Rudolf Berghammer

Rudolf Berghammer, Jules Desharnais, Michael Winter

Given a finite and non-empty set $X$ and randomly selected specific functions and relations on $X$, we investigate the existence and non-existence of fixed points and reflexive points, respectively. First, we consider the class of functions, weaken it to the classes of partial functions, total relations and general relations and also strengthen it to the class of permutations. Then we investigate the class of involutions and the subclass of proper involutions. Finally, we treat idempotent functions, partial idempotent functions and related concepts. We count relations, calculate corresponding probabilities and also calculate the limiting values of the latter in case that the cardinality of $X$ tends to infinity. All these results have been motivated and also supported by numerous experiments performed with the RelView tool.

Rudolf Berghammer, Henning Schnoor

We present a relation-algebraic model of Condorcet voting and, based on it, relation-algebraic solutions of the constructive control problem via the removal of voters. We consider two winning conditions, viz. to be a Condorcet winner and to be in the (Gilles resp. upward) uncovered set. For the first condition the control problem is known to be NP-hard; for the second condition the NP-hardness of the control problem is shown in the paper. All relation-algebraic specifications we will develop in the paper immediately can be translated into the programming language of the BDD-based computer system RelView. Our approach is very flexible and especially appropriate for prototyping and experimentation, and as such very instructive for educational purposes. It can easily be applied to other voting rules and control problems.