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

Automated Generation of Non-Linear Loop Invariants Utilizing Hypergeometric Sequences

Analyzing and reasoning about safety properties of software systems becomes an especially challenging task for programs with complex flow and, in particular, with loops or recursion. For such programs one needs additional information, for example in the form of loop invariants, expressing properties to hold at intermediate program points. In this paper we study program loops with non-trivial arithmetic, implementing addition and multiplication among numeric program variables. We present a new approach for automatically generating all polynomial invariants of a class of such programs. Our approach turns programs into linear ordinary recurrence equations and computes closed form solutions of these equations. These closed forms express the most precise inductive property, and hence invariant. We apply Gr\"obner basis computation to obtain a basis of the polynomial invariant ideal, yielding thus a finite representation of all polynomial invariants. Our work significantly extends the class of so-called P-solvable loops by handling multiplication with the loop counter variable. We implemented our method in the Mathematica package Aligator and showcase the practical use of our approach.Comment: A revised version of this paper is published in the proceedings of ISSAC 201

Solving Robust Glucose-Insulin Control by Dixon Resultant Computations

We present a symbolic approach towards solving the Bergman three-state minimal patient model of glucose metabolism. Our work first translates the Bergman three-state minimal patient model into the modified control algebraic Riccati equation. Next, the modified control algebraic Ricatti equation is reduced to a system of polynomial equations, and an optimal (minimal) solution of these polynomials is computed using Dixon resultants. We demonstrate the use of our method by reporting on three case studies over glucose metabolism

Automated Sensitivity Analysis for Probabilistic Loops

We present an exact approach to analyze and quantify the sensitivity of higher moments of probabilistic loops with symbolic parameters, polynomial arithmetic and potentially uncountable state spaces. Our approach integrates methods from symbolic computation, probability theory, and static analysis in order to automatically capture sensitivity information about probabilistic loops. Sensitivity information allows us to formally establish how value distributions of probabilistic loop variables influence the functional behavior of loops, which can in particular be helpful when choosing values of loop variables in order to ensure efficient/expected computations. Our work uses algebraic techniques to model higher moments of loop variables via linear recurrence equations and introduce the notion of sensitivity recurrences. We show that sensitivity recurrences precisely model loop sensitivities, even in cases where the moments of loop variables do not satisfy a system of linear recurrences. As such, we enlarge the class of probabilistic loops for which sensitivity analysis was so far feasible. We demonstrate the success of our approach while analyzing the sensitivities of probabilistic loops

Strong Invariants Are Hard: On the Hardness of Strongest Polynomial Invariants for (Probabilistic) Programs

We show that computing the strongest polynomial invariant for single-path loops with polynomial assignments is at least as hard as the Skolem problem, a famous problem whose decidability has been open for almost a century. While the strongest polynomial invariants are computable for affine loops, for polynomial loops the problem remained wide open. As an intermediate result of independent interest, we prove that reachability for discrete polynomial dynamical systems is Skolem-hard as well. Furthermore, we generalize the notion of invariant ideals and introduce moment invariant ideals for probabilistic programs. With this tool, we further show that the strongest polynomial moment invariant is (i) uncomputable, for probabilistic loops with branching statements, and (ii) Skolem-hard to compute for polynomial probabilistic loops without branching statements. Finally, we identify a class of probabilistic loops for which the strongest polynomial moment invariant is computable and provide an algorithm for it

Hazai tapasztalatok a transznacionális önkormányzati együttműködések terén

This paper explores the Hungarian local government’s involvement into transnational, mainly European local authority networks. For local authorities – as a result of globalization and Europeanization – there is an increasingly important role of learning from each other. It can be related, that the phenomenon of local government networking is an increasingly dynamic one. In Europe, likewise the other parts of the world, the number of cooperating cities is growing in an extraordinary measure. The study considers to what extent the Hungarian local authorities are involved, and also how they are able to use these collaborations. To map the Hungarian local government’s involvement into transnational local authority networks and to measure the conduciveness of their international cooperation, both quantitative (online questionnaire, sample of 263) and qualitative (7 interviews) approaches were used. In this paper the emphasis is put on the latter one