On the practical side, we have corpora with CCG derivations for each sentence (Hockenmaier and Steedman, 2007), a wide-coverage parser trained on that corpus (Clark and Curran, 2007) and a system for converting CCG derivations into semantic representations (Bos et al., 2004).
despite being treated as a single unified grammar formalism, each of these authors use variations of CCG which differ primarily on which combinators are included in the grammar and the restrictions that are put on them.