One Context Unification Problems Solvable in Polynomial Time

One context unification extends first-order unification by introducing a single context variable, possibly with multiple occurrences. One context unification is known to be in NP, but it is not known to be solvable in polynomial time. In this paper, we present a polynomial time algorithm for certain interesting classes of the one context unification problem. Our algorithm is presented as an inference system that non-trivially extends the usual inference rules for first-order unification. The algorithm is of independent value as it can be used, with slight modifications, to solve other problems, such as the first-order unification problem that tolerates one clash.