Search results for: automated theorem proving

Authors: Jiaqi Huang, Yuheng Wang

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Premises selection, the task of selecting a set of axioms for proving a given conjecture, is a major bottleneck in automated theorem proving. An array of deep-learning-based methods has been established for premises selection, but a perfect performance remains challenging. Our study examines the inaccuracy of deep neural networks in premises selection. Through training network models using encoded conjecture and axiom pairs from the Mizar Mathematical Library, two potential biases are found: the network models classify more premises as necessary than unnecessary, referred to as the ‘positive bias’, and the network models perform better in proving conjectures that paired with more axioms, referred to as ‘length bias’. The ‘positive bias’ and ‘length bias’ discovered could inform the limitation of existing deep neural networks.

Keywords: automated theorem proving, premises selection, deep learning, interpreting deep learning

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Authors: Fahad Suleiman, Sammani Abdullahi

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The study depicts that a Wedderburn Artin – theorem for rings is considered to be a semisimple ring R which is isomorphic to a product of finitely many mi by mi matrix rings over division rings Di, for some integers n_i, both of which are uniquely determined up to permutation of the index i. It has been concluded that when R is simple the Wedderburn – Artin theorem is known as Wedderburn’s theorem.

Keywords: Commutativity, Wedderburn theorem, Semisimple ring, R module

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Authors: Ayubi Wirara, Ardya Suryadinata

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Improper application of the RSA algorithm scheme can cause vulnerability to attacks. The attack utilizes the relationship between broadcast messages sent to the user with some fixed polynomial functions that belong to each user. Scheme attacks carried out by applying the Chinese Remainder Theorem to obtain a general polynomial equation with the same modulus. The formation of the general polynomial becomes a first step to get back the original message. Furthermore, to solve these equations can use Coppersmith's theorem.

Keywords: RSA algorithm, broadcast message, Chinese Remainder Theorem, Coppersmith’s theorem

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Authors: Mohammed Husein Mohammad Rashid

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For a bounded linear operator T acting on a complex infinite dimensional Hilbert space ℋ, we say that T is ∗-class A operator (abbreviation T∈A*) if |T²|≥ |T*|². In this article, we prove the following assertions:(i) we establish some conditions which imply the normality of ∗-class A; (ii) we consider ∗-class A operator T ∈ ℬ(ℋ) with reducing kernel such that TX = XS for some X ∈ ℬ(K, ℋ) and prove the Fuglede-Putnam type theorem when adjoint of S ∈ ℬ(K) is dominant operators; (iii) furthermore, we extend the asymmetric Putnam-Fuglede theorem the class of ∗-class A operators.

Keywords: fuglede-putnam theorem, normal operators, ∗-class a operators, dominant operators

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Authors: Nicodemus M. J. Mbwambo, Yu-Shan Sun, Murali Sitaraman, Joan Krone

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This paper presents a generic data abstraction that captures a navigable tree position. The mathematical modeling of the abstraction encapsulates the current tree position, which can be used to navigate and modify the tree. The encapsulation of the tree position in the data abstraction specification avoids the use of explicit references and aliasing, thereby simplifying verification of (imperative) client code that uses the data abstraction. To ease the tasks of such specification and verification, a general tree theory, rich with mathematical notations and results, has been developed. The paper contains an example to illustrate automated verification ramifications. With sufficient tree theory development, automated proving seems plausible even in the absence of a special-purpose tree solver.

Keywords: automation, data abstraction, maps, specification, tree, verification

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Authors: Ivan Stanev, Maria Koleva

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A common platform for automated programming (CPAP) is defined in details. Two versions of CPAP are described: Cloud-based (including the set of components for classic programming, and the set of components for combined programming) and KBASE based (including the set of components for automated programming, and the set of components for ontology programming). Four KBASE products (module for automated programming of robots, intelligent product manual, intelligent document display, and intelligent form generator) are analyzed and CPAP contributions to automated programming are presented.

Keywords: automated programming, cloud computing, knowledge based software engineering, service oriented architecture

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Authors: Yu Lu

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Fixed point theory is a basic tool for the study of the existence of Nash equilibria in game theory. This paper presents a significant generalization of the Veinott-Zhou fixed point theorem for increasing correspondences, which serves as an essential framework for investigating the existence of Nash equilibria in supermodular and quasisupermodular games. To establish our proofs, we explore different conceptions of multivalued increasingness and provide comprehensive results concerning the existence of the largest/least fixed point. We provide two distinct approaches to the proof, each offering unique insights and advantages. These advancements not only extend the applicability of the Veinott-Zhou theorem to a broader range of economic scenarios but also enhance the theoretical framework for analyzing equilibrium behavior in complex game-theoretic models. Our findings pave the way for future research in the development of more sophisticated models of economic behavior and strategic interaction.

Keywords: fixed-point, Tarski’s fixed-point theorem, Nash equilibrium, supermodular game

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Authors: Ivan Stanev, Maria Koleva

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Automated software development issues are addressed in this paper. Layers and packages of a Common Platform for Automated Programming (CPAP) are defined based on Service Oriented Architecture, Cloud computing, Knowledge based automated software engineering (KBASE) and Method of automated programming. Tools of seven leading companies (AWS of Amazon, Azure of Microsoft, App Engine of Google, vCloud of VMWare, Bluemix of IBM, Helion of HP, OCPaaS of Oracle) are analyzed in the context of CPAP. Based on the results of the analysis CPAP requirements are formulated

Keywords: automated programming, cloud computing, knowledge based software engineering, service oriented architecture

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Authors: M. H. M. Rashid

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Given H a Hilbert space and B(H) the algebra of bounded linear operator in H, let δAB denote the generalized derivation defined by A and B. The main objective of this article is to study Weyl type theorems for generalized derivation for (A,B) satisfying a couple of Fuglede.

Keywords: Fuglede Property, Weyl’s theorem, generalized derivation, Aluthge transform

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Authors: Tze Jin Wong, Lee Feng Koo, Pang Hung Yiu

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Lenstra’s attack uses Chinese remainder theorem as a tool and requires a faulty signature to be successful. This paper reports on the security responses of fourth and sixth order Lucas based (LUC_4,6) cryptosystem under the Lenstra’s attack as compared to the other two Lucas based cryptosystems such as LUC and LUC₃ cryptosystems. All the Lucas based cryptosystems were exposed mathematically to the Lenstra’s attack using Chinese Remainder Theorem and Dickson polynomial. Result shows that the possibility for successful Lenstra’s attack is less against LUC_4,6 cryptosystem than LUC₃ and LUC cryptosystems. Current study concludes that LUC_4,6 cryptosystem is more secure than LUC and LUC₃ cryptosystems in sustaining against Lenstra’s attack.

Keywords: Lucas sequence, Dickson polynomial, faulty signature, corresponding signature, congruence

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Authors: David A. Harness

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Natural evolution of ATP cognitive systems is to meet AI peer review standards. ATP process of axiom selection from Mizar to prove a conjecture would be further refined, as in all human and machine learning, by solving the real world problem of the proposed AI peer review challenge: Determine which conjecture forms the higher confidence level constructive proof between Standard Model of Physics SU(n) lattice gauge group operation vs. present non-standard 4D GEM EOS SU(n) lattice gauge group spatially extended operation in which the photon and electron are the first two trace angular momentum invariants of a gravitoelectromagnetic (GEM) energy momentum density tensor wavetrain integration spin-stress pressure-volume equation of state (EOS), initiated via 32 lines of Mathematica code. Resulting gravitoelectromagnetic spectrum ranges from compressive through rarefactive of the central cosmological constant vacuum energy density in units of pascals. Said self-adjoint group operation exclusively operates on the stress energy momentum tensor of the Einstein field equations, introducing quantization directly on the 4D spacetime level, essentially reformulating the Yang-Mills virtual superpositioned particle compounded lattice gauge groups quantization of the vacuum—into a single hyper-complex multi-valued GEM U(1) × SU(1,3) lattice gauge group Planck spacetime mesh quantization of the vacuum. Thus the Mizar corpus already contains all of the axioms required for relevant DeepMath premise selection and unambiguous formal natural language parsing in context deep learning.

Keywords: automated theorem proving, constructive quantum field theory, information theory, neural networks

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Authors: Mohamed Houas, Maamar Benbachir

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In this paper, under weak assumptions, we study the existence and uniqueness of solutions for a nonlinear fractional boundary value problem. New existence and uniqueness results are established using Banach contraction principle. Other existence results are obtained using scheafer and krasnoselskii's fixed point theorem. At the end, some illustrative examples are presented.

Keywords: caputo derivative, boundary value problem, fixed point theorem, local conditions

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Authors: Christian Rathgeber, Franz Winkler, Xiaoyu Kang, Steffen Müller

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In this contribution two approaches for calculating optimal trajectories for highly automated vehicles are presented and compared. The first one is based on a non-linear vehicle model, used for evaluation. The second one is based on a simplified model and can be implemented on a current ECU. In usual driving situations both approaches show very similar results.

Keywords: trajectory planning, direct method, indirect method, highly automated driving

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Authors: Thanin Sitthiwirattham, Jiraporn Reunsumrit

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We study the existence of positive solutions to the three points difference summation boundary value problem. We show the existence of at least one positive solution if f is either superlinear or sublinear by applying the fixed point theorem due to Krasnoselskii in cones.

Keywords: positive solution, boundary value problem, fixed point theorem, cone

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Authors: Paola Cattabriga

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According to Gödel’s first incompleteness argument it is possible to construct a formally undecidable proposition in Principia mathematica, a statement that, although true, turns out to be neither provable nor refutable for the system, making therefore incomplete any formal system suitable for the arithmetic of integers. Its features and limitation effects are today widespread basics throughout whole scientific thought. This article brings Gödel’s achievement into question by the definition of the refutability predicate as a number-theoretical statement. We develop proof of invalidity of Theorem VI in Gödel’s 1931, the so-called Gödel’s first incompleteness theorem, in two steps: defining refutability within the same recursive status as provability and showing that as a consequence propositions (15) and (16), derived from definition 8.1 in Gödel’s 1931, are false and unacceptable for the system. The achievement of their falsity blocks the derivation of Theorem VI, which turns out to be therefore invalid, together with all the depending theorems. This article opens up thus new perspectives for mathematical research and for the overall scientific reasoning.

Keywords: Gödel numbering, incompleteness, provability predicate, refutability predicate

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Authors: Hao-Hsi Tseng, Hsin-Yun Lee

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The Most Advantageous Tender (MAT) has been criticized for its susceptibility to dictatorial situations and for its processing of same score, same rank issues. This study applies the four criteria from Arrow's Impossibility Theorem to construct a mechanism for revealing illegitimate scores in scoring methods. While commonly be used to improve on problems resulting from extreme scores, ranking methods hide significant defects, adversely affecting selection fairness. To address these shortcomings, this study relies mainly on the overall evaluated score method, using standardized scores plus normal cumulative distribution function conversion to calculate the evaluation of vender preference. This allows for free score evaluations, which reduces the influence of dictatorial behavior and avoiding same score, same rank issues. Large-scale simulations confirm that this method outperforms currently used methods using the Impossibility Theorem.

Keywords: Arrow’s impossibility theorem, cumulative normal distribution function, most advantageous tender, scoring method

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Authors: Ivan Stanev, Maria Koleva

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Described are challenges to the Bulgarian e-Customs (BeC) related to low level of interoperability and standardization, inefficient use of available infrastructure, lack of centralized identification and authorization, extremely low level of software process automation, and insufficient quality of data stored in official registers. The technical requirements for BeC are prepared with a focus on domain independent common platform, specialized customs and excise components, high scalability, flexibility, and reusability. The Knowledge Based Automated Software Engineering (KBASE) Common Platform for Automated Programming (CPAP) is selected as an instrument covering BeC requirements for standardization, programming automation, knowledge interpretation and cloud computing. BeC stage 3 results are presented and analyzed. BeC.S3 development trends are identified.

Keywords: service oriented architecture, cloud computing, knowledge based automated software engineering, common platform for automated programming, e-customs

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Authors: Manish Jain, Dinesh Gopalani

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Growing dependency of mankind on software technology increases the need for thorough testing of the software applications and automated testing techniques that support testing activities. We have outlined our testing strategy for performing various types of automated testing of Java applications using AspectJ which has become the de-facto standard for Aspect Oriented Programming (AOP). Likewise JUnit, a unit testing framework is the most popular Java testing tool. In this paper, we have evaluated our proposed AOP approach for automated testing and JUnit on various parameters. First we have provided the similarity between the two approaches and then we have done a detailed comparison of the two testing techniques on factors like lines of testing code, learning curve, testing of private members etc. We established that our AOP testing approach using AspectJ has got several advantages and is thus particularly more effective than JUnit.

Keywords: aspect oriented programming, AspectJ, aspects, JU-nit, software testing

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Authors: Mukhtari Hussaini Muhammad, Abba Adamu

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The paper will examine the approaches and solutions that will be offered by Senior Secondary School II Students (Demonstration Secondary School, Azare Bauchi State Northern Nigeria – Hausa/ Fulani predominant area) toward solving exercises related to the circle theorem. The angle that an arc of a circle subtends at the center is twice that which it subtends at any point on the remaining part of the circumference. The Students will be divided in to 2 groups by given them numbers 1, 2; 1, 2; 1, 2, then all 1s formed group I and all 2s formed group II. Group I will be considered as control group in which the traditional method will be applied during instructions. Thus, the researcher will revise the concept of circle, state the theorem, prove the theorem and then solve examples. Group II, experimental group in which the concept of circle will be revised to the students and then the students will be asked to draw different circles, mark arcs, draw angle at the center, angle at the circumference then measure the angles constructed. The students will be asked to explain what they can infer/deduce from the angles measured and lastly, examples will be solved. During the next contact day, both groups will be subjected to solving exercises in the classroom related to the theorem. The angle that an arc of a circle subtends at the center is twice that which it subtends at any point on the remaining part of circumference. The solution to the exercises will be marked, the scores compared/analysed using relevant statistical tool. It is expected that group II will perform better because of the method/ technique followed during instructions is more learner-centered. By exploiting the talents of the individual learners through listening to the views and asking them how they arrived at a solution will really improve learning and understanding.

Keywords: circle theorem, control group, experimental group, traditional method

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Authors: Cristian Pauna

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With few steps away from the 2020, investments in financial markets is a common activity nowadays. In the electronic trading environment, the automated investment software has become a major part in the business intelligence system of any modern financial company. The investment decisions are assisted and/or made automatically by computers using mathematical algorithms today. The complexity of these algorithms requires computer assistance in the investment process. This paper will present several investment strategies that can be automated with algorithmic trading for Deutscher Aktienindex DAX30. It was found that, based on several price action mathematical models used for high-frequency trading some investment strategies can be optimized and improved for automated investments with good results. This paper will present the way to automate these investment decisions. Automated signals will be built using all of these strategies. Three major types of investment strategies were found in this study. The types are separated by the target length and by the exit strategy used. The exit decisions will be also automated and the paper will present the specificity for each investment type. A comparative study will be also included in this paper in order to reveal the differences between strategies. Based on these results, the profit and the capital exposure will be compared and analyzed in order to qualify the investment methodologies presented and to compare them with any other investment system. As conclusion, some major investment strategies will be revealed and compared in order to be considered for inclusion in any automated investment system.

Keywords: Algorithmic trading, automated investment systems, limit conditions, trading principles, trading strategies

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Authors: Hemant Kumar Pathak

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In 1994, Matthews (S.G. Matthews, Partial metric topology, in: Proc. 8th Summer Conference on General Topology and Applications, in: Ann. New York Acad. Sci., vol. 728, 1994, pp. 183-197) introduced the concept of a partial metric as a part of the study of denotational semantics of data flow networks. He gave a modified version of the Banach contraction principle, more suitable in this context. In fact, (complete) partial metric spaces constitute a suitable framework to model several distinguished examples of the theory of computation and also to model metric spaces via domain theory. In this paper, we introduce the concept of almost partial Hausdorff metric. We prove a fixed point theorem for multi-valued mappings on partial metric space using the concept of almost partial Hausdorff metric and prove an analogous to the well-known Nadler’s fixed point theorem. In the sequel, we derive a homotopy result as an application of our main result.

Keywords: fixed point, partial metric space, homotopy, physical sciences

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Authors: Ti-Jun Xiao, Zhe Xu

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Controllability is one of the fundamental issues in control systems. In this paper, we study the controllability of second order evolutional control systems in Hilbert spaces with memory and boundary controls, which model dynamic behaviors of some viscoelastic materials. Transferring the control problem into a moment problem and showing the Riesz property of a family of functions related to Cauchy problems for some integrodifferential equations, we obtain a general boundary controllability theorem for these second order evolutional control systems. This controllability theorem is applicable to various concrete 1D viscoelastic systems and recovers some previous related results. It is worth noting that Riesz sequences can be used for numerical computations of the control functions and the identification of new Riesz sequence is of independent interest for the basis-function theory. Moreover, using the Riesz sequences, we obtain the existence and uniqueness of (weak) solutions to these second order evolutional control systems in Hilbert spaces. Finally, we derive the exact boundary controllability of a viscoelastic beam equation, as an application of our abstract theorem.

Keywords: evolutional control system, controllability, boundary control, existence and uniqueness

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Authors: Aprilia, Wei Liang Keith Nguyen, Shu Beng Tor, Gerald Gim Lee Seet, Chee Kai Chua

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Automated remanufacturing process is of great interest in today’s marine and offshore industry. Most of the current remanufacturing processes are carried out manually and hence they are error prone, labour-intensive and costly. In this paper, a conceptual framework for automated remanufacturing is presented. This framework involves the integration of 3D non-contact digitization, adaptive surface reconstruction, additive manufacturing and machining operation. Each operation is operated and interconnected automatically as one system. The feasibility of adaptive surface reconstruction on marine and offshore engineering components is also discussed. Several engineering components were evaluated and the results showed that this proposed system is feasible. Conclusions are drawn and further research work is discussed.

Keywords: adaptive surface reconstruction, automated remanufacturing, automatic repair, reverse engineering

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Authors: D. S. Palimkar

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Random fixed point theory has received much attention in recent years, and it is needed for the study of various classes of random equations. The study of random fixed point theorems was initiated by the Prague school of probabilistic in the 1950s. The existence and uniqueness of fixed points for the self-maps of a metric space by altering distances between the points with the use of a control function is an interesting aspect in the classical fixed point theory. In a new category of fixed point problems for a single self-map with the help of a control function that alters the distance between two points in a metric space which they called an altering distance function. In this paper, we prove the results of existence of random common fixed point and its uniqueness for a pair of random mappings under weakly contractive condition for generalizing alter distance function in polish spaces using Random Common Fixed Point Theorem for Generalized Weakly Contractions.

Keywords: Polish space, random common fixed point theorem, weakly contractive mapping, altering function

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Authors: Eugene Benilov, Mikhail Benilov

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The Enskog-Vlasov (EV) equation is a widely used semi-phenomenological model of gas/liquid phase transitions. We show that it does not generally conserve energy, although there exists a restriction on its coefficients for which it does. Furthermore, if an energy-preserving version of the EV equation satisfies an H-theorem as well, it can be used to rigorously derive the so-called Maxwell construction which determines the parameters of liquid-vapor equilibria. Finally, we show that the EV model provides an accurate description of the thermodynamics of noble fluids, and there exists a version simple enough for use in applications.

Keywords: Enskog collision integral, hard spheres, kinetic equation, phase transition

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Authors: Ather Saeed, Arif Khan, Jeffrey Gosper

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Sensor devices are prone to errors and sudden node failures, which are difficult to detect in a timely manner when deployed in real-time, hazardous, large-scale harsh environments and in medical emergencies. Therefore, the loss of data can be life-threatening when the sensed phenomenon is not disseminated due to sudden node failure, battery depletion or temporary malfunctioning. We introduce a set of partial differential equations for localizing faults, similar to Green’s and Maxwell’s equations used in Electrostatics and Electromagnetism. We introduce a node organization and clustering scheme for self-stabilizing sensor networks. Green’s theorem is applied to regions where the curve is closed and continuously differentiable to ensure network connectivity. Experimental results show that the proposed GTFD (Green’s Theorem fault-detection and Self-stabilization) protocol not only detects faulty nodes but also accurately generates network stability graphs where urgent intervention is required for dynamically self-stabilizing the network.

Keywords: Green’s Theorem, self-stabilization, fault-localization, RSSI, WSN, clustering

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Authors: Hichem Ramoul

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In this work, we introduce a new type of contractive maps and we establish a new fixed point theorem for the class of almost θ-contractions (more general than the class of almost contractions) in a complete generalized metric space. The major novelty of our work is to prove a new fixed point result by weakening some hypotheses imposed on the function θ which will change completely the classical technique used in the literature review to prove fixed point theorems for almost θ-contractions in a complete generalized metric space.

Keywords: almost contraction, almost θ-contraction, fixed point, generalized metric space

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Authors: Muhammad Sufian Jusoh, Mesliza Mohamed

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In this paper, we investigate the existence of positive solutions for a Dirichlet second order boundary value problem by applying the Krasnosel'skii fixed point theorem on compression and expansion of cones.

Keywords: Krasnosel'skii fixed point theorem, positive solutions, Dirichlet boundary value problem, Dirichlet second order boundary problem

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Authors: Hsiao-Chuan Huang, King-Lung Kuo, Mei-Hsin Lo, Hsiao-Yun Chou, Yusen Lin

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The most robust and economical method for laboratory diagnosis of TB is to identify mycobacterial bacilli (AFB) under acid-fast staining despite its disadvantages of low sensitivity and labor-intensive. Though digital pathology becomes popular in medicine, an automated microscopic system for microbiology is still not available. A new AI-assisted automated microscopic system, consisting of a microscopic scanner and recognition program powered by big data and deep learning, may significantly increase the sensitivity of TB smear microscopy. Thus, the objective is to evaluate such an automatic system for the identification of AFB. A total of 5,930 smears was enrolled for this study. An intelligent microscope system (TB-Scan, Wellgen Medical, Taiwan) was used for microscopic image scanning and AFB detection. 272 AFB smears were used for transfer learning to increase the accuracy. Referee medical technicians were used as Gold Standard for result discrepancy. Results showed that, under a total of 1726 AFB smears, the automated system's accuracy, sensitivity and specificity were 95.6% (1,650/1,726), 87.7% (57/65), and 95.9% (1,593/1,661), respectively. Compared to culture, the sensitivity for human technicians was only 33.8% (38/142); however, the automated system can achieve 74.6% (106/142), which is significantly higher than human technicians, and this is the first of such an automated microscope system for TB smear testing in a controlled trial. This automated system could achieve higher TB smear sensitivity and laboratory efficiency and may complement molecular methods (eg. GeneXpert) to reduce the total cost for TB control. Furthermore, such an automated system is capable of remote access by the internet and can be deployed in the area with limited medical resources.

Keywords: TB smears, automated microscope, artificial intelligence, medical imaging

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Authors: Anas M. Fares

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Reinforced concrete shear walls are the most frequently used forms of lateral resisting structural elements. These walls may take many forms due to their functions and locations in the building. In Palestine, the most lateral resisting forces construction forms is the cantilever shear walls system. It is thus of prime importance to study the rigidity of these walls. The virtual work theorem is used to derive the total lateral deflection of cantilever shear walls due to flexural and shear deformation. The case of neglecting the shear deformation in the walls is also studied, and it is found that the wall height to length aspect ratio (H/B) plays a major role in calculating the lateral deflection and the rigidity of such walls. When the H/B is more than or equal to 3.7, the shear deformation may be neglected from the calculation of the lateral deflection. Moreover, the walls with the same material properties, same lateral load value, and same aspect ratio, shall have the same of both the lateral deflection and the rigidity. Finally, an equation to calculate the total rigidity and total deflection of such walls is derived by using the virtual work theorem for a cantilever beam.

Keywords: cantilever shear walls, flexural deformation, lateral deflection, lateral loads, reinforced concrete shear walls, rigidity, shear deformation, virtual work theorem

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