Deciding definability by deterministic regular expressions

We investigate the complexity of deciding whether a given regular language can be expressed by a deterministic regular expression. Our main technical result shows that deciding if the language of a given regular expression (or, non-deterministic finite automaton) can be defined by a deterministic regular expression is PSPACE-complete. The problem becomes EXPSPACE-complete if the input language is represented as a regular expression with counters and NL-hard if the input language is given by a minimal deterministic finite automaton.