203
Logic Programming and Artificial Neural Networks
1. Approximate the embedded T P -operator using constant pieces. As before, we start by constructing P n for a given level n. After embedding
the approximating operator T Pn , we find that the resulting function is
a piecewise constant function. Due to the finiteness of the resulting program, we obtain the greatest relevant input level by taking the maximal
level of an atom occurring in any of the bodies. Since no atom of a
greater level influences the result of the T P -operator, we see that it is a
piecewise constant function.
2. Approximating the embedded T P -operator using threshold functions. Obviously, every piecewise constant function R → R can be represented as
a sum of (parametrized) threshold functions. To approximate the embedded T P -operator of Program 7.5.1 up to level 3, we need the three
functions: H
0.016
0.042 , H
−0.078
0.016
y
0.167 , H 0.292 , where H p (x) := y · H(x − p) denotes
an h-step at position p.
3. Approximating the embedded T P -operator using sigmoidal functions. To
enable the construction of sigmoidal networks, we need to replace the
threshold functions with sigmoidal functions. This can be done because
(a) we are only interested in the approximation of embedded interpretations, and (b) we can place the threshold functions so that the jumps
are located between two embedded interpretations. First, we construct
the threshold approximation not for the greatest relevant input level
n as introduced earlier, but up to level n + 1. Every approximation of
this function up to ε
' := b
−(n+1) results in a sufficient approximation
of the embedded T P -operator. Under these conditions, we can replace
the threshold functions by appropriately set up sigmoidal functions. We
just need to make sure that the sigmoidal functions approximate the
threshold functions on all embedded interpretations up to ε
' . For the
example of Program 7.5.1, see also Example 7.5.7, we obtain the following sigmoidal functions: S
0.016
0.167,53.864 , S
0.016
where
0.042,135.994 , S
−0.078
0.292,135.994 ,
S
h
h
(x) :=
p,s
1+e −s(x−p) .
4. Approximating the embedded T P -operator using a sigmoidal network.
The approximating sigmoidal functions constructed in Step 3 can easily
be embedded into a standard 3-layer sigmoidal network as follows: the
input and output layer contain exactly one unit computing the identity
function. The hidden layer contains a sigmoidal unit for every sigmoidal
function constructed in Step 3. The weights from input to hidden layer
are set up such that they represent the steepness of the constructed sigmoidal. The thresholds of the hidden layer correspond to the locations
of the sigmoidal functions, and the weights from hidden to output layer
coincide with the step width of the underlying threshold functions.
Figure 7.9 shows the resulting network for ε = 0.04 corresponding to an
approximation of the T P -operator up to level 3.
Logic Programming and Artificial Neural Networks
1. Approximate the embedded T P -operator using constant pieces. As before, we start by constructing P n for a given level n. After embedding
the approximating operator T Pn , we find that the resulting function is
a piecewise constant function. Due to the finiteness of the resulting program, we obtain the greatest relevant input level by taking the maximal
level of an atom occurring in any of the bodies. Since no atom of a
greater level influences the result of the T P -operator, we see that it is a
piecewise constant function.
2. Approximating the embedded T P -operator using threshold functions. Obviously, every piecewise constant function R → R can be represented as
a sum of (parametrized) threshold functions. To approximate the embedded T P -operator of Program 7.5.1 up to level 3, we need the three
functions: H
0.016
0.042 , H
−0.078
0.016
y
0.167 , H 0.292 , where H p (x) := y · H(x − p) denotes
an h-step at position p.
3. Approximating the embedded T P -operator using sigmoidal functions. To
enable the construction of sigmoidal networks, we need to replace the
threshold functions with sigmoidal functions. This can be done because
(a) we are only interested in the approximation of embedded interpretations, and (b) we can place the threshold functions so that the jumps
are located between two embedded interpretations. First, we construct
the threshold approximation not for the greatest relevant input level
n as introduced earlier, but up to level n + 1. Every approximation of
this function up to ε
' := b
−(n+1) results in a sufficient approximation
of the embedded T P -operator. Under these conditions, we can replace
the threshold functions by appropriately set up sigmoidal functions. We
just need to make sure that the sigmoidal functions approximate the
threshold functions on all embedded interpretations up to ε
' . For the
example of Program 7.5.1, see also Example 7.5.7, we obtain the following sigmoidal functions: S
0.016
0.167,53.864 , S
0.016
where
0.042,135.994 , S
−0.078
0.292,135.994 ,
S
h
h
(x) :=
p,s
1+e −s(x−p) .
4. Approximating the embedded T P -operator using a sigmoidal network.
The approximating sigmoidal functions constructed in Step 3 can easily
be embedded into a standard 3-layer sigmoidal network as follows: the
input and output layer contain exactly one unit computing the identity
function. The hidden layer contains a sigmoidal unit for every sigmoidal
function constructed in Step 3. The weights from input to hidden layer
are set up such that they represent the steepness of the constructed sigmoidal. The thresholds of the hidden layer correspond to the locations
of the sigmoidal functions, and the weights from hidden to output layer
coincide with the step width of the underlying threshold functions.
Figure 7.9 shows the resulting network for ε = 0.04 corresponding to an
approximation of the T P -operator up to level 3.