Behaviour/structure transformations under uncertainty

Abstract

This paper analyses the problem of determining a structure for an automaton, optimal in some sense, from observations of its behaviour which are themselves uncertain. It is shown that extension of deterministic modelling techniques based on the Nerode equivalence to probabilistic sources gives meaningless results. The problem of approximate modelling with nondeterministic structures is rigorously formulated leading to the concept of a space of admissible models . The special case where the observed behaviour may be represented as a symbol string is then analysed in terms of measures of string approximation . It is shown that appropriate measures lead to the poorness-of-fit of admissible models of a probabilistic source being an entropy for that source. The formulation is consistent with a computational complexity basis for probability theory and leads to natural expressions for the surprise at each observation and the uncertainty as to the next observation. An implemented algorithm for this modelling process is then described with examples of its application to: probabilistic sources; sampled deterministic sources; grammatical inference; human behaviour; and program derivation from traces.