Optimal diagnostic questionnaires which allow less than truthful responses

This paper examines the problem of determining a sequence of questions, or a questionnaire, in order to optimally identify the unknown state of a system for the case where the answers to the questions asked are not necessarily truthful. The problem is modeled by a slight generalization of the partially observed Markov optimization problem for the special case where the underlying process is not allowed to change state during the period of questioning. In this formulation, each question is considered equivalent to a discrete memoryless channel and its associated channel matrix. Recent results in questionnaire theory and an application of restriction set theory to diagnosis are shown to be essentially equivalent to the case where all available channel matrices are deterministic, i.e., all questions receive truthful replies, and further results for this particular case are determined. When answers to questions are not necessarily truthful, a charge per question of the channel capacity of the associated channel and an appropriate terminal charge are shown to imply that the average cost of a questionnaire is bounded below by the Shannon entropy of a given a priori density, thus further exploring the relationship between optimal diagnostic questioning and information theory. Conditions sufficient for an optimal questionnaire to attain this lower limit are then presented.