Assistance to people suffering from cognitive deficiencies in a smart home raises complex issues. Plan recognition is one of them. We propose a formal framework for the recognition process based on lattice theory and action description logic. The framework minimizes the uncertainty about the prediction of the observed agent's behaviour by dynamically generating new implicit extra-plans. This approach offers an effective solution to actual plan recognition problem in a smart home, in order to provide assistance to persons suffering from cognitive deficits. An implementation of this model was incorporated in our smart home laboratory, in order to validate the approach. We currently planning the experimentation phase of the system, which will be based on a set of real case scenarios.