We have two polynomial time results for the uniform word problem for a quasivariety Q: The uniform word problem for Q can be solved in polynomial time iff one can find a certain congruence on finite partial algebras in polynomial time. Let Q* be the relational class determined by Q. If any universal Horn class between the universal closure S and the weak embedding closure S̄ of Q* is finitely axiomatizable then the uniform word problem for Q is solvable in (...) polynomial time. This covers Skolem's 1920 solution to the uniform word problem for lattices and Evans' 1953 applications of the weak embeddability property for finite partial V algebras. (shrink)