Roland Glück

Towards Interactive Verification of Programmable Logic Controllers Using Modal Kleene Algebra and KIV

In this paper we develop an approach to interactive verification of programmable logic controllers which often serve as controllers in safety critical systems and hence need thorough verification. As a verification tool we use the KIV system, whereas the formalization is done in modal Kleene algebra. We first prove a bunch of theorems from modal Kleene algebra in KIV, subsequently translate the desired properties of a program for a programmable logic controller in modal Kleene algebra, and finally prove these encoded properties interactively with KIV.

Algebraic Investigation of Connected Components

This paper characterizes connected components of both directed and undirected graphs as atomic fixpoints. As algebraic structure for our investigations we combine complete Boolean algebras with the well-known theory of Kleene Algebra with domain. Using diamond operators as an algebraic generalization of relational image and preimage we show how connected components can be modeled as atomic fixpoints of functions operating on tests and prove some advanced theorems concerning connected components.

Algebraic Derivation of Until Rules and Application to Timer Verification

Using correspondences between linear temporal logic and modal Kleene Algebra, we prove in an algebraic manner rules of linear temporal logic involving the until operator. These can be used to verify programmable logic controllers; as a case study we use a part of the control of pedestrian lights, verified with the interactive tool KIV.

Distances, Norms and Error Propagation in Idempotent Semirings

Error propagation and perturbation theory are well-investigated areas of mathematics dealing with the influence of errors and perturbations of input quantities on output quantities. However, these methods are restricted to quantities relying on real numbers under traditional addition and multiplication. We aim to present first steps of an analogous theory on idempotent semirings, so we define distances and norms on idempotent semirings and matrices over them. These concepts are used to derive inequalities characterizing the influence of changes in the input quantities on the output quantities of some often used semiring expressions.

Covering Polygons with Rectangles

A well-known and well-investigated family of hard optimization problems deals with nesting, i.e., the non-overlapping placing of polygons to be cut from a rectangle or the plane whilst minimizing the waste. Here we consider the in some sense inverse problem of a subsequent step in production technology: given a set of polygons in the plane and an axis-aligned rectangle (modeling a gripping device), we seek the minimum number of copies of the rectangle such that every polygon is completely covered by at least one copy of the rectangle. As motions of the given rectangle for obtaining the copies we investigate the cases of translation in x-direction, of arbitrary translation and of arbitrary translation combined with rotation. We give a generic algorithm for all three cases which leads to a polynomial-time algorithm for the first case. The other two cases are NP-hard so we introduce a rather straightforward algorithm for the second case and two different approaches to the third one. Finally, we give experimental results and compare them to the theoretical analysis done before.

Towards Interactive Verification ofProgrammable Logic Controllers using ModalKleene Algebra and KIV

In this paper we develop an approach to interactive verification of programmable logic controllers which often serve as controllers in safety critical systems and hence need thorough verification. As a verification tool we use the KIV system, whereas the formalization is done in modal Kleene algebra. We first prove a bunch of theorems from modal Kleene algebra in KIV, subsequently translate the desired properties of a program for a programmable logic controller in modal Kleene algebra, and finally prove these encoded properties interactively with KIV.Formalisms that admit calculational reasoning with a relation-algebraic flavour are finding applications in numerous areas of computer science. As a kind of “linear algebra of logic”, these formalisms, including different kinds of allegories and Kleene algebras, enable methods of concise formalisation and reasoning in a “point-free” style with a high-level of abstraction. This facilitates not only high-confidence manual reasoning with good readability, but also automated tool support. These methods are the focus of a series of international conferences that started out using the name “Relational Methods in Computer Science” (RelMiCS), soon joined by “Applications of Kleene Algebra” (AKA); since 2011, the unified name “Relational and Algebraic Methods in Computer Science” (RAMiCS) has been adopted. The 13th International Conference on Relational and Algebraic Methods in Computer Science (RAMiCS 2012) was held in Cambridge, UK, September 17–20, 2012; the proceedings of this conference have been published as Lecture Notes in Computer Science (LNCS), volume 7560, which contains 23 contributed papers, each reviewed by at least three referees. The programme committee of RAMiCS 2012 selected ten papers presented at the conference; we invited their authors and the three keynote speakers to submit an extended version for this special issue of the Journal of Logic and Algebraic Methods in Programming. Each of the ten substantially extended and revised submissions we received was again evaluated by at least two referees. The resulting collection of papers nicely illustrates the wide spectrum of different formalisms sharing that common “relation-algebraic flavour”, and includes a number of different application areas.