Is it possible to make use of a pseudo-random permutation to construct a one-way compression function?
This is possible in the ideal cipher model, and is the basis of the Davies-Meyer and other hashes; see https://en.wikipedia.org/wiki/One-way_compression_function. There has been a lot of work looking at this, but I think a good place to start is the paper Black-Box Analysis of the Block-Cipher-Based Hash-Function Constructions from PGV. It is important, however, to be aware that the ideal cipher model is very strong. For example, AES suffers from related-key attacks and so is not a good basis for such constructions. In addition, due to DES weak keys, some of these constructions are completely broken with DES. However, it does explain the rationale behind the SHA256 and other SHA constructions that essentially construct a type of block cipher and apply a Davies-Meyer type transformation.