects such as logistic handling, decentralized applications, business process management, automated payments, etc. Key drivers for adoption are supply chain pressure, competitive pressure, top management support, simple layout, reduction in risks of clients, clarity in responsibility and risk allocation, whereas the key barriers include insecurity, limited observability, incompatibility, inactive government collaboration and limited storage capacity. Key strategies to enhance the application of SC in construction include integrating theorems proving symbolic execution, using the selective transparency method and lock fund system, testing the integration of SCs with other systems at the initial stage, incorporating semi-automated consensus mechanisms for payments, constructing a mechanism to actively engage with government bodies, etc. Full article

A micro air vehicle (MAV) is physically lightweight, such that even a slight perturbation could affect its attitude and position tracking. To attain better autonomous flight system performance, MAVs require good control strategies to maintain their attitude stability during translational movement. However, the available control methods nowadays have fixed gain, which is associated with the chattering phenomenon and is not robust enough. To overcome the aforementioned issues, an adaptive proportional integral derivative (PID) control scheme is proposed. An adaptive mechanism based on a second-order sliding mode control is used to tune the parameter gains of the PID controller, and chattering phenomena are reduced by a fuzzy compensator. The Lyapunov stability theorem and gradient descent approach were the basis for the automated tuning. Comparisons between the proposed scheme against SMC-STA and SMC-TanH were also made. MATLAB Simulink simulation results showed the overall favourable performance of the proposed scheme. Finally, the proposed scheme was tested on a model-based platform to prove its effectiveness in a complex real-time embedded system. Orbit and waypoint followers in the platform simulation showed satisfactory performance for the MAV in completing its trajectory with the environment and sensor models as perturbation. Both tests demonstrate the advantages of the proposed scheme, which produces better transient performance and fast convergence towards stability. Full article

The fault detection system using automated concepts is a crucial aspect of the industrial process. The automated system can contribute efficiently in minimizing equipment downtime therefore improving the production process cost. This paper highlights a novel model based fault detection (FD) approach combined with an interval type-2 (IT2) Takagi–Sugeno (T–S) fuzzy system for fault detection in the drilling process. The system uncertainty is considered prevailing during the process, and type-2 fuzzy methodology is utilized to deal with these uncertainties in an effective way. Two theorems are developed; Theorem 1, which proves the stability of the fuzzy modeling, and Theorem 2, which establishes the fault detector algorithm stability. A Lyapunov stabilty analysis is implemented for validating the stability criterion for Theorem 1 and Theorem 2. In order to validate the effective implementation of the complex theoretical approach, a numerical analysis is carried out at the end. The proposed methodology can be implemented in real time to detect faults in the drilling tool maintaining the stability of the proposed fault detection estimator. This is critical for increasing the productivity and quality of the machining process, and it also helps improve the surface finish of the work piece satisfying the customer needs and expectations. Full article

Recently developed GeoGebra tools for the automated deduction and discovery of geometric statements combine in a unique way computational (real and complex) algebraic geometry algorithms and graphic features for the introduction and visualization of geometric statements. In our paper we will explore the capabilities and limitations of these new tools, through the case study of a classic geometric inequality, showing how to overcome, by means of a double approach, the difficulties that might arise attempting to ‘discover’ it automatically. On the one hand, through the introduction of the dynamic color scanning method, which allows to visualize on GeoGebra the set of real solutions of a given equation and to shed light on its geometry. On the other hand, via a symbolic computation approach which currently requires the (tricky) use of a variety of real geometry concepts (determining the real roots of a bivariate polynomial $p(x,y)$ by reducing it to a univariate case through discriminants and Sturm sequences, etc.), which leads to a complete resolution of the initial problem. As the algorithmic basis for both instruments (scanning, real solving) are already internally available in GeoGebra (e.g., via the Tarski package), we conclude proposing the development and merging of such features in the future progress of GeoGebra automated reasoning tools. Full article

Omnidirectional honeybee traffic is the number of bees moving in arbitrary directions in close proximity to the landing pad of a beehive over a period of time. Automated video analysis of such traffic is critical for continuous colony health assessment. In our previous research, we proposed a two-tier algorithm to measure omnidirectional bee traffic in videos. Our algorithm combines motion detection with image classification: in tier 1, motion detection functions as class-agnostic object location to generate regions with possible objects; in tier 2, each region from tier 1 is classified by a class-specific classifier. In this article, we present an empirical and theoretical comparison of random reinforced forests and shallow convolutional networks as tier 2 classifiers. A random reinforced forest is a random forest trained on a dataset with reinforcement learning. We present several methods of training random reinforced forests and compare their performance with shallow convolutional networks on seven image datasets. We develop a theoretical framework to assess the complexity of image classification by a image classifier. We formulate and prove three theorems on finding optimal random reinforced forests. Our conclusion is that, despite their limitations, random reinforced forests are a reasonable alternative to convolutional networks when memory footprints and classification and energy efficiencies are important factors. We outline several ways in which the performance of random reinforced forests may be improved. Full article

We report, through different examples, the current development in GeoGebra, a widespread Dynamic Geometry software, of geometric automated reasoning tools by means of computational algebraic geometry algorithms. Then we introduce and analyze the case of the degeneracy conditions that so often arise in the automated deduction in geometry context, proposing two different ways for dealing with them. One is working with the saturation of the hypotheses ideal with respect to the ring of geometrically independent variables, as a way to globally handle the statement over all non-degenerate components. The second is considering the reformulation of the given hypotheses ideal—considering the independent variables as invertible parameters—and developing and exploiting the specific properties of this zero-dimensional case to analyze individually the truth of the statement over the different non-degenerate components. Full article

As the scale and complexity of safety-critical software continue to grow, it is necessary to ensure safety and reliability to avoid minor errors leading to catastrophic disasters. Meantime, the traditional method, such as testing and simulation alone is insufficient to ensure the correctness of systems. This leads to using formal methods to provide sufficient evidence for systems. However, design a high assurance safety-critical system by formal methods is challenging due to the complexity of operating systems. In addition, the traditional interactive theorem prover used in system verification requires hand-written proofs, which are more expensive. Therefore, the efforts of providing a standardized formal framework as well as safety proofs, are notable for the develop a safety-critical system. The purpose of this paper is to provide a safety framework to establish a highly reliable and safety-critical operating system based on the ARINC653 standard, a multilevel and standardized formal model. To verify the functional correctness of this model, we propose a context-based formal proof method for programs. To achieve this goal, we first model 57 core services of ARINC653 and define the high-level requirements as pre-and post-conditions. Then, we construct a set of specification statements a formal axiom system transformed into logical sentences, and the core service model is transformed into a logical sentence sequence to be proved. Finally, a context-based formal proof system for specification correctness is developed. We have verified the correctness of safety-critical operating system core services with this system. Experience shows that the verification system we developed can be achieved the functional correctness of a complete OS with a low implement burden, and that can simplify the difficulty of automated verification and increase the degree of automation of proof. Full article

First-order logic is an important part of mathematical logic, and automated theorem proving is an interdisciplinary field of mathematics and computer science. The paper presents an automated theorem prover for first-order logic, called $CSE\_E$ 1.0, which is a combination of two provers contradiction separation extension (CSE) and E, where CSE is based on the recently-introduced multi-clause standard contradiction separation (S-CS) calculus for first-order logic and E is the well-known equational theorem prover for first-order logic based on superposition and rewriting. The motivation of the combined prover $CSE\_E$ 1.0 is to (1) evaluate the capability, applicability and generality of $CSE\_E$ , and (2) take advantage of novel multi-clause S-CS dynamic deduction of CSE and mature equality handling of E to solve more and harder problems. In contrast to other improvements of E, $CSE\_E$ 1.0 optimizes E mainly from the inference mechanism aspect. The focus of the present work is given to the description of $CSE\_E$ including its S-CS rule, heuristic strategies, and the S-CS dynamic deduction algorithm for implementation. In terms of combination, in order not to lose the capability of E and use $CSE\_E$ to solve some hard problems which are unsolved by E, $CSE\_E$ 1.0 schedules the running of the two provers in time. It runs plain E first, and if E does not find a proof, it runs plain CSE, then if it does not find a proof, some clauses inferred in the CSE run as lemmas are added to the original clause set and the combined clause set handed back to E for further proof search. $CSE\_E$ 1.0 is evaluated through benchmarks, e.g., CASC-26 (2017) and CASC-J9 (2018) competition problems (FOFdivision). Experimental results show that $CSE\_E$ 1.0 indeed enhances the performance of E to a certain extent. Full article

The high degree of miniaturization in the electronics industry has been, for several years, a driver to push embedded systems to different fields and applications. One example is safety-critical systems, where the compactness in the form factor helps to reduce the costs and allows for the implementation of new techniques. The automotive industry is a great example of a safety-critical area with a great rise in the adoption of microelectronics. With it came the creation of the ISO 26262 standard with the goal of guaranteeing a high level of dependability in the designs. Other areas in the safety-critical applications domain have similar standards. However, these standards are mostly guidelines to make sure that designs reach the desired dependability level without explicit instructions. In the end, the success of the design to fulfill the standard is the result of a thorough verification process. Naturally, the goal of any verification team dealing with such important designs is complete coverage as well as standards conformity, but as these are complex hardware, complete functional verification is a difficult task. From the several techniques that exist to verify hardware, where each has its pros and cons, we studied six well-established in academia and in industry. We can divide them into two categories: simulation, which needs extremely large amounts of time, and formal verification, which needs unrealistic amounts of resources. Therefore, we conclude that a hybrid approach offers the best balance between simulation (time) and formal verification (resources). Full article