A relation-algebraic approach to simple games

Simple games are a powerful tool to analyze decision - making and coalition formation in social and political life. In this paper, we present relation-algebraic models of simple games and develop relational specifications for solving some basic problems of them. In particular, we test certain fundamental properties of simple games and compute specific players and coalitions. We also apply relation algebra to determine power indices. This leads to relation-algebraic specifications, which can be evaluated with the help of the BDD-based tool RelView after a simple translation into the tool's programming language. In order to demonstrate the visualization facilities of RelView, we consider an example of the Catalonian Parliament after the 2003 election.Relation algebra ; RelView ; simple game ; winning coalition ; swinger ; dominant player ; central player ; power index

A Relation-algebraic Approach to Simple Games

Simple games are a powerful tool to analyze decision-making and coalition formation in social and political life. In this paper, we present relation-algebraic models of simple games and develop relational algorithms for solving some basic problems of them. In particular, we test certain fundamental properties of simple games (being monotone, proper, respectively strong) and compute specific players (dummies, dictators, vetoers, null players) and coalitions (minimal winning coalitions and vulnerable winning coalitions). We also apply relation-algebra to determine central and dominant players, swingers and power indices (the Banzhaf, Holler-Packel and Deegan-Packel indices). This leads to relation-algebraic specifications, which can be executed with the help of the BDD-based tool RelView after a simple translation into the tool's programming language. In order to demonstrate the visualization facilities of RelView we consider an example of the Catalonian Parliament after the 2003 election.relation algebra; RelView; simple game; winning coalition; swinger; dominant player; central player; power index

A relation-algebraic approach to simple games

International audienceSimple games are a powerful tool to analyze decision - making and coalition formation in social and political life. In this paper, we present relation-algebraic models of simple games and develop relational specifications for solving some basic problems of them. In particular, we test certain fundamental properties of simple games and compute specific players and coalitions. We also apply relation algebra to determine power indices. This leads to relation-algebraic specifications, which can be evaluated with the help of the BDD-based tool RelView after a simple translation into the tool's programming language. In order to demonstrate the visualization facilities of RelView, we consider an example of the Catalonian Parliament after the 2003 election

Relation-algebraic specification and solution of special university timetabling problems

AbstractIn this paper, we are concerned with a special timetabling problem. It was posed to us by the administration of our university and stems from the adoption of the British-American system of university education in Germany. This change led to the concrete task of constructing a timetable that enables the undergraduate education of secondary school teachers within three years in the “normal case” and within four years in the case of exceptional combinations of subjects. We develop two relation-algebraic models of the timetabling problem and in each case algorithms for computing solutions. The latter easily can be implemented in the Kiel RelView tool showing that RelView can be used for timetabling

Relation-algebraic modeling and solution of chessboard independence and domination problems

AbstractWe describe a simple computing technique for solving independence and domination problems on rectangular chessboards. It rests upon relational modeling and uses the BDD-based specific purpose computer algebra system RelView for the evaluation of the relation-algebraic expressions that specify the problems’ solutions and the visualization of the computed results. The technique described in the paper is very flexible and especially appropriate for experimentation. It can easily be applied to other chessboard problems

RelMDD-A Library for Manipulating Relations Based on MDDs

Relation algebras is one of the state-of-the-art means used by mathematicians and computer scientists for solving very complex problems. As a result, a computer algebra system for relation algebras called RelView has been developed at Kiel University. RelView works within the standard model of relation algebras. On the other hand, relation algebras do have other models which may have different properties. For example, in the standard model we always have L;L=L (the composition of two (heterogeneous) universal relations yields a universal relation). This is not true in some non-standard models. Therefore, any example in RelView will always satisfy this property even though it is not true in general. On the other hand, it has been shown that every relation algebra with relational sums and subobjects can be seen as matrix algebra similar to the correspondence of binary relations between sets and Boolean matrices. The aim of my research is to develop a new system that works with both standard and non-standard models for arbitrary relations using multiple-valued decision diagrams (MDDs). This system will implement relations as matrix algebras. The proposed structure is a library written in C which can be imported by other languages such as Java or Haskell

Entwurf funktionaler Implementierungen von Graphalgorithmen

Classic graph algorithms are usually presented and analysed in imperative programming languages. Imperative programming languages are well-suited for the description of a program flow, in which the order in which the operations are performed is important. One common example of such a description is the successive, typically destructive modification of objects. This kind of iteration often occurs in the context of graph algorithms that deal with a certain kind of optimisation. In functional programming, the order of execution is abstracted and problem solutions are described as compositions of intermediate solutions. Additionally, functional programming languages are referentially transparent and thus destructive updates of objects are discouraged. The development of purely functional graph algorithms begins with the decomposition of a given problem into simpler problems. In many cases the solutions of these partial problems can be used to solve different problems as well. What is more, this compositionality allows exchanging functions for more efficient or more comprehensible versions with little effort. An algebraic approach with a focus on relation algebra as defined by Tarski is used as an intermediate step in this dissertation. One advantage of this approach is the formality of the resulting specifications. Despite their formality, the resulting expressions are still readable, because the algebraic operations have intuitive interpretations. Another advantage is that the specification is executable, once the necessary operations are implemented. This dissertation presents the basics of the algebraic approach in the functional programming language Haskell. Using this foundation, some exemplary graph-theoretic problems are solved in the presented framework. Finally, optimisations of the presented implementations are discussed and pointers are provided to further problems that can be solved using the above methods.Klassische Graphalgorithmen werden üblicherweise in imperativen Programmiersprachen beschrieben und analysiert. Imperative Programmiersprachen eignen sich gut, um Programmabläufe zu beschreiben, in welchen die Reihenfolge der Operationen wichtig ist. Dies betrifft insbesondere die schrittweise, in der Regel destruktive Veränderung von Objekten, wie sie häufig im Falle von Optimierungsproblemen auf Graphen vorkommt. In der funktionalen Programmierung abstrahiert man von einer festen Berechnungsreihenfolge und beschreibt Problemlösungen als Kompositionen von Teillösungen. Ferner sind funktionale Programmiersprachen referentiell transparent, sodass destruktive Veränderungen nur bedingt möglich sind. Die Entwicklung rein funktionaler Graphalgorithmen setzt bei der Zerlegung der bestehenden Probleme in einfachere Probleme an. Oftmals können Lösungen dieser Teilprobleme auch in anderen Situationen eingesetzt werden. Darüber hinaus erlaubt es diese Kompositionalität, einzelne Funktionen mit wenig Aufwand durch effizientere oder verständlichere Fassungen auszutauschen. Als Zwischenschritt in der Entwicklung wird in dieser Dissertation ein algebraischer Ansatz basierend auf der Relationenalgebra im Sinne von Tarski verwendet. Ein Vorteil dieses Ansatzes ist die Formalität der entstehenden Spezifikationen. Trotz ihrer Formalität bleiben die entstehenden Ausdrücke oft leserlich, weil die algebraischen Operationen anschauliche Interpretationen zulassen. Ein weiterer Vorteil ist, dass Spezifikationen ausführbar werden, sobald bestimmte Basisoperationen implementiert sind. In dieser Dissertation werden Grundlagen einer Implementierung des algebraischen Ansatzes in der funktionalen Programmiersprache Haskell behandelt. Ausgehend hiervon werden exemplarisch einige Probleme der Graphentheorie gelöst. Schließlich werden Optimierungen der vorgestellten Implementierungen und weitere Probleme, welche mit den obigen Methoden lösbar sind, diskutiert