Protein signaling networks are static views of dynamic processes where proteins go through many biochemical modifications such as ubiquitination and phosphorylation to propagate signals that regulate cells and can act as feed-back systems. Understanding the precise mechanisms underlying protein interactions can elucidate how signaling and cell cycle progression occur within cells in different diseases such as cancer. Large-scale protein signaling networks contain an important number of experimentally verified protein relations but lack the capability to predict the outcomes of the system, and therefore to be trained with respect to experimental measurements. Boolean Networks (BNs) are a simple yet powerful framework to study and model the dynamics of the protein signaling networks. While many BN approaches exist to model biological systems, they focus mainly on system properties, and few exist to integrate experimental data in them. In this work, we show an application of a method conceived to integrate time series phosphoproteomic data into protein signaling networks. We use a large-scale real case study from the HPN-DREAM Breast Cancer challenge. Our efficient and parameter-free method combines logic programming and model-checking to infer a family of BNs from multiple perturbation time series data of four breast cancer cell lines given a prior protein signaling network. Because each predicted BN family is cell line specific, our method highlights commonalities and discrepancies between the four cell lines. Our models have a Root Mean Square Error (RMSE) of 0.31 with respect to the testing data, while the best performant method of this HPN-DREAM challenge had a RMSE of 0.47. To further validate our results, BNs are compared with the canonical mTOR pathway showing a comparable AUROC score (0.77) to the top performing HPN-DREAM teams. In addition, our approach can also be used as a complementary method to identify erroneous experiments. These results prove our methodology as an efficient dynamic model discovery method in multiple perturbation time course experimental data of large-scale signaling networks. The software and data are publicly available at https://github.com/misbahch6/caspo-ts.