  The word inference problem is to determine languages such that the
information on the number of occurrences of those subwords in the language can
uniquely identify a word. A considerable amount of work has been done on this
problem, but the same cannot be said for circular words despite growing
interests on the latter due to their applications -- for example, in splicing
systems. Meanwhile, Parikh matrices are useful tools and well established in
the study of subword occurrences. In this work, we propose two ways of counting
subword occurrences in circular words. We then extend the idea of Parikh
matrices to the context of circular words and investigate this extension.
Motivated by the word inference problem, we study ambiguity in the
identification of a circular word by its Parikh matrix. Accordingly, two
rewriting rules are developed to generate ternary circular words which share
the same Parikh matrix.
