You can trim/cut off the affected fronds and it should be OK - just don't cut off all of them. The more intact leaves you leave on, the faster the palm will recuperate.

I guess it depends on what you mean by "chewed in half" -- if it was just the foliage, and the meristem (growing point) down inside the trunk is intact, it's fine, it will pop out new growth from the center.

If the trunk itself was chewed, it just depends. A palm actually has sort of the same structure as an onion, where the meristem sits atop the a "basal plate" type structure. If the dog chewed it above this point, it will grow, below, it's toast.

Thank you so much for the comments, after he chewed it there is about 2' left above ground. I will see what happens.

Well, that trunk is probably dead now, but if there were multiple palms in the group, they should grow back. Some P roebellenii are suckering and will grow back anyway. Some are not and chewing off the head will kill the palm.