  A grammar logic refers to an extension to the multi-modal logic K in which
the modal axioms are generated from a formal grammar. We consider a proof
theory, in nested sequent calculus, of grammar logics with converse, i.e.,
every modal operator [a] comes with a converse. Extending previous works on
nested sequent systems for tense logics, we show all grammar logics (with or
without converse) can be formalised in nested sequent calculi, where the axioms
are internalised in the calculi as structural rules. Syntactic cut-elimination
for these calculi is proved using a procedure similar to that for display
logics. If the grammar is context-free, then one can get rid of all structural
rules, in favor of deep inference and additional propagation rules. We give a
novel semi-decision procedure for context-free grammar logics, using nested
sequent calculus with deep inference, and show that, in the case where the
given context-free grammar is regular, this procedure terminates. Unlike all
other existing decision procedures for regular grammar logics in the
literature, our procedure does not assume that a finite state automaton
encoding the axioms is given.
