How to construct a collision resistant hash function that is not a one-way function? How to construct a CRHF (collision resistant hash function) that is not a OWF (one-way function)? Not sure but I think it probably needs another CRHF?
It is going to be pretty hard to achieve collision resistance without one-wayness. Indeed, negation of one-wayness means that for a given output, you can find a corresponding input. So a collision is easily obtained by simply choosing a random input m, hashing it into output x, then finding a preimage m' for the obtained output x. The only way for such a method not to work is to have an output space at least as large as the input space, so that, with high probability, the input m' is identical to m. However, since hash functions have a much larger input space than output space (e.g. SHA-256 output space has size 2256, but its input space has size 218446744073709551616-1, which is substantially greater), chances are that the m' you get from leveraging the non-one-wayness will be distinct from the m you started with, and that yields a collision.