\item The hypo function

One experimental algorithm for hypo would be defined as follows:
{\small
\begin{verbatim}
--------------------------------------------------------------------
function hypo(relationship, background, predicates) 
         returns a set of Horn clauses
  inputs: relationship, the given relationship, e.g. (uncle ?x ?y)
          background,   the given background knowledge
          predicates,   the related predicates, e.g. parent, male, female
begin
  binary-subset = get all the binary relationship from the background
  unary-subset  = get all the unary relationship from the background
  relation-set  = get all the relation pairs from the background
  allpath = get all path which was related with relationship 
  maximumset = get all the relationship set
  generalizableset = substitute the variable to the maximumset eliminate 
                     the duplicates
  return generalizableset 
end
--------------------------------------------------------------------
\end{verbatim}}
The {\bf hypo} can propose a most specific list of (conjunction of) conditions 
for a given {\sl relationship}, based upon the content of a given list
of {\sl background} descriptions in terms of the given list of {\sl predicates}.
The algorithm should do this by finding a maximal set of common, generalizable 
set of conditions in the background that is common to
all individual terms seen to share the given relationship. 

\item The generalize condition function

The generalize condition function will substitute the variables to the maximumset which 
eliminate the duplicates. If the input is as follows:
\begin{verbatim}
((grandfather philip william)
  (parent charles william)
  (parent philip charles)
  (male charles)
  (male william)
  (male philip))
\end{verbatim}
The output will be as follows:
\begin{verbatim}
(parent ?v1 ?y) (parent ?x ?v1) (male ?v1) (male ?y) (male ?x))
\end{verbatim}


\item Results

What is the result of 

(hypo '(uncle ?x ?y) background '(parent male female))? 

It would be nice if it were something like 

((parent ?v1 ?y) (parent ?v2 ?v1) (parent ?v2 ?x) (male ?x))



\item Kinship relationships.

The background is a complete set of kinship relationships for a particular 
set of people that is sufficient to illustrate the following relationships 
between people: parent, son, daughter, sister, brother, uncle, aunt, cousin, 
grandparent, and grandchild; information regarding the unary predicates male 
and female is included. 

\item Extra Credit Part

Theorically, it is possible to combine a set of rule conditions 
to simplify the background, using one rule conclusion as a rule condition for another. 
For example, if we have the background:
$$ father(a,b) \wedge father(b,c) \wedge father(c,d) $$
then by combine a set of rule conditions: 
$$ father(a,b) \wedge father(b,c) \Rightarrow grandfather(a,c) $$
we can simplify the background to as follows:
$$ grandfather(a,c) \wedge father(c,d) $$