Progress in advancing our understanding of biological systems is limited by their sheer complexity, the cost of laboratory materials and equipment, and limitations of current laboratory technology. Computational and mathematical modeling provide ways to address these obstacles through hypothesis generation and testing without experimentation---allowing researchers to analyze system structure and dynamics in silico and, then, design lab experiments that yield desired information about phenomena of interest. These models, however, are only as accurate and complete as the data used to build them. Currently, most models are constructed from quantitative experimental data. However, since accurate quantitative measurements are hard to obtain and difficult to adapt from literature and online databases, new sources of data for building models need to be explored. In my work, I have designed methods for building and executing computational models of cellular network dynamics based on qualitative experimental data, which are more abundant, easier to obtain, and reliably reproducible. Such executable models allow for in silico perturbation, simulation, and exploration of biological systems. In this thesis, I present two general strategies for building and executing tokenized models of biochemical networks using only qualitative data. Both methods have been successfully used to model and predict the dynamics of signaling networks in normal and cancer cell lines, rivaling the accuracy of existing methods trained on quantitative data. I have implemented these methods in the software tools PathwayOracle and Monarch, making the new techniques I present here accessible to experimental biologists and other domain experts in cellular biology.