Relativization makes contradictions harder for Resolution

We provide a number of simplified and improved separations between pairs of Resolution-with-bounded-conjunction refutation systems, Res(d)Res(d), as well as their tree-like versions, Res⁎(d)Res⁎(d). The contradictions we use are natural combinatorial principles: the Least number principle  , LNPnLNPn and an ordered variant thereof, the Induction principle  , IPnIPn.LNPnLNPn is known to be easy for Resolution. We prove that its relativization is hard for Resolution, and more generally, the relativization of LNPnLNPn iterated d   times provides a separation between Res(d)Res(d) and Res(d+1)Res(d+1). We prove the same result for the iterated relativization of IPnIPn, where the tree-like variant Res⁎(d)Res⁎(d) is considered instead of Res(d)Res(d).We go on to provide separations between the parameterized versions of Res(1)Res(1) and Res(2)Res(2). Here we are able again to use the relativization of the LNPnLNPn, but the classical proof breaks down and we are forced to use an alternative. Finally, we separate the parameterized versions of Res⁎(1)Res⁎(1) and Res⁎(2)Res⁎(2). Here, the relativization of IPnIPn will not work as it is, and so we make a vectorizing amendment to it in order to address this shortcoming.