Embeddings between state and action labeled probabilistic systems

Abstract

This paper proposes a formal framework which allows one to move from action labeled discrete time Markov chains (ADTMCs) to state labeled discrete time Markov chains (SDTMCs) and vice versa. We define the embeddings sld and ald which construct an SDTMC from an ADTMC and an ADTMC from an SDTMC, respectively. We prove that forward bisimulation, backward bisimulation and trace equivalence are preserved by both these embeddings. Next, we define the reversibility criteria and the inverse of the embeddings sld, i.e. sld-1, and ald, i.e. ald-1. Our inverse embeddings are not only unique but also revert back to the original ADTMC (SDTMC, respectively). We show that reversibility is preserved w.r.t. forward and backward bisimulation. Finally, we prove that a model can be minimized in one setting by minimizing its embedded model in the other setting and taking the inverse of the embedding. Our results enable one to use the state-of-the-art tools developed in one setting for model minimization and analysis in the other setting.