Decidable and Undecidable Second-Order Unification Problems

There is a close relationship between word unification and second-order unification. This similarity has been exploited for instance for proving decidability of monadic second-order unification. Word unification can be easily decided by transformation rules (similar to the ones applied in higher-order unification procedures) when variables are restricted to occur at most twice. Hence a well-known open question was the decidability of second-order unification under this same restriction. Here we answer this question negatively by reducing simultaneous rigid E-unification to second-order unification. This reduction, together with an inverse reduction found by Degtyarev and Voronkov, states an equivalence relationship between both unification problems. Our reduction is in some sense reversible, providing decidability results for cases when simultaneous rigid E-unification is decidable. This happens, for example, for one-variable problems where the variable occurs at most twice (because rigid E-unification is decidable for just one equation). We also prove decidability when no variable occurs more than once, hence significantly narrowing the gap between decidable and undecidable second-order unification problems with variable occurrence restrictions.