  In this paper, we investigate problems which are dual to the unification
problem, namely the Fixed Point (FP) problem, Common Term (CT) problem and the
Common Equation (CE) problem for string rewriting systems. Our main motivation
is computing fixed points in systems, such as loop invariants in programming
languages. We show that the fixed point (FP) problem is reducible to the common
term problem. Our new results are: (i) the fixed point problem is undecidable
for finite convergent string rewriting systems whereas it is decidable in
polynomial time for finite, convergent and dwindling string rewriting systems,
(ii) the common term problem is undecidable for the class of dwindling string
rewriting systems, and (iii) for the class of finite, monadic and convergent
systems, the common equation problem is decidable in polynomial time but for
the class of dwindling string rewriting systems, common equation problem is
undecidable.
