Search Results (8)

The p-median problem is one of the earliest location-allocation models used in spatial analysis and GIS. It involves locating a set of central facilities (the location decision) and allocating customers to these facilities (the allocation decision) so as to minimize the total transportation cost. It is important not only because of its wide use in spatial analysis but also because of its role as a unifying location model in GIS. A classical way of solving the p-median problem (dating back to the 1970s) is to formulate it as an Integer Linear Program (ILP), and then solve it using off-the-shelf solvers. Two fundamental properties of the p-median problem (and its variants) are the integral assignment property and the closest assignment property. They are the basis for the efficient formulation of the problem, and are important for studying the p-median problems and other location-allocation models. In this paper, we demonstrate that these fundamental properties of the p-median can be proven mechanically using integer linear programming and theorem provers under the program-as-proof paradigm. While these theorems have been proven informally, mechanized proofs using computers are fail-safe and contain no ambiguity. The presented proof method based on ILP and the associated definitions of problem data are general, and we expect that they can be generalized and extended to prove the theoretical properties of other spatial-optimization models, old or new. 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

An important development in geometric algebra in recent years is the new system known as point geometry, which treats points as direct objects of operations and considerably simplifies the process of geometric reasoning. In this paper, we provide a complete formal description of the point geometry theory architecture and give a rigorous and reliable formal verification of the point geometry theory based on the theorem prover Coq. Simultaneously, a series of tactics are also designed to assist in the proof of geometric propositions. Based on the theoretical architecture and proof tactics, a universal and scalable interactive point geometry machine proof system, PointGeo, is built. In this system, any arbitrary point-geometry-solvable geometric statement may be proven, along with readable information about the solution’s procedure. Additionally, users may augment the rule base by adding trustworthy rules as needed for certain issues. The implementation of the system expands the library of Coq resources on geometric algebra, which will become a significant research foundation for the fields of geometric algebra, computer science, mathematics education, and other related fields. 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

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

The formalization of mathematics based on theorem prover becomes increasingly important in mathematics and computer science, and, particularly, formalizing fundamental mathematical theories becomes especially essential. In this paper, we describe the formalization in Coq of eight very representative completeness theorems of real numbers. These theorems include the Dedekind fundamental theorem, Supremum theorem, Monotone convergence theorem, Nested interval theorem, Finite cover theorem, Accumulation point theorem, Sequential compactness theorem, and Cauchy completeness theorem. We formalize the real number theory strictly following Landau’s Foundations of Analysis where the Dedekind fundamental theorem can be proved. We extend this system and complete the related notions and properties for finiteness and sequence. We prove these theorems in turn from Dedekind fundamental theorem, and finally prove the Dedekind fundamental theorem by the Cauchy completeness theorem. The full details of formal proof are checked by the proof assistant Coq, which embodies the characteristics of reliability and interactivity. This work can lay the foundation for many applications, especially in calculus and topology. Full article

Cost and utility modeling of economics agents based on the differential theory is fundamental to the analysis of the microeconomics models. In particular, the first and second-order derivative tests are used to specify the desired properties of the cost and utility models. Traditionally, paper-and-pencil proof methods and computer-based tools are used to investigate the mathematical properties of these models. However, these techniques do not provide an accurate analysis due to their inability to exhaustively specify and verify the mathematical properties of the cost and utility models. Additionally, these techniques cannot accurately model and analyze pure continuous behaviors of the economic agents due to the utilization of computer arithmetic. On the other hand, an accurate analysis is direly needed in many safety and cost-critical microeconomics applications, such as agriculture and smart grids. To overcome the issues pertaining to the above-mentioned techniques, in this paper, we propose a theorem proving based methodology to formally analyze and specify the mathematical properties of functions used in microeconomics modeling. The proposed methodology is primarily based on a formalization of the derivative tests and root analysis of the polynomial functions, within the sound core of the HOL-Light theorem prover. We also provide a formalization of the first-order condition, which is used to analyze the maximum of the profit function in a higher-order-logic theorem prover. We then present the formal analysis of the utility, cost and first-order condition based on the polynomial functions. To illustrate the usefulness of proposed formalization, the proposed formalization is used to formally analyze and verify the quadratic cost and utility functions, which have been used in an optimal power flow problem and demand response (DR) program, respectively. 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