Axiomatizations of signed discrete Choquet integrals

We study the so-called signed discrete Choquet integral (also called non-monotonic discrete Choquet integral) regarded as the Lov\'asz extension of a pseudo-Boolean function which vanishes at the origin. We present axiomatizations of this generalized Choquet integral, given in terms of certain functional equations, as well as by necessary and sufficient conditions which reveal desirable properties in aggregation theory

Modal µ-Calculus, Model Checking and Gauß Elimination

In this paper we present a novel approach for solving Boolean equation systems with nested minimal and maximal fixpoints. The method works by successively eliminating variables and reducing a Boolean equation system similar to Gauß elimination for linear equation systems. It does not require backtracking techniques. Within one framework we suggest a global and a local algorithm. In the context of model checking in the modal-calculus the local algorithm is related to the tableau methods, but has a better worst case complexity

Computational core and fixed-point organisation in Boolean networks

In this paper, we analyse large random Boolean networks in terms of a constraint satisfaction problem. We first develop an algorithmic scheme which allows to prune simple logical cascades and under-determined variables, returning thereby the computational core of the network. Second we apply the cavity method to analyse number and organisation of fixed points. We find in particular a phase transition between an easy and a complex regulatory phase, the latter one being characterised by the existence of an exponential number of macroscopically separated fixed-point clusters. The different techniques developed are reinterpreted as algorithms for the analysis of single Boolean networks, and they are applied to analysis and in silico experiments on the gene-regulatory networks of baker's yeast (saccaromices cerevisiae) and the segment-polarity genes of the fruit-fly drosophila melanogaster.Comment: 29 pages, 18 figures, version accepted for publication in JSTA