Static Analysis Problems Research Articles

Program Analysis via Multiple Context Free Language Reachability

Context-free language (CFL) reachability is a standard approach in static analyses, where the analysis question (e.g., is there a dataflow from x to y ?) is phrased as a language reachability problem on a graph G wrt a CFL L . However, CFLs lack the expressiveness needed for high analysis precision. On the other hand, common formalisms for context-sensitive languages are too expressive, in the sense that the corresponding reachability problem becomes undecidable. Are there useful context-sensitive language-reachability models for static analysis? In this paper, we introduce Multiple Context-Free Language (MCFL) reachability as an expressive yet tractable model for static program analysis. MCFLs form an infinite hierarchy of mildly context sensitive languages parameterized by a dimension d and a rank r . Larger d and r yield progressively more expressive MCFLs, offering tunable analysis precision. We showcase the utility of MCFL reachability by developing a family of MCFLs that approximate interleaved Dyck reachability, a common but undecidable static analysis problem. Given the increased expressiveness of MCFLs, one natural question pertains to their algorithmic complexity, i.e., how fast can MCFL reachability be computed? We show that the problem takes O ( n 2 d +1 ) time on a graph of n nodes when r =1, and O ( n d ( r +1) ) time when r >1. Moreover, we show that when r =1, even the simpler membership problem has a lower bound of n 2 d based on the Strong Exponential Time Hypothesis, while reachability for d =1 has a lower bound of n 3 based on the combinatorial Boolean Matrix Multiplication Hypothesis. Thus, for r =1, our algorithm is optimal within a factor n for all levels of the hierarchy based on the dimension d (and fully optimal for d =1). We implement our MCFL reachability algorithm and evaluate it by underapproximating interleaved Dyck reachability for a standard taint analysis for Android. When combined with existing overapproximate methods, MCFL reachability discovers all tainted information on 8 out of 11 benchmarks, while it has remarkable coverage (confirming 94.3% of the reachable pairs reported by the overapproximation) on the remaining 3. To our knowledge, this is the first report of high and provable coverage for this challenging benchmark set.

Penalty 4-Node Quadrilateral Element Formulation for Axisymmetric Couple Stress Problems.

To address the issue of size effects in axisymmetric deformation of small-scale solids, this work proposes a 4-node 12-DOF element for axisymmetric problems based on the consistent couple stress theory (CCST), following the framework of the unsymmetric finite element method. With the use of the penalty function method, an independently assumed rotational field is introduced into the virtual work principle to approximate the physical rotation, ensuring the satisfaction of the C1 continuity requirement of the CCST in a weak form. As a benefit, the enriched C0 isoparametric-based interpolation is employed to construct the test functions for displacement and rotation. Furthermore, the force-stress field that satisfies the equilibrium equations related to axisymmetric deformation is employed as the element's force-stress trial function. In order to circumvent locking issues, reduced integration is employed in the penalty stiffness integration process. The numerical results demonstrate that the new element exhibits high computational accuracy and convergence rate in both static and modal analysis problems, effectively capturing size-dependent phenomena.

Languages Generated by Conjunctive Query Fragments of FC[REG]

FC is a finite model variant on the theory of concatenation, FC[REG] extends FC with regular constraints. This paper considers the languages generated by their conjunctive query fragments, FC-CQ and FC[REG]-CQ. We compare the expressive power of FC[REG]-CQ to that of various related language generators, such as regular expressions, patterns, and typed patterns. We then consider decision problems for FC-CQ and FC[REG]-CQ, and show that certain static analysis problems (such as equivalence and regularity) are undecidable. While this paper defines FC-CQ based on the logic FC, it can equally be understood as synchronized intersections of pattern languages, or as systems of restricted word equations.

Continuum and Discrete Analytical Methods for Vibration and Buckling Eigenvalues Shape Sensitivities

AbstractGradient-based optimization techniques need precise and efficient sensitivities. Numerical sensitivity methods such as finite differences are easy to implement but imprecise and computationally inefficient. In contrast, analytical sensitivity methods, such as the discrete and continuum ones are highly accurate and efficient. Continuum Sensitivity Analysis (CSA) is an analytical method used to calculate derivatives of shape or value parameters. While CSA has been successfully applied in static analysis and dynamic problems in the time domain, this work presents an extension of the approach to eigenvalue sensitivities for the first time. However, CSA revealed limitations, prompting the exploration of an alternative approach based on discrete analytical differentiation. This method is employed for the first time in shape sensitivities. The derivatives of the stiffness and mass matrices required by the method are calculated analytically, resulting in high accuracy and computational efficiency. In addition, an element agnostic approach has been developed leveraging primary analysis matrices to calculate their derivative. This characteristic, along with the nonintrusivity, makes the method employable with standard commercial software. Both approaches have been applied and validated in a wide range of scenarios, involving vibration and buckling problems.

Vibration of solid and thin-walled slender structures made of soft materials by high-order beam finite elements

In this work, high-order beam (1D) finite element models for the modal analysis of structures made of compressible and nearly-incompressible hyperelastic soft materials are presented in the well-established framework of Carrera Unified Formulation (CUF). In this investigation, the modal behavior of soft structures subjected to progressively increasing loads can be correctly predicted by higher-order structural theories, and the influence of pre-stress conditions applied on the modal response of structures is investigated. The mathematical formulation of hyperelastic isotropic materials is presented in terms of invariants of the right Cauchy–Green strain tensor, obtaining the most general expression of the Piola–Kirchoff 2 stress tensor and tangent elasticity tensor, both independent of the model adopted for the material strain energy function. Governing equations in matrix forms for the static nonlinear analysis and subsequent vibration problem around non-trivial equilibrium states are derived through the Principle of Virtual Displacements (PVD) under a total Lagrangian formulation, defining the fundamental nuclei of stiffness matrices and internal and external forces vector, all independent of expansion theories and kinematic models adopted in the mathematical modeling of finite elements. Actual numerical results are obtained by an iterative Newton–Raphson linearized scheme coupled with line-search algorithms, and they are compared with results obtained by the commercial code ABAQUS. Our proposed models are tested with large strain problems involving hyperelastic slender and thin-walled structures, for which mode aberration such as crossing or bearing are observed.

Numerical investigation on the premature and extended contact behaviour of engineering thermoplastic gears and its effect in gear kinematics

PurposeThis research work aims to determine the maximum load a thermoplastic gear can withstand without the occurrence of extended contact. The extended contact of polymer gears is usually overlooked in basic design calculations, although it considerably affects the gear's load-carrying ability. Although various researchers highlighted the phenomenon, an extensive investigation of the extended contact behaviour is limited. Hence the work aims to investigate the premature and extended contact behaviour of thermoplastic gears and its effect in the gear kinematics, bending stiffness, stresses induced and the roll angle subtended by the gear pair.Design/methodology/approachThe work uses finite element method to perform quasi-static two-dimensional analysis of the meshing gear teeth. The FE model was developed in AutoCAD and analysed using ANSYS 19.1 simulation package. A three-dimensional gear model with all the teeth is computationally intensive for solving a static analysis problem. Hence, planar analysis with a reduced number of teeth is considered to reduce the computational time and difficulty.FindingsThe roll angle subtended at the centre by the path of approach is higher than the path of recess because of the increased load sharing. The contact stress profile followed a unique R-F-R-F pattern in the premature and extended contact regions due to the driven tip-driver flank surface contact. A non-dimensional parameter was formulated correlating the young's modulus, the load applied and deflection induced that can be utilised to predict the occurrence of premature and extended contact in thermoplastic gears.Originality/valueThe gear rating standards for polymer gears are formulated from the conventional metal gears which does not include the effect of gear tooth deflection. The work attempts to explain the gear tooth deflection for various standard thermoplastics and its effect in kinematics. Likewise, a new dimensionless number was introduced to predict the extended contact that will help in appropriate selection of load reducing the possibility of wear.

Multiple context-free path querying by matrix multiplication

Many graph analysis problems can be formulated as formal language-constrained path querying problems where the formal languages are used as constraints for navigational path queries. Recently, the context-free language (CFL) reachability formulation has become very popular and can be used in many areas, for example, querying graph databases, Resource Description Framework (RDF) analysis. However, the generative capacity of context-free grammars (CFGs) is too weak to generate some complex queries, for example, from natural languages, and the various extensions of CFGs have been proposed. Multiple context-free grammar (MCFG) is one of such extensions of CFGs. Despite the fact that, to the best of our knowledge, there is no algorithm for MCFL-reachability, this problem is known to be decidable. This paper is devoted to developing the first such algorithm for the MCFL-reachability problem. The essence of the proposed algorithm is to use a set of Boolean matrices and operations on them to find paths in a graph that satisfy the given constraints. The main operation here is Boolean matrix multiplication. As a result, the algorithm returns a set of matrices containing all information needed to solve the MCFL-reachability problem. The presented algorithm is implemented in Python using GraphBLAS API. An analysis of real RDF data and synthetic graphs for some MCFLs is performed. The study showed that using a sparse format for matrix storage and parallel computing for graphs with tens of thousands of edges the analysis time can be 10–20 minutes. The result of the analysis provides tens of millions of reachable vertex pairs. The proposed algorithm can be applied in problems of static code analysis, bioinformatics, network analysis, as well as in graph databases when a path query cannot be expressed using context-free grammars. The provided algorithm is linear algebra-based, hence, it allows one to use high-performance libraries and utilize modern parallel hardware.

The Fine-Grained Complexity of CFL Reachability

Many problems in static program analysis can be modeled as the context-free language (CFL) reachability problem on directed labeled graphs. The CFL reachability problem can be generally solved in time O ( n 3 ), where n is the number of vertices in the graph, with some specific cases that can be solved faster. In this work, we ask the following question: given a specific CFL, what is the exact exponent in the monomial of the running time? In other words, for which cases do we have linear, quadratic or cubic algorithms, and are there problems with intermediate runtimes? This question is inspired by recent efforts to classify classic problems in terms of their exact polynomial complexity, known as fine-grained complexity. Although recent efforts have shown some conditional lower bounds (mostly for the class of combinatorial algorithms), a general picture of the fine-grained complexity landscape for CFL reachability is missing. Our main contribution is lower bound results that pinpoint the exact running time of several classes of CFLs or specific CFLs under widely believed lower bound conjectures (e.g., Boolean Matrix Multiplication, k -Clique, APSP, 3SUM). We particularly focus on the family of Dyck- k languages (which are strings with well-matched parentheses), a fundamental class of CFL reachability problems. Remarkably, we are able to show a Ω( n 2.5 ) lower bound for Dyck-2 reachability, which to the best of our knowledge is the first super-quadratic lower bound that applies to all algorithms, and shows that CFL reachability is strictly harder that Boolean Matrix Multiplication. We also present new lower bounds for the case of sparse input graphs where the number of edges m is the input parameter, a common setting in the database literature. For this setting, we show a cubic lower bound for Andersen’s Pointer Analysis which significantly strengthens prior known results.

Collapse capacity of masonry domes under horizontal loads: A static limit analysis approach

A static limit analysis approach is proposed for assessing the collapse capacity of axisymmetric masonry domes subject to horizontal forces. The problem formulation is based on the sound theoretical framework provided by the classical statics of shells. After introducing the shell stress resultants on the dome mid-surface, integral equilibrium equations are enforced for its typical part. Heyman’s assumptions of infinite compressive and vanishing tensile strengths are made, with cohesive-frictional behavior governing the shear strength, to characterize the admissible stress states in the dome. An original computational strategy is developed to address the resulting static limit analysis problem, involving the introduction of a mesh on the dome mid-surface, the interpolation of the physical components of the shell stress tensors on the element boundaries, and the imposition of equilibrium and admissibility conditions respectively for the elements and at the nodes of the mesh. The descending discrete second-order cone programming problem is solved by standard and effective optimization tools, automatically providing collapse multiplier of horizontal forces, incipient collapse mechanism and expected crack pattern. Convergence analysis, validation with experimental results available in the literature, and parametric analyses with respect to geometric parameters and material shear response, are presented for spherical and ellipsoidal masonry domes, proving the reliability of the proposed approach for estimating the pseudo-static seismic resistance of masonry domes. Finally, as an application to a real structure, the pseudo-static seismic assessment of the central dome of the Taj-Mahal is presented.

A finite difference method for the static limit analysis of masonry domes under seismic loads

The static limit analysis of axially symmetric masonry domes subject to pseudo-static seismic forces is addressed. The stress state in the dome is represented by the shell stress resultants (normal-force tensor, bending-moment tensor, and shear-force vector) on the dome mid-surface. The classical differential equilibrium equations of shells are resorted to for imposing the equilibrium of the dome. Heyman’s assumptions of infinite compressive and vanishing tensile strength, alongside with cohesive-frictional shear response, are adopted for imposing the admissibility of the stress state. A finite difference method is proposed for the numerical discretization of the problem, based on the use of two staggered rectangular grids in the parameter space generating the dome mid-surface. The resulting discrete static limit analysis problem results to be a second-order cone programming problem, to be effectively solved by available convex optimization softwares. In addition to a convergence analysis, numerical simulations are presented, dealing with the parametric analysis of the collapse capacity under seismic forces of spherical and ogival domes with parameterized geometry. In particular, the influence that the shear response of masonry material and the distribution of horizontal forces along the height of the dome have on the collapse capacity is explored. The obtained results, that are new in the literature, show the computational merit of the proposed method, and quantitatively shed light on the seismic resistance of masonry domes.

Calculational design of a regular model checker by abstract interpretation

Security monitors have been used to check for safety program properties at runtime, that is for any given execution trace. Such security monitors check a safety temporal property specified by a finite automaton or, equivalently, a regular expression. Checking this safety temporal specification for all possible execution traces, that is the program semantics, is a static analysis problem, more precisely a model checking problem, since model checking specializes in temporal properties. We show that the model checker can be formally designed by calculus, by abstract interpretation of a formal trace semantics of the programming language. The result is a structural sound and complete model checker, which proceeds by induction on the program syntax (as opposed to the more classical approach using computation steps formalized by a transition system). By Rice theorem, further hypotheses or abstractions are needed to get a realistic model checking algorithm.

The fine-grained and parallel complexity of andersen’s pointer analysis

Pointer analysis is one of the fundamental problems in static program analysis. Given a set of pointers, the task is to produce a useful over-approximation of the memory locations that each pointer may point-to at runtime. The most common formulation is Andersen’s Pointer Analysis (APA), defined as an inclusion-based set of m pointer constraints over a set of n pointers. Scalability is extremely important, as points-to information is a prerequisite to many other components in the static-analysis pipeline. Existing algorithms solve APA in O ( n 2 · m ) time, while it has been conjectured that the problem has no truly sub-cubic algorithm, with a proof so far having remained elusive. It is also well-known that APA can be solved in O ( n 2 ) time under certain sparsity conditions that hold naturally in some settings. Besides these simple bounds, the complexity of the problem has remained poorly understood. In this work we draw a rich fine-grained and parallel complexity landscape of APA, and present upper and lower bounds. First, we establish an O ( n 3 ) upper-bound for general APA, improving over O ( n 2 · m ) as n = O ( m ). Second, we show that even on-demand APA (“may a specific pointer a point to a specific location b ?”) has an Ω( n 3 ) (combinatorial) lower bound under standard complexity-theoretic hypotheses. This formally establishes the long-conjectured “cubic bottleneck” of APA, and shows that our O ( n 3 )-time algorithm is optimal. Third, we show that under mild restrictions, APA is solvable in Õ( n ω ) time, where ω<2.373 is the matrix-multiplication exponent. It is believed that ω=2+ o (1), in which case this bound becomes quadratic. Fourth, we show that even under such restrictions, even the on-demand problem has an Ω( n 2 ) lower bound under standard complexity-theoretic hypotheses, and hence our algorithm is optimal when ω=2+ o (1). Fifth, we study the parallelizability of APA and establish lower and upper bounds: (i) in general, the problem is P-complete and hence unlikely parallelizable, whereas (ii) under mild restrictions, the problem is parallelizable. Our theoretical treatment formalizes several insights that can lead to practical improvements in the future.

Выполнимость мю-исчисления с арифметическими ограничениями

The propositional modal μ-calculus is a well-known specification language for labeled transition systems. In this work, we study an extension of this logic with converse modalities and Presburger arithmetic constraints, interpreted over tree models. We describe a satisfiability algorithm based on breadth-first construction of Fischer-Lardner models. An implementation together several experiments are also reported. Furthermore, we also describe an application of the algorithm to solve static analysis problems over semi-structured data.

Mu-Calculus Satisfiability with Arithmetic Constraints

The propositional modal μ-calculus is a well-known specification language for labeled transition systems. In this work, we study an extension of this logic with converse modalities and Presburger arithmetic constraints, interpreted over tree models. We describe a satisfiability algorithm based on breadth-first construction of Fischer-Lardner models. An implementation together several experiments are also reported. Furthermore, we also describe an application of the algorithm to solve static analysis problems over semi-structured data.

Cognitive Regionology: The Experience of Modeling Regional Socio-Economic Processes

Introduction. In the current situation, the society is subject to extremely dynamic changes and strategic developments are becoming obsolete more rapidly. It is cognitive modeling of complex semi-structured systems that is one of the modern methods that presents technological solutions in response to the challenge of obsolescence of strategies. The practice of strategizing Russia’s regions makes it possible to single out the regional dimension as a self-sufficient subject of cognitive modeling. The objective of the article is to summarize many years of experience of applying the method of cognitive modeling of regional socio-economic processes and the application of the obtained models in educational, scientific and administrative activities of the region on the basis of the study conducted. Materials and Methods. The database of Analytic software application was used as the information resource. Cognitive modeling was the main method employed and was considered in more detail; statistical methods, comparative analysis, and an expert survey were also used. Results. Specific examples of research conducted in the Nizhny Novgorod Region, the Samara Region, and the Republic of Mordovia have shown the advantages of using cognitive modeling technology in the educational, scientific, and administrative activities in a region. The factor-digital cognitive model of a region becomes the basis for organizing trainings on the strategy of sustainable development at the regional and interregional levels. Analytic software application supports the research process and is integrated into the electronic information and educational environment of regional universities. Discussion and Conclusion. The cognitive modeling method makes it possible to solve the problems of static and dynamic analysis of a region as a complex system in various subject areas. The factor-digital models of regions obtained using Analytic software application are of a universal nature and are relatively easily modified under the framework conditions of any other region of Russia on remote access platforms.

Systemizing Interprocedural Static Analysis of Large-scale Systems Code with Graspan

There is more than a decade-long history of using static analysis to find bugs in systems such as Linux. Most of the existing static analyses developed for these systems are simple checkers that find bugs based on pattern matching. Despite the presence of many sophisticated interprocedural analyses, few of them have been employed to improve checkers for systems code due to their complex implementations and poor scalability. In this article, we revisit the scalability problem of interprocedural static analysis from a “Big Data” perspective. That is, we turn sophisticated code analysis into Big Data analytics and leverage novel data processing techniques to solve this traditional programming language problem. We propose Graspan , a disk-based parallel graph system that uses an edge-pair centric computation model to compute dynamic transitive closures on very large program graphs. We develop two backends for Graspan, namely, Graspan-C running on CPUs and Graspan-G on GPUs, and present their designs in the article. Graspan-C can analyze large-scale systems code on any commodity PC, while, if GPUs are available, Graspan-G can be readily used to achieve orders of magnitude speedup by harnessing a GPU’s massive parallelism. We have implemented fully context-sensitive pointer/alias and dataflow analyses on Graspan. An evaluation of these analyses on large codebases written in multiple languages such as Linux and Apache Hadoop demonstrates that their Graspan implementations are language-independent, scale to millions of lines of code, and are much simpler than their original implementations. Moreover, we show that these analyses can be used to uncover many real-world bugs in large-scale systems code.

Generalized model of software code`s static analysis based on machine learning for vulnerabilitys search

Over the past years, the use of unsafe software, the search for vulnerabilities in which relies on static and dynamic analysis, continues to be the main threat to the infosphere. The manual form of conducting static analysis is extremely time-consuming and requires the involvement of highly qualified, and therefore deficient specialists. An alternative is the automation of the process based on artificial intelligence. This work is aimed at finding solutions for the use of machine learning methods at all stages of the static analysis of program code, for which the formal needs of the stages and the possibilities of the methods are studied and correlated. The main result of the study is a generalized domain model, and private — 14 solutions to the “key” problems of static analysis of program code using machine learning methods.

Extended layerwise method for laminated piezoelectric and composite plates with delaminations, cracks or debonding of a piezoelectric patch

Extended layerwise method (XLWM) is applied to the static analysis problems of laminated piezoelectric plates and laminated composite plates with a piezoelectric patch in this paper. A piezoelectric XLWM is first developed for the laminated piezoelectric plates with multiple delaminations and transverse cracks. The discontinuities of displacements and electric potential resulted from delamination are described in the assumptions along the thickness direction. In the in-plane displacement and electric potential fields, the transverse cracks are modeled by using the extended finite element method (XFEM). The coupled electromechanical variational principle is employed to derive the Euler equations, boundary conditions and finite element formulations. And then, this piezoelectric XLWM is employed to establish a coupling analysis model for the damaged laminated composite plates with a piezoelectric patch. The damaged composite plate is discretized by the XLWM as well, so the surface displacement freedoms are included into the governing equations of composite plate and piezoelectric patch. The displacement continuity and internal force equilibrium conditions at the interface of the laminated composite plate and piezoelectric patch are considered in the final governing equations.

Rigid block modelling of historic masonry structures using mathematical programming: a unified formulation for non-linear time history, static pushover and limit equilibrium analysis

A unified formulation is presented for non-linear time history, static pushover and limit analysis of historic masonry structures modelled as 2D assemblages of rigid blocks interacting at no-tension, frictional contact interfaces. The dynamic, incremental static and limit analysis problems are formulated as mathematical programming problems which are equivalent to the equations system governing equilibrium, kinematics and contact failure. Available algorithms from the field of mathematical programming, contact dynamics and limit analysis are used to tackle the contact problems between rigid blocks in a unified framework. To evaluate the accuracy and computational efficiency of the implemented formulation, applications to numerical case studies from the literature are presented. The case studies comprise rigid blocks under earthquake excitation, varying lateral static loads and sliding motion. A set of two leaves wall panels and an arch-pillars system are also analysed to compare failure mechanisms, displacement capacity and magnitudes of lateral loads promoting the collapse.

ABOUT WAVELET-BASED COMPUTATIONAL BEAM ANALYSIS WITH THE USE OF DAUBECHIES SCALING FUNCTIONS

The first part of the distinctive paper contains brief review of wavelet-based numerical and semianalytical analysis, particularly with the use of Daubechies scaling functions. The second part of the paper is devoted to numerical solution of the problem of static analysis of beam on elastic foundation within Winkler model. Finite element method (FEM) and wavelet analysis (Daubechies scaling functions) are used. Variational formulation and approximation of the problem are under consideration. Numerical sample is presented as well. The third part of the paper is dedicated to wavelet based discrete-continual finite element method of beam analysis with allowance for impulse load. Daubechies scaling functions are used as well.