The authors study irreducible word problems of finitely presented groups over a group/monoid generating set in connection with language theory. It is well known [R. H. Haring-Smith, Trans. Amer. Math. Soc. {\bf 279} (1983), no.~1, 337--356; MR0704619 (85b:20045)] that a group has a finite irreducible word problem over a finite group generating set if and only if it is a plain group. First, the authors remark that a group with a finite irreducible word problem over a finite monoid generating set need not be plain. Next, they study groups with a context-free irreducible word problem over a group/monoid generating set. They show for example that any infinite context-free group has no-context-free irreducible word problems on some group generating set.