Given a ranked alphabet, context are terms with a single occurrence of a special symbol • (outside of the alphabet), which represents a missing subterm. One can naturally build equations over contexts: the context variables are treated as symbols of arity one and a substitution S assigns to each such a variable a context S(X). A substitution S is extended to terms with context variables in a natural way: S(X(t)) is a context S(X) in which the unique occurrence of • is replaced with S(t). For historical reasons, the satisfiability of context equations is usually referred to as context unification. Context unification generalizes word equations and first-order term unification (which are decidable) and is subsumed by second order unification (which is undecidable) and its decidability status remained open for almost two decades. In this paper I will sketch a PSPACE algorithm for this problem. The idea is to apply simple compression rules (replacing pairs of neighbouring function symbols) to the solution of the context equation; to this end we appropriately modify the equation (without the knowledge of the actual solution) so that compressing the solution can be simulated by compressing parts of the equation. When the compression operations are appropriately chosen, then the size of the instance is polynomial during the whole algorithm, thus giving a PSPACE-upper bound. The best known lower bounds are as for word equations, i.e. NP. The method can be extended to the scenario in which tree-regular constraints for the variables are present, in which case the problem is EXPTIME-complete.
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