Artificial Intelligence and Soft Computing: Behavioral ... - Arteimi.info

Here ai , b, ci , and xi ∈ ℜ and one has to determine xi s for known constants
ai and b, so that Z=∑ci xi is maximized. The search space of the solution xi s,
here, is constrained by the set of linear inequalities. Making a change for a
10$ note is also a CSP problem, where the constraints for the solution xi s
are given by ∑ ci xi = 10$ and ∃xi ≥ 0, where xi denotes the valid notes 1$, 2$
and 5$ and ci denotes the count of xi. For example, if we consider x1,x2 and x5
to correspond to 1$, 2$ and 5$ notes and c1,c2, c5 be their respective counts,
then a few possible solutions are {x5 =2} or {x5= 1 ,x2 =2,x1= 1} or {x2=5} or
{x1 =10}or {x2= 4,x1= 2}. The last two examples involve the algebraic
constraints. Many formulations of the problems, however, require satisfying
logical or geometric constraints. For example, finding the grand-fatherhood
relationship between X3 (=d) and X1 (=l) from the definition of grandfather
and father, discussed below, is a CSP. To be more specific, suppose, given the
following set of constraints:
1.Grandfather (X3,X1) :-
Father (X2,,X1),
Father (X3,X2).
2. Father (d, r).
3. Father (r, l).
We want to determine: Grandfather (X3, X1) ?
The above problem can be solved by constructing an SLD-resolution
tree, which finally yields the solution: Grand-father(d, l). Problems that deal
with such logical constraints also belong to the category of CSP. Before
switching to the next section, we present an example of geometric constraint
satisfaction problem. Suppose, we want a set of three dimensional objects
like a i) Box , ii) Pyramid, iii) Cone (see fig.19.1) and we recognize the box
among them by a set of geometric constraints, presented below.
1. (Object = Box) :-
Has-no-of- vertices = 8, No-of-planes-meeting-at-each-vertex =3,
Angle-between-any- two-planes-meeting- long-an-edge =90 0 .
2. (Object = Pyramid) :-
Has-no-of- vertices =4,
No-of-planes- meeting- at-each-vertex ≥ 3,
Angle-between-any-two-planes-meeting-along-an-edge