Formal Verification of Nonlinear Inequalities with Taylor Interval Approximations

We present a formal tool for verification of multivariate nonlinear inequalities. Our verification method is based on interval arithmetic with Taylor approximations. Our tool is implemented in the HOL Light proof assistant and it is capable to verify multivariate nonlinear polynomial and non-polynomial inequalities on rectangular domains. One of the main features of our work is an efficient implementation of the verification procedure which can prove non-trivial high-dimensional inequalities in several seconds. We developed the verification tool as a part of the Flyspeck project (a formal proof of the Kepler conjecture). The Flyspeck project includes about 1000 nonlinear inequalities. We successfully tested our method on more than 100 Flyspeck inequalities and estimated that the formal verification procedure is about 3000 times slower than an informal verification method implemented in C++. We also describe future work and prospective optimizations for our method.Comment: 15 page

Evaluation of the Effect of Ascorbic Acid and Sucrose Foliar Application on some Quantitative and Qualitative Characteristics of Cucurbita pepo var. Styriaca

Introduction Pumpkin (Cucurbita pepo) is a medicinal plant belonging to the Cucurbitaceae family and the order Cucurbitals. The seeds of this plant are a rich source of essential oils and proteins for the body. It is used in the production of various drugs such as Peponen, Pepostrin, Grunfig and treatment of prostate swelling, urinary tract inflammation, atherosclerosis, gastrointestinal regulation, etc. Since the components of medicinal plants are low at natural condition, and could be increased by means of different environmental conditions, nutrition or application of elicitors; thus, it is necessary to work on mentioned parameters effects on quantitative and qualitative attributes of medicinal plants. Recent years, many researches have been done based on natural components for increment of yield and secondary metabolites of medicinal plants. Ascorbic acid is one of these materials which its effect on plant growth has been validated. In the management of agricultural inputs, especially in the cultivation of medicinal plants, the application of substances that have the least harmful side effects on human health and the environment is recommended. Meanwhile, sucrose and ascorbic acid are healthy substances to improve growth and increase crop yield. Therefore, the aim of the present study was to determine the effect of these two substances on yield, yield components and phytochemical characteristics of pumpkins. Materials and Methods  Pumpkin seeds were prepared from Pakan Bazr Esfahan by purity of 99%. Then, planted in a farm of 500 m2 at Behshar. After plant growth, spray treatments were conducted at three times as before flowering, onset of flowering and fruit set stages. This experiment was conducted in factorial with sucrose factor at four levels (0, 5, 10, 15 g.l-1) and ascorbic acid factor at four levels (0, 15, 30, 45 mM), based on a randomized complete block design with three replications. The studied characteristics included number of leaves and fruits, plant yield, 1000-seed weight, total number of seeds, number of healthy seeds, percentage of healthy seeds, number of blank (deaf) seeds, percentage of blank seeds, total chlorophyll, antioxidant activity, phenol, flavonoids, protein and oil percentage. Statistical analysis of data was performed using SAS statistical software and comparison of mean was performed using the least significant difference (LSD) at the level of 5% probability. Figures were graphed with Excel software. Results and Discussion  According to this study results, the effect of foliar application of sucrose and ascorbic acid and their interaction on most of the studied traits was significant. Application of 15 g.l-1 sucrose with 15 mM ascorbic acid increased the number of fruits to 1.68 per plant, which showed an increase compared to the control treatment. The highest total number of seeds with an average of 464 seeds per fruit was obtained by applying 5 g.l-1 sucrose with 45 mM ascorbic acid, which compared to the control (247.33) recorded an increase of 87.60%. The highest total chlorophyll content was measured with an average of 2.081 (mg.g-1 fresh weight) using 5 g.l-1 sucrose with 15 mM ascorbic acid, which showed an increase of 1.81% compared to the control treatment (2.044). Also, application of 15 g.l-1 sucrose along with 15 mM ascorbic acid increased protein by 40.03%, which showed an increase of 79.26% compared to the control (22.33). Other results indicate that increasing the amount of seed oil up to 44.50% is available with the application of 15 g.l-1 sucrose with 30 mM ascorbic acid and also with the application of 10 g.l-1 sucrose with 45 mM ascorbic acid; which had an increase of 16.61% compared to the control (38.16). The results of the present study showed that the application of combined ratios of sucrose and ascorbic acid has been effective in improving the quantitative and qualitative attributes of pumpkin, including protein content and percentage of pumpkin seed oil. Conclusion  Since the treatment of sucrose 10 g.l-1 with 45 mM ascorbic acid significantly affected most of important attributes such as total antioxidant activity, total flavonoids, protein content and high oil content, therefore, this combination of treatment can be applied to increase the quality of pumpkin seeds. However, if only quantity is important, the treatment of sucrose 15 g.l-1 with 15 mM ascorbic acid, which caused the highest number of fruits per plant, the highest yield as well as the highest protein, can be recommended

Satisfiability Modulo Transcendental Functions via Incremental Linearization

In this paper we present an abstraction-refinement approach to Satisfiability Modulo the theory of transcendental functions, such as exponentiation and trigonometric functions. The transcendental functions are represented as uninterpreted in the abstract space, which is described in terms of the combined theory of linear arithmetic on the rationals with uninterpreted functions, and are incrementally axiomatized by means of upper- and lower-bounding piecewise-linear functions. Suitable numerical techniques are used to ensure that the abstractions of the transcendental functions are sound even in presence of irrationals. Our experimental evaluation on benchmarks from verification and mathematics demonstrates the potential of our approach, showing that it compares favorably with delta-satisfiability /interval propagation and methods based on theorem proving

Certification of Bounds of Non-linear Functions: the Templates Method

The aim of this work is to certify lower bounds for real-valued multivariate functions, defined by semialgebraic or transcendental expressions. The certificate must be, eventually, formally provable in a proof system such as Coq. The application range for such a tool is widespread; for instance Hales' proof of Kepler's conjecture yields thousands of inequalities. We introduce an approximation algorithm, which combines ideas of the max-plus basis method (in optimal control) and of the linear templates method developed by Manna et al. (in static analysis). This algorithm consists in bounding some of the constituents of the function by suprema of quadratic forms with a well chosen curvature. This leads to semialgebraic optimization problems, solved by sum-of-squares relaxations. Templates limit the blow up of these relaxations at the price of coarsening the approximation. We illustrate the efficiency of our framework with various examples from the literature and discuss the interfacing with Coq.Comment: 16 pages, 3 figures, 2 table

Proving Tight Bounds on Univariate Expressions with Elementary Functions in Coq

International audienceThe verification of floating-point mathematical libraries requires computing numerical bounds on approximation errors. Due to the tightness of these bounds and the peculiar structure of approximation errors, such a verification is out of the reach of generic tools such as computer algebra systems. In fact, the inherent difficulty of computing such bounds often mandates a formal proof of them. In this paper, we present a tactic for the Coq proof assistant that is designed to automatically and formally prove bounds on univariate expressions. It is based on a formalization of floating-point and interval arithmetic, associated with an on-the-fly computation of Taylor expansions. All the computations are performed inside Coq's logic, in a reflexive setting. This paper also compares our tactic with various existing tools on a large set of examples

Formalization of Transform Methods using HOL Light

Transform methods, like Laplace and Fourier, are frequently used for analyzing the dynamical behaviour of engineering and physical systems, based on their transfer function, and frequency response or the solutions of their corresponding differential equations. In this paper, we present an ongoing project, which focuses on the higher-order logic formalization of transform methods using HOL Light theorem prover. In particular, we present the motivation of the formalization, which is followed by the related work. Next, we present the task completed so far while highlighting some of the challenges faced during the formalization. Finally, we present a roadmap to achieve our objectives, the current status and the future goals for this project.Comment: 15 Pages, CICM 201

Polynomial function intervals for floating-point software verification

The focus of our work is the verification of tight functional properties of numerical programs, such as showing that a floating-point implementation of Riemann integration computes a close approximation of the exact integral. Programmers and engineers writing such programs will benefit from verification tools that support an expressive specification language and that are highly automated. Our work provides a new method for verification of numerical software, supporting a substantially more expressive language for specifications than other publicly available automated tools. The additional expressivity in the specification language is provided by two constructs. First, the specification can feature inclusions between interval arithmetic expressions. Second, the integral operator from classical analysis can be used in the specifications, where the integration bounds can be arbitrary expressions over real variables. To support our claim of expressivity, we outline the verification of four example programs, including the integration example mentioned earlier. A key component of our method is an algorithm for proving numerical theorems. This algorithm is based on automatic polynomial approximation of non-linear real and real-interval functions defined by expressions. The PolyPaver tool is our implementation of the algorithm and its source code is publicly available. In this paper we report on experiments using PolyPaver that indicate that the additional expressivity does not come at a performance cost when comparing with other publicly available state-of-the-art provers. We also include a scalability study that explores the limits of PolyPaver in proving tight functional specifications of progressively larger randomly generated programs

Psychometric Properties of the Farsi Version of Posttraumatic Growth Inventory for Children-Revised in Iranian Children with Cancer

Objective: Coping with childhood cancer, as a stressful incident, can lead to a growth in various aspects of the child's life. Therefore, this study aims to validate Posttraumatic Growth Inventory for Children-Revised (PTGI-C-R) in children with cancer. Methods: This methodological research was carried out in referral children hospitals in Tehran. PTGI-C-R was translated and back-translated. Content and face validity were assessed. Confirmatory factor analysis (CFA) was performed on 200 children with inclusion criteria, using LISREL V8.5. Due to the rejection of the model, an exploratory factor analysis (EFA) was done, using SPSS V21. The correlation of posttraumatic growth (PTG) with the variables, i.e., age and gender, was investigated. Results: Some writing changes were made in phrases in the sections concerning face and content validity. CFA rejected the five-factor model due to the undesirable fit indices. Therefore, an EFA was used and the three-factor model was not approved, either despite the statistical appropriateness or due to the lack of similarity between the items loaded on factors. The results also indicated a significant relationship between PTG and age (r = 0.13, P = 0.05). There is no significant relationship between PTG and gender (z = -1.35, P = 0.83). Conclusions: PTGI-C-R does not have desirable psychometric properties in Iranian children with cancer and may not be able to reflect all the aspects of PTG experienced by them. Therefore, it cannot be used as an appropriate scale, and it is necessary to develop and validate a specific tool through a qualitative study. © 2021 Wolters Kluwer Medknow Publications. All rights reserved