Machine Learning - DISCo

Given:
rn Instance space X: Each instance describes a pair of objects represented by the predicates Type,
Color, Volume, Owner, Material, Density, and On.
rn Hypothesis space H: Each hypothesis is a set of Horn clause rules. The head of each Horn
clause is a literal containing the target predicate SafeToStack. The body of each Horn clause
is a conjunction of literals based on the same predicates used to describe the instances, as
well as the predicates LessThan, Equal, GreaterThan, and the functions plus, minus, and
times. For example, the following Horn clause is in the hypothesis space:
Sa f eToStack(x, y) t Volume(x, vx) r\ Volurne(y, vy) A LessThan(vx, vy)
rn Target concept: SafeToStack(x,y)
rn Training Examples: A typical positive example, SafeToStack(Obj1, ObjZ), is shown below:
On(Objl.Obj2)
Owner(0bj I, Fred)
Type(0bj I, Box)
Owner(Obj2, Louise)
Type(Obj2, Endtable) Density(0bj 1 ,0.3)
Color(Obj1, Red)
Material(Obj1, Cardboard)
Color(Obj2, Blue)
Material (Obj2, Wood)
Volume(Objl,2)
Domain Theory B:
SafeToStack(x, y) c -Fragile(y)
SafeToStack(x, y) c Lighter(x, y)
Lighter@, y) c Weight(x, wx) A Weight(y, wy) r\ LessThan(wx, wy)
Weight(x, w) c Volume(x, v) A Density(x,d) A Equal(w, times(v, d))
Weight(x, 5) c Type(x, Endtable)
Fragile(x) c Material (x, Glass)
Determine:
rn A hypothesis from H consistent with the training examples and domain theory.
TABLE 11.1
An analytical learning problem: SafeToStack(x,y).
As shown in Table 11.1, we have chosen a hypothesis space H in which
each hypothesis is a set of first-order if-then rules, or Horn clauses (throughout
this chapter we follow the notation and terminology for first-order Horn clauses
summarized in Table 10.3). For instance, the example Horn clause hypothesis
shown in the table asserts that it is SafeToStack any object x on any object y, if
the Volume of x is LessThan the Volume of y (in this Horn clause the variables
vx and vy represent the volumes of x and y, respectively). Note the Horn clause
hypothesis can refer to any of the predicates used to describe the instances, as well
as several additional predicates and functions. A typical positive training example,
SafeToStack(Obj1, Obj2), is also shown in the table.
To formulate this task as an analytical learning problem we must also provide
a domain theory sufficient to explain why observed positive examples satisfy the
target concept. In our earlier chess example, the domain theory corresponded to
knowledge of the legal moves in chess, from which we constructed explanations