Computing the smallest fixed point of order-preserving nonexpansive mappings arising in positive stochastic games and static analysis of programs

The problem of computing the smallest fixed point of an order-preserving map arises in the study of zero-sum positive stochastic games. It also arises in static analysis of programs by abstract interpretation. In this context, the discount rate may be negative. We characterize the minimality of a fixed point in terms of the nonlinear spectral radius of a certain semidifferential. We apply this characterization to design a policy iteration algorithm, which applies to the case of finite state and action spaces. The algorithm returns a locally minimal fixed point, which turns out to be globally minimal when the discount rate is nonnegative.Comment: 26 pages, 3 figures. We add new results, improvements and two examples of positive stochastic games. Note that an initial version of the paper has appeared in the proceedings of the Eighteenth International Symposium on Mathematical Theory of Networks and Systems (MTNS2008), Blacksburg, Virginia, July 200

Get rid of inline assembly through verification-oriented lifting

Formal methods for software development have made great strides in the last two decades, to the point that their application in safety-critical embedded software is an undeniable success. Their extension to non-critical software is one of the notable forthcoming challenges. For example, C programmers regularly use inline assembly for low-level optimizations and system primitives. This usually results in driving state-of-the-art formal analyzers developed for C ineffective. We thus propose TInA, an automated, generic, trustable and verification-oriented lifting technique turning inline assembly into semantically equivalent C code, in order to take advantage of existing C analyzers. Extensive experiments on real-world C code with inline assembly (including GMP and ffmpeg) show the feasibility and benefits of TInA

Affine Disjunctive Invariant Generation with Farkas' Lemma

Invariant generation is the classical problem that aims at automated generation of assertions that over-approximates the set of reachable program states in a program. We consider the problem of generating affine invariants over affine while loops (i.e., loops with affine loop guards, conditional branches and assignment statements), and explore the automated generation of disjunctive affine invariants. Disjunctive invariants are an important class of invariants that capture disjunctive features in programs such as multiple phases, transitions between different modes, etc., and are typically more precise than conjunctive invariants over programs with these features. To generate tight affine invariants, existing constraint-solving approaches have investigated the application of Farkas' Lemma to conjunctive affine invariant generation, but none of them considers disjunctive affine invariants

Automatic modular abstractions for template numerical constraints

We propose a method for automatically generating abstract transformers for static analysis by abstract interpretation. The method focuses on linear constraints on programs operating on rational, real or floating-point variables and containing linear assignments and tests. In addition to loop-free code, the same method also applies for obtaining least fixed points as functions of the precondition, which permits the analysis of loops and recursive functions. Our algorithms are based on new quantifier elimination and symbolic manipulation techniques. Given the specification of an abstract domain, and a program block, our method automatically outputs an implementation of the corresponding abstract transformer. It is thus a form of program transformation. The motivation of our work is data-flow synchronous programming languages, used for building control-command embedded systems, but it also applies to imperative and functional programming

On the Practice and Application of Context-Free Language Reachability

The Context-Free Language Reachability (CFL-R) formalism relates to some of the most important computational problems facing researchers and industry practitioners. CFL-R is a generalisation of graph reachability and language recognition, such that pairs in a labelled graph are reachable if and only if there is a path between them whose labels, joined together in the order they were encountered, spell a word in a given context-free language. The formalism finds particular use as a vehicle for phrasing and reasoning about program analysis, since complex relationships within the data, logic or structure of computer programs are easily expressed and discovered in CFL-R. Unfortunately, The potential of CFL-R can not be met by state of the art solvers. Current algorithms have scalability and expressibility issues that prevent them from being used on large graph instances or complex grammars. This work outlines our efforts in understanding the practical concerns surrounding CFL-R, and applying this knowledge to improve the performance of CFL-R applications. We examine the major difficulties with solving CFL-R-based analyses at-scale, via a case-study of points-to analysis as a CFL-R problem. Points-to analysis is fundamentally important to many modern research and industry efforts, and is relevant to optimisation, bug-checking and security technologies. Our understanding of the scalability challenge motivates work in developing practical CFL-R techniques. We present improved evaluation algorithms and declarative optimisation techniques for CFL-R, capitalising on the simplicity of CFL-R to creating fully automatic methodologies. The culmination of our work is a general-purpose and high-performance tool called Cauliflower, a solver-generator for CFL-R problems. We describe Cauliflower and evaluate its performance experimentally, showing significant improvement over alternative general techniques

Détermination de propriétés de flot de données pour améliorer les estimations de temps d'exécution pire-cas

La recherche d'une borne supérieure au temps d'exécution d'un programme est une partie essentielle du processus de vérification de systèmes temps-réel critiques. Les programmes de tels systèmes ont généralement des temps d'exécution variables et il est difficile, voire impossible, de prédire l'ensemble de ces temps possibles. Au lieu de cela, il est préférable de rechercher une approximation du temps d'exécution pire-cas ou Worst-Case Execution Time (WCET). Une propriété cruciale de cette approximation est qu'elle doit être sûre, c'est-à-dire qu'elle doit être garantie de majorer le WCET. Parce que nous cherchons à prouver que le système en question se termine en un temps raisonnable, une surapproximation est le seul type d'approximation acceptable. La garantie de cette propriété de sûreté ne saurait raisonnablement se faire sans analyse statique, un résultat se basant sur une série de tests ne pouvant être sûr sans un traitement exhaustif des cas d'exécution. De plus, en l'absence de certification du processus de compilation (et de transfert des propriétés vers le binaire), l'extraction de propriétés doit se faire directement sur le code binaire pour garantir leur fiabilité. Toutefois, cette approximation a un coût : un pessimisme - écart entre le WCET estimé et le WCET réel - important entraîne des surcoûts superflus de matériel pour que le système respecte les contraintes temporelles qui lui sont imposées. Il s'agit donc ensuite, tout en maintenant la garantie de sécurité de l'estimation du WCET, d'améliorer sa précision en réduisant cet écart de telle sorte qu'il soit suffisamment faible pour ne pas entraîner des coûts supplémentaires démesurés. Un des principaux facteurs de surestimation est la prise en compte de chemins d'exécution sémantiquement impossibles, dits infaisables, dans le calcul du WCET. Ceci est dû à l'analyse par énumération implicite des chemins ou Implicit Path Enumeration Technique (IPET) qui raisonne sur un surensemble des chemins d'exécution. Lorsque le chemin d'exécution pire-cas ou Worst-Case Execution Path (WCEP), correspondant au WCET estimé, porte sur un chemin infaisable, la précision de cette estimation est négativement affectée. Afin de parer à cette perte de précision, cette thèse propose une technique de détection de chemins infaisables, permettant l'amélioration de la précision des analyses statiques (dont celles pour le WCET) en les informant de l'infaisabilité de certains chemins du programme. Cette information est passée sous la forme de propriétés de flot de données formatées dans un langage d'annotation portable, FFX, permettant la communication des résultats de notre analyse de chemins infaisables vers d'autres analyses. Les méthodes présentées dans cette thèse sont inclues dans le framework OTAWA, développé au sein de l'équipe TRACES à l'IRIT. Elles usent elles-mêmes d'approximations pour représenter les états possibles de la machine en différents points du programme. Ce sont des abstractions maintenues au fil de l'analyse, et dont la validité est assurée par des outils de la théorie de l'interprétation abstraite. Ces abstractions permettent de représenter de manière efficace - mais sûre - les ensembles d'états pour une classe de chemins d'exécution jusqu'à un point du programme, et de détecter d'éventuels points du programme associés à un ensemble d'états possibles vide, traduisant un (ou plusieurs) chemin(s) infaisable(s). L'objectif de l'analyse développée, la détection de tels cas, est rendue possible par l'usage de solveurs SMT (Satisfiabilité Modulo des Théories). Ces solveurs permettent essentiellement de déterminer la satisfiabilité d'un ensemble de contraintes, déduites à partir des états abstraits construits. Lorsqu'un ensemble de contraintes, formé à partir d'une conjonction de prédicats, s'avère insatisfiable, aucune valuation des variables de la machine ne correspond à un cas d'exécution possible, et la famille de chemins associée est donc infaisable. L'efficacité de cette technique est soutenue par une série d'expérimentations sur divers suites de benchmarks, reconnues dans le domaine du WCET statique et/ou issues de cas réels de l'industrie. Des heuristiques sont configurées afin d'adoucir la complexité de l'analyse, en particulier pour les applications de plus grande taille. Les chemins infaisables détectés sont injectés sous la forme de contraintes de flot linéaires dans le système de Programmation Linéaire en Nombres Entiers ou Integer Linear Programming (ILP) pilotant le calcul final de l'analyse WCET d'OTAWA. Selon le programme analysé, cela peut résulter en une réduction du WCET estimé, et donc une amélioration de sa précision.The search for an upper bound of the execution time of a program is an essential part of the verification of real-time critical systems. The execution times of the programs of such systems generally vary a lot, and it is difficult, or impossible, to predict the range of the possible times. Instead, it is better to look for an approximation of the Worst-Case Execution Time (WCET). A crucial requirement of this estimate is that it must be safe, that is, it must be guaranteed above the real WCET. Because we are looking to prove that the system in question terminates reasonably quickly, an overapproximation is the only acceptable form of approximation. The guarantee of such a safety property could not sensibly be done without static analysis, as a result based on a battery of tests could not be safe without an exhaustive handling of test cases. Furthermore, in the absence of a certified compiler (and tech- nique for the safe transfer of properties to the binaries), the extraction of properties must be done directly on binary code to warrant their soundness. However, this approximation comes with a cost : an important pessimism, the gap between the estimated WCET and the real WCET, would lead to superfluous extra costs in hardware in order for the system to respect the imposed timing requirements. It is therefore important to improve the precision of the WCET by reducing this gap, while maintaining the safety property, as such that it is low enough to not lead to immoderate costs. A major cause of overestimation is the inclusion of semantically impossible paths, said infeasible paths, in the WCET computation. This is due to the use of the Implicit Path Enumeration Technique (IPET), which works on an superset of the possible execution paths. When the Worst-Case Execution Path (WCEP), corresponding to the estimated WCET, is infeasible, the precision of that estimation is negatively affected. In order to deal with this loss of precision, this thesis proposes an infeasible paths detection technique, enabling the improvement of the precision of static analyses (namely for WCET estimation) by notifying them of the infeasibility of some paths of the program. This information is then passed as data flow properties, formatted in the FFX portable annotation language, and allowing the communication of the results of our infeasible path analysis to other analyses. The methods hereafter presented are included in the OTAWA framework, developed in TRACES team at the IRIT lab. They themselves make use of approximations in order to represent the possible states of the machine in various program points. These approximations are abstractions maintained throughout the analysis, and which validity is ensured by abstract interpretation tools. They enable us to represent the set of states for a family of execution paths up to a given program point in an efficient - yet safe - way, and to detect the potential program points associated to an empty set of possible states, signalling one (or several) infeasible path(s). As the end goal of the developed analysis, the detection of such cases is made possible by the use of Satisfiability Modulo Theory (SMT) solvers. Those solvers are notably able to determine the satisfiability of a set of contraints, which we deduct from the abstract states. If a set of constraints, derived from a conjonction of predicates, is unsatisfiable, then there exists no valuation of the machine variables that match a possible execution case, and thus the associated infeasible paths are infeasible. The efficiency of this technique is asserted by a series of experiments on various benchmarks suites, some of which widely recognized in the domain of static WCET, some others derived from actual industrial applications. Heuristics are set up in order to soften the complexity of the analysis, especially for the larger applications. The detected infeasible paths are injected as Integer Linear Programming (ILP) linear data flow constraints in the final computation for the WCET estimation in OTAWA. Depending on the analysed program, this can result in a reduction of the estimated WCET, thereby improving its precision

Design and implementation of WCET analyses : including a case study on multi-core processors with shared buses

For safety-critical real-time embedded systems, the worst-case execution time (WCET) analysis — determining an upper bound on the possible execution times of a program — is an important part of the system verification. Multi-core processors share resources (e.g. buses and caches) between multiple processor cores and, thus, complicate the WCET analysis as the execution times of a program executed on one processor core significantly depend on the programs executed in parallel on the concurrent cores. We refer to this phenomenon as shared-resource interference. This thesis proposes a novel way of modeling shared-resource interference during WCET analysis. It enables an efficient analysis — as it only considers one processor core at a time — and it is sound for hardware platforms exhibiting timing anomalies. Moreover, this thesis demonstrates how to realize a timing-compositional verification on top of the proposed modeling scheme. In this way, this thesis closes the gap between modern hardware platforms, which exhibit timing anomalies, and existing schedulability analyses, which rely on timing compositionality. In addition, this thesis proposes a novel method for calculating an upper bound on the amount of interference that a given processor core can generate in any time interval of at most a given length. Our experiments demonstrate that the novel method is more precise than existing methods.Die Analyse der maximalen Ausführungszeit (Worst-Case-Execution-Time-Analyse, WCET-Analyse) ist für eingebettete Echtzeit-Computer-Systeme in sicherheitskritischen Anwendungsbereichen unerlässlich. Mehrkernprozessoren erschweren die WCET-Analyse, da einige ihrer Hardware-Komponenten von mehreren Prozessorkernen gemeinsam genutzt werden und die Ausführungszeit eines Programmes somit vom Verhalten mehrerer Kerne abhängt. Wir bezeichnen dies als Interferenz durch gemeinsam genutzte Komponenten. Die vorliegende Arbeit schlägt eine neuartige Modellierung dieser Interferenz während der WCET-Analyse vor. Der vorgestellte Ansatz ist effizient und führt auch für Computer-Systeme mit Zeitanomalien zu korrekten Ergebnissen. Darüber hinaus zeigt diese Arbeit, wie ein zeitkompositionales Verfahren auf Basis der vorgestellten Modellierung umgesetzt werden kann. Auf diese Weise schließt diese Arbeit die Lücke zwischen modernen Mikroarchitekturen, die Zeitanomalien aufweisen, und den existierenden Planbarkeitsanalysen, die sich alle auf die Kompositionalität des Zeitverhaltens verlassen. Außerdem stellt die vorliegende Arbeit ein neues Verfahren zur Berechnung einer oberen Schranke der Menge an Interferenz vor, die ein bestimmter Prozessorkern in einem beliebigen Zeitintervall einer gegebenen Länge höchstens erzeugen kann. Unsere Experimente zeigen, dass das vorgestellte Berechnungsverfahren präziser ist als die existierenden Verfahren.Deutsche Forschungsgemeinschaft (DFG) as part of the Transregional Collaborative Research Centre SFB/TR 14 (AVACS