Unifiability is Complete for co-NLogSpace

Unification between first-order terms is a fundamental problem in mechanical logic [3] and a classic problem in algorithm design. Paterson and Wegman 121 showed that with appropriate choice of data structures for maintaining equivalence relations, the problem of determining whether two terms are unifiable can be solved in time linear in the size of the input. Moreover, the general problem of simultaneously unifying sets of terms is reducible in logarithmic space and linear time to the probiem of unifying a single pair. In this note we shcw that the unification problem is complete for llondeterministic logarithmic space (NLogSpace) with respect to logspace reductions (see [l] for definitions). It follows that the unification problem cannot be solved in logarithmic space (LogSpace), unless NLogSpace = LogSpace. Thus we establish a likely lower bound on the size of the data structures used by any unification algorithm. We begin by showing the upper bound, that is, membership of the set of ununifiable pairs of terms in NLogSpace. The method WC describe might be viewed as a space-efficient nondeterministic implementation of the naive Algorithm A of [2]. For the sake of clarity and explicitness, however, we intrduce some graph-theoretic formalism beyond that given in [2].