Intelligent Problem Solvers in Education: Design Method and Applications - Intelligent Systems - page 132

3.2.1 Computational networks with simple valued variables
In this part a simple model of computational networks will be presented together related
problems and techniques for solving them. Although this model is not very complicated, but
it is a very useful tool for designing many knowledge base systems in practice.

Definition 3.3: A computational network (Com-Net) with simple valued variables is a pair (M, F),
in which M = x 1 , x 2 , ..., x n  is a set of variables with simple values (or unstructured values),
and F = f 1 , f 2 , ..., f m  is a set of computational relations over the variables in the set M. Each
computational relation f  F has the following form:

i. An equation over some variables in M, or

ii. Deductive rule f : u(f)  v(f), with u(f)  M, v(f)  M, and there are corresponding

formulas to determine (or to compute) variables in v(f) from variables in u(f).We also
define the set M(f) = u(f)  v(f).

Remark: In many applications equations can be represented as deduction rules.

Problems: Given a computational net (M, F). The popular problem arising from reality
applications is that to find a solution to determine a set H  M from a set G  M. This
problem is denoted by the symbol HG, H is the hypothesis and G is the goal of the
problem. To solve the problem we have to answer two questions below:

Q1: Is the problem solvable based on the knowledge K = (M, F)?

Q2: How to obtain the goal G from the hypothesis H based on the knowledge K = (M, F) in
case the problem is solvable?

Definition 3.4: Given a computational net K = (M, F).

i. For each A  M and f  F, denote f(A) = A  M(f) be the set obtained from A by

applying f. Let S = [f 1 , f 2 , ..., f k ] be a list consisting relations in F, the notation S(A) =
f k (f k-1 (… f 2 (f 1 (A)) … )) is used to denote the set of variables obtained from A by applying
relations in S.

ii. The list S = [f 1 , f 2 , ..., f k ] is called a solution of the problem HG if S(H)  G. Solution S

is called a good solution if there is not a proper sublist S' of S such that S' is also a
solution of the problem. The problem is solvable if there is a solution to solve it.

Definition 3.5: Given a computational net K = (M, F). Let A be a subset of M. It is easy to
verify that there exists a unique set A  M such that the problem A A is solvable; the set
A is called the closure of A.

The following are some algorithms and results that show methods and techniques for
solving the above problems on computational nets.

Theorem 3.1: Given a computational net K = (M, F). The following statements are
equivalent.

i. Pr oblem HG is solvable.
ii. H  G.
iii. There exists a list of relations S such that S(H)  G.