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

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

Investigating the prevalence of sleep disorder and the impact of sweet almond on the quality of sleep in students of Tehran, Iran

Background: Insomnia is an important problem in medical sciences students and has implications for their educational progress. The current study aimed to estimate the prevalence of sleep disorders and investigating the impact of sweet almond on quality of sleep in students of the Tehran University of Medical Sciences (TUMS), Tehran, Iran who live in dormitories. Methods: This is a before-after study conducted in 2017. At first, using the ISI questionnaire prevalence of sleep disorders was determined. Sweet almond was the study intervention. Each day, 10 almonds were given to 446 students for 14 d. At the end of the second week, again ISI questionnaire was filled. SPSS was used to analyze data. The McNemar, Wilcoxson Signed Ranks, and Repeated Measures tests were used. Results: Out of 442 participants, 217 (49.1) were female. Before intervention, 343 (77.6) had insomnia and 99 (22.4) had normal sleep. After intervention, 306 (69.2) had insomnia and 136 (30.8) had normal sleep. Having sweet almond for two weeks is associated with reducing insomnia (P<0.05). Investigating the almond impact in different categories also showed that it has a reducing impact on severe, mild, weak and normal sleep categories (P<0.05). Conclusion: Sweet almond has impacts on quality of sleep of those students of the TUMS that are living in dormitories. Intervention programs to improve quality of sleep are necessary and with regard to the high prevalence of insomnia, students must be protected, guided and consulted. © 2019, Iranian Journal of Public Health. All rights reserved

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

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

Real Algebraic Strategies for MetiTarski Proofs

Abstract. MetiTarski [1] is an automatic theorem prover that can prove inequalities involving sin, cos, exp, ln, etc. During its proof search, it generates a series of subproblems in nonlinear polynomial real arithmetic which are reduced to true or false using a decision procedure for the theory of real closed fields (RCF). These calls are often a bottleneck: RCF is fundamentally infeasible. However, by studying these subproblems, we can design specialised variants of RCF decision procedures that run faster and improve MetiTarski’s performance.

Computation in Real Closed Infinitesimal and Transcendental Extensions of the Rationals.

Abstract. Recent applications of decision procedures for nonlinear real arithmetic (the theory of real closed fields, or RCF) have presented a need for reasoning not only with polynomials but also with transcendental constants and infinitesimals. In full generality, the algebraic setting for this reasoning consists of real closed transcendental and infinitesimal extensions of the rational numbers. We present a library for computing over these extensions. This library contains many contributions, including a novel combination of Thom’s Lemma and interval arithmetic for representing roots, and provides all core machinery required for building RCF decision procedures. We describe the abstract algebraic setting for computing with such field extensions, present our concrete algorithms and optimizations, and illustrate the library on a collection of examples. 1 Overview and Related Work Decision methods for nonlinear real arithmetic are essential to the formal verification of cyber-physical systems and formalized mathematics. Classically, thes