We present polynomial-time algorithms for the uniform word problem and for the generator problem for lattices. The algorithms are derived from novel, prooftheoretic approaches. We prove that both problems are log-space complete for P, but can be solved in deterministic logarithmic space in the case of free lattices. We also show that the more general problem of testing whether a given open sentence is true in all lattices is co-NP complete. © 1988.