  We introduce regular language states, a family of quantum many-body states.
They are built from a special class of formal languages, called regular, which
has been thoroughly studied in the field of computer science. They can be
understood as the superposition of all the words in a regular language and
encompass physically relevant states such as the GHZ-, W- or Dicke-states. By
leveraging the theory of regular languages, we develop a theoretical framework
to describe them. First, we express them in terms of matrix product states,
providing efficient criteria to recognize them. We then develop a canonical
form which allows us to formulate a fundamental theorem for the equivalence of
regular language states, including under local unitary operations. We also
exploit the theory of tensor networks to find an efficient criterion to
determine when regular languages are shift-invariant.
