  An efficient and flexible engine for computing fixed points is critical for
many practical applications. In this paper, we firstly present a goal-directed
fixed point computation strategy in the logic programming paradigm. The
strategy adopts a tabled resolution (or memorized resolution) to mimic the
efficient semi-naive bottom-up computation. Its main idea is to dynamically
identify and record those clauses that will lead to recursive variant calls,
and then repetitively apply those alternatives incrementally until the fixed
point is reached. Secondly, there are many situations in which a fixed point
contains a large number or even infinite number of solutions. In these cases, a
fixed point computation engine may not be efficient enough or feasible at all.
We present a mode-declaration scheme which provides the capabilities to reduce
a fixed point from a big solution set to a preferred small one, or from an
infeasible infinite set to a finite one. The mode declaration scheme can be
characterized as a meta-level operation over the original fixed point. We show
the correctness of the mode declaration scheme. Thirdly, the mode-declaration
scheme provides a new declarative method for dynamic programming, which is
typically used for solving optimization problems. There is no need to define
the value of an optimal solution recursively, instead, defining a general
solution suffices. The optimal value as well as its corresponding concrete
solution can be derived implicitly and automatically using a mode-directed
fixed point computation engine. Finally, this fixed point computation engine
has been successfully implemented in a commercial Prolog system. Experimental
results are shown to indicate that the mode declaration improves both time and
space performances in solving dynamic programming problems.
