We propose a new algebraic four-level expression called k-EXOR-projected sum of products (kEP-SOP). The optimization of a kEP-SOP is NPNP-hard, but can be approximated within a fixed performance guarantee in polynomial time. Moreover, fully testable circuits under the stuck-at-fault model can be derived from kEP-SOPs by adding at most a constant number of multiplexer gates. The experiments show that the computational time is very short and the results are most of the time optimal with respect to the number of products involved. kEP-SOPs also prove experimentally a good starting point for general multilevel logic synthesis.