Abstract questionnaires and FS-decision digraphs

A questionnaire is a sequence of multiple choice questions aiming to collect data on a population. We define an abstract questionnaire as an ordered pair $(N,{\cal M})$, where $N$ is a positive integer and ${\cal M}=(m_0,m_1,\ldots,m_{N-1})$ is an $N$-tuple of positive integers, with $m_i$, for $i \in \{0, 1, \ldots, N-1 \}$, as the number of possible answers to question $i$. An abstract questionnaire may be endowed with a skip-list (which tells us which questions to skip based on the sequence of answers to the earlier questions) and a flag-set (which tells us which sequences of answers are of special interest). An FS-decision tree is a decision tree of an abstract questionnaire that also incorporates the information contained in the skip-list and flag-set. The main objective of this paper is to represent the abstract questionnaire using a directed graph, which we call an FS-decision digraph, that contains the full information of an FS-decision tree, but is in general much more concise. We present an algorithm for constructing a fully reduced FS-decision digraph, and develop the theory that supports it. In addition, we show how to generate all possible orderings of the questions in an abstract questionnaire that respect a given precedence relation.