Простой алгоритм решения задачи покрытия для монотонных счетчиковых систем

An algorithm for solving the coverability problem for monotonic counter systems is presented. The solvability of this problem is well-known, but the algorithm is interesting due to its simplicity. The algorithm has emerged as a simplification of a certain procedure of a supercompiler application (a program specializer based on V.F. Turchin's supercompilation) to a program encoding a monotonic counter system along with initial and target sets of states and from the proof that under some conditions the procedure terminates and solves the coverability problem.Предложен алгоритм решения задачи покрытия для монотонных счетчиковых систем. Разрешимость этой задачи хорошо известна, но данный алгоритм интересен своей простотой. Он возник из упрощения некоторой итеративной процедуры применения суперкомпилятора (специализатора программ, основанного на методе суперкомпиляции В.Ф. Турчина) к программе, кодирующей счетчиковую систему и начальное и целевое множества состояний, и из доказательства, что при определенных условиях эта процедура завершается и решает задачу покрытия

Compaction of Church Numerals for Higher-Order Compression

In this study, we address the problem of compacting Church numerals. Church numerals appear as a representation of the repetitive part of data in higher-order compression. We propose a novel decomposition scheme for a natural number using tetration, which leads to a compact representation of $\lambda$-terms equivalent to the original Church numerals. For natural number $n$, we prove that the size of the $\lambda$-term obtained by the proposed method is $O(({\rm slog}_{2}n)^{\log n/ \log \log n})$. Moreover, we quantitatively confirmed experimentally that the proposed method outperforms a binary expression of Church numerals when $n$ is less than approximately 10000

Quasi-friendly sup-interpretations

In a previous paper, the sup-interpretation method was proposed as a new tool to control memory resources of first order functional programs with pattern matching by static analysis. Basically, a sup-interpretation provides an upper bound on the size of function outputs. In this former work, a criterion, which can be applied to terminating as well as non-terminating programs, was developed in order to bound polynomially the stack frame size. In this paper, we suggest a new criterion which captures more algorithms computing values polynomially bounded in the size of the inputs. Since this work is related to quasi-interpretations, we compare the two notions obtaining two main features. The first one is that, given a program, we have heuristics for finding a sup-interpretation when we consider polynomials of bounded degree. The other one consists in the characterizations of the set of function computable in polynomial time and in polynomial space

Refactoring OCL annotated UML class diagrams

Refactoring of UML class diagrams is an emerging research topic and heavily inspired by refactoring of program code written in object-oriented implementation languages. Current class diagram refactoring techniques concentrate on the diagrammatic part but neglect OCL constraints that might become syntactically incorrect by changing the underlying class diagram. This paper formalizes the most important refactoring rules for class diagrams and classifies them with respect to their impact on attached OCL constraints. For refactoring rules that have an impact on OCL constraints, we formalize the necessary changes of the attached constraints. Our refactoring rules are specified in a graph-grammar inspired formalism. They have been implemented as QVT transformation rules. We finally discuss for our refactoring rules the problem of syntax preservation and show, by using the KeY-system, how this can be resolve

Invariant Generation for Multi-Path Loops with Polynomial Assignments

Program analysis requires the generation of program properties expressing conditions to hold at intermediate program locations. When it comes to programs with loops, these properties are typically expressed as loop invariants. In this paper we study a class of multi-path program loops with numeric variables, in particular nested loops with conditionals, where assignments to program variables are polynomial expressions over program variables. We call this class of loops extended P-solvable and introduce an algorithm for generating all polynomial invariants of such loops. By an iterative procedure employing Gr\"obner basis computation, our approach computes the polynomial ideal of the polynomial invariants of each program path and combines these ideals sequentially until a fixed point is reached. This fixed point represents the polynomial ideal of all polynomial invariants of the given extended P-solvable loop. We prove termination of our method and show that the maximal number of iterations for reaching the fixed point depends linearly on the number of program variables and the number of inner loops. In particular, for a loop with m program variables and r conditional branches we prove an upper bound of m*r iterations. We implemented our approach in the Aligator software package. Furthermore, we evaluated it on 18 programs with polynomial arithmetic and compared it to existing methods in invariant generation. The results show the efficiency of our approach

The Parma Polyhedra Library: Toward a Complete Set of Numerical Abstractions for the Analysis and Verification of Hardware and Software Systems

Since its inception as a student project in 2001, initially just for the handling (as the name implies) of convex polyhedra, the Parma Polyhedra Library has been continuously improved and extended by joining scrupulous research on the theoretical foundations of (possibly non-convex) numerical abstractions to a total adherence to the best available practices in software development. Even though it is still not fully mature and functionally complete, the Parma Polyhedra Library already offers a combination of functionality, reliability, usability and performance that is not matched by similar, freely available libraries. In this paper, we present the main features of the current version of the library, emphasizing those that distinguish it from other similar libraries and those that are important for applications in the field of analysis and verification of hardware and software systems.Comment: 38 pages, 2 figures, 3 listings, 3 table

Model refactoring using transformations

Modern software is reaching levels of complexity encountered in biological systems; sometimes comprising systems of systems each of which may include tens of millions of lines of code. Model Driven Engineering (MDE) advocates raising the level of abstraction as an instrument to deal with software complexity. It promotes usage of software models as primary artifacts in a software development process. Traditionally, these MDE models are specified by Unified Modeling Language (UML) or by a modeling language created for a specific domain. However, in the vast area of software engineering there are other techniques used to improve quality of software under development. One of such techniques is refactoring which represents introducing structured changes in software in order to improve its readability, extensibility, and maintainability, while preserving behavior of the software. The main application area for refactorings is still programming code, despite the fact that modeling languages and techniques has significantly gained in popularity, in recent years. The main topic of this thesis is making an alliance between the two virtually orthogonal techniques: software modeling and refactoring. In this thesis we have investigated how to raise the level of abstraction of programming code refactorings to the modeling level. This resulted in a catalog of model refactorings each specified as a model transformation rule. In addition, we have investigated synchronization problems between different models used to describe one software system, i.e. when one model is refactored what is the impact on all dependent models and how this impact can be formalized. We have concentrated on UML class diagrams as domain of refactorings. As models dependent on class diagrams, we have selected Object Constraint Language (OCL) annotations, and object diagrams. This thesis formalizes the most important refactoring rules for UML class diagrams and classifies them with respect to their impact on object diagrams and annotated OCL constraints. For refactoring rules that have an impact on dependent artifacts we formalize the necessary changes of these artifacts. Moreover, in this thesis, we present a simple criterion and a proof technique for the semantic preservation of refactoring rules that are defined for UML class and object diagrams, and OCL constraints. In order to be able to prove semantic preservation, we propose a model transformation approach to specify the semantics of constraint languages

Global Guidance for Local Generalization in Model Checking

SMT-based model checkers, especially IC3-style ones, are currently the most effective techniques for verification of infinite state systems. They infer global inductive invariants via local reasoning about a single step of the transition relation of a system, while employing SMT-based procedures, such as interpolation, to mitigate the limitations of local reasoning and allow for better generalization. Unfortunately, these mitigations intertwine model checking with heuristics of the underlying SMT-solver, negatively affecting stability of model checking. In this paper, we propose to tackle the limitations of locality in a systematic manner. We introduce explicit global guidance into the local reasoning performed by IC3-style algorithms. To this end, we extend the SMT-IC3 paradigm with three novel rules, designed to mitigate fundamental sources of failure that stem from locality. We instantiate these rules for the theory of Linear Integer Arithmetic and implement them on top of SPACER solver in Z3. Our empirical results show that GSPACER, SPACER extended with global guidance, is significantly more effective than both SPACER and sole global reasoning, and, furthermore, is insensitive to interpolation.Comment: Published in CAV 202