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A serious obstacle in applying computational models of genomic regulation is their complexity. Thus, there is a need for size reducing mappings that preserve biologically meaningful properties of the models. There are several available reduction mappings for the PBN model that are capable of preserving important structural or dynamical properties of the network. However, the cost of applying such mappings has been largely ignored. This paper studies how the notion of stochastic complexity can be used to measure the cost of reduction in the case of a specific class of constrained reduction mappings.