"Don't care" modeling: A logical framework for developing predictive system models

Analysis of biological data often requires an understanding of components of pathways and/or networks and their mutual dependency relationships. Such systems are often analyzed and understood from datasets made up of the states of the relevant components and a set of discrete outcomes or results. The analysis of these systems can be assisted by models that are consistent with the available data while being maximally predictive for untested conditions. Here, we present a method to construct such models for these types of systems. To maximize predictive capability, we introduce a set of "don't care" (dc) Boolean variables that must be assigned values in order to obtain a concrete model. When a dc variable is set to 1, this indicates that the information from the corresponding component does not contribute to the observed result. Intuitively, more dc variables that are set to 1 maximizes both the potential predictive capability as well as the possibility of obtaining an inconsistent model. We thus formulate our problem as maximizing the number of dc variables that are set to 1, while retaining a model solution that is consistent and can explain all the given known data. This amounts to solving a quantified Boolean formula (QBF) with three levels of quantifier alternations, with a maximization goal for the dc variables. We have developed a prototype implementation to support our new modeling approach and are applying our method to part of a classical system in developmental biology describing fate specification of vulval precursor cells in the C. elegans nematode. Our work indicates that biological instances can serve as challenging and complex benchmarks for the formal-methods research community.