Minimising Simple XPath Expressions

We consider a subset of XPath expressions, called simple XPath expressions, which correspond to a class of conjunctive queries. We show that, in the absence of a DTD, each simple XPath expression has a unique minimal equivalent expression which can be found in polynomial time. We then consider D-equivalence, the equivalence of expressions with respect to the set of documents valid for a given DTD D. We show that a simple XPath expression P does not necessarily have a unique minimal D-equivalent expression. However, if P is reduced (there are no wildcards in it), then there is a unique minimal equivalent expression, but we show that deciding whether two reduced expressions are D-equivalent is coNP-hard.