It is well-known that CCG can generate languages that are not context-free (which is necessary to capture natural languages), but can still be parsed in polynomial time.
vijay-Shanker and Weir (1994) identified a version of CCG that is weakly equivalent to Tree Adjoining Grammar (TAG) (Joshi and Schabes, 1997) and other mildly context-sensitive grammar formalisms, and can generate non-context-free languages such as a n b n c n .