Halliday & Matthiessen (1999: 174):
The circumstance of Accompaniment does not correspond to any one particular participant rôle. Rather, it corresponds to an extending of the participant itself, by addition or variation: John came with Mary is agnate to both John and Mary came; Mary came without John is agnate to Mary but not John came; Mary came instead of John is agnate to not John but Mary came. Grammatically, the analogous type of participant is one represented by a nominal group complex.