  In this article, we study "questionable representations" of (partial or
total) orders, introduced in our previous article "A class of orders with
linear? time sorting algorithm". (Later, we consider arbitrary binary
functional/relational structures instead of orders.) A "question" is the first
difference between two sequences (with ordinal index) of elements of
orders/sets. In finite width "questionable representations" of an order O,
comparison can be solved by looking at the "question" that compares elements of
a finite order O'. A corollary of a theorem by Cantor (1895)is that all
countable total orders have a binary (width 2) questionable representation. We
find new classes of orders on which testing isomorphism or counting the number
of linear extensions can be done in polynomial time. We also present a
generalization of questionable-width, called balanced tree-questionable-width,
and show that if a class of binary structures has bounded tree-width or
clique-width, then it has bounded balanced tree-questionable-width. But there
are classes of graphs of bounded balanced tree-questionable-width and unbounded
tree-width or clique-width.
