Action Codes
Abstract
We provide a new perspective on the problem how highlevel state machine models with abstract actions can be related to lowlevel models in which these actions are refined by sequences of concrete actions. We describe the connection between highlevel and lowlevel actions using \emph{action codes}, a variation of the prefix codes known from coding theory. For each action code ${\mathcal{R}}$, we introduce a \emph{contraction} operator $\alpha_{\mathcal{R}}$ that turns a lowlevel model $\mathcal{M}$ into a highlevel model, and a \emph{refinement} operator $\rho_{\mathcal{R}}$ that transforms a highlevel model $\mathcal{N}$ into a lowlevel model. We establish a Galois connection $\rho_{\mathcal{R}}(\mathcal{N}) \sqsubseteq \mathcal{M} \Leftrightarrow \mathcal{N} \sqsubseteq \alpha_{\mathcal{R}}(\mathcal{M})$, where $\sqsubseteq$ is the wellknown simulation preorder. For conformance, we typically want to obtain an overapproximation of model $\mathcal{M}$. To this end, we also introduce a \emph{concretization} operator $\gamma_{\mathcal{R}}$, which behaves like the refinement operator but adds arbitrary behavior at intermediate points, giving us a second Galois connection $\alpha_{\mathcal{R}}(\mathcal{M}) \sqsubseteq \mathcal{N} \Leftrightarrow \mathcal{M} \sqsubseteq \gamma_{\mathcal{R}}(\mathcal{N})$. Action codes may be used to construct adaptors that translate between concrete and abstract actions during learning and testing of Mealy machines. If Mealy machine $\mathcal{M}$ models a blackbox system then $\alpha_{\mathcal{R}}(\mathcal{M})$ describes the behavior that can be observed by a learner/tester that interacts with this system via an adaptor derived from code ${\mathcal{R}}$. Whenever $\alpha_{\mathcal{R}}(\mathcal{M})$ implements (or conforms to) $\mathcal{N}$, we may conclude that $\mathcal{M}$ implements (or conforms to) $\gamma_{\mathcal{R}} (\mathcal{N})$.
 Publication:

arXiv eprints
 Pub Date:
 December 2022
 DOI:
 10.48550/arXiv.2301.00199
 arXiv:
 arXiv:2301.00199
 Bibcode:
 2023arXiv230100199V
 Keywords:

 Computer Science  Formal Languages and Automata Theory;
 Computer Science  Information Theory