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<Chapter Label="thr_elements">
<Heading>Threshold Elements</Heading>

<Section Label="thr_basic">
<Heading>Basic Operations</Heading>

For a given real vector <Alt Only="LaTeX"><M>w=(w_1, \ldots, w_n) \in {\mathbb{R}}^n</M></Alt> and a threshold  <Alt Only="LaTeX"><M>T \in \mathbb{R}</M></Alt>, the <A>threshold element</A> is a function <M>f: \mathbb{Z}_2^n \to \mathbb{Z}_2</M> defined by the following relations:

<Display>
f(x_1,\dots,x_n) = 1, \quad \textrm{if} \quad \sum_{i = 1}^{n} w_i x_i \geq T, \quad \textrm{and} \quad f(x_1,\dots,x_n) = 0 \quad \textrm{otherwise},
</Display>

in which <M>f(x_1,\dots,x_n)</M> is the binary output (valued 0 or 1), each variable <M>x_i</M> is the i-th input (valued 0 or 1), and <M>n</M> is the number of inputs. <P/>

The vector <M>w</M> is the <A>weight</A> vector, and the <M>x=(x_1, \ldots, x_n)</M> is the <A>input</A> vector.
The vector <M>(w_1, \ldots, w_n;T)</M> is called the <A>structure vector</A> (or simply the
<A>structure</A>) of the threshold element.

<ManSection>
<Func Name="ThresholdElement" Arg="Weights, Threshold"/>
<Description>

For the list of rational numbers <C>Weights</C> and the rational <C>Threshold</C>
the function <C>ThresholdElement</C> returns a threshold element with the
number of inputs equal to the length of the <C>Weights</C> list.

<Example>
<![CDATA[
gap> te:=ThresholdElement([1,2],3);
< threshold element with weight vector [ 1, 2 ] and threshold 3 >
gap> Display(te);
Weight vector = [ 1, 2 ], Threshold = 3.
Threshold Element realizes the function f :
Boolean function of 2 variables.
[ 0, 0 ] || 0
[ 0, 1 ] || 0
[ 1, 0 ] || 0
[ 1, 1 ] || 1
Sum of Products:[ 3 ]
]]>
</Example>

The function <C>Display</C> outputs the stucture of the given threshold element <C>ThrEl</C> and
the Sum of Products or Product of Sums representation of the function realized by <C>ThrEl</C>.
For threshold elements of <M>n \leq 4</M> variables it also prints the truth table of the realized Boolean function.

<Example>
<![CDATA[
gap> w:=[1,2,4,-4,6,8,10,-25,6,32];;
gap> T:=60;;
gap> te:=ThresholdElement(w,T);
< threshold element with weight vector [ 1, 2, 4, -4, 6, 8, 10, -25, 6, 32
] and threshold 60 >
gap> Display(te);
Weight vector = [ 1, 2, 4, -4, 6, 8, 10, -25, 6, 32 ], Threshold = 60.
Threshold Element realizes the function f :
Sum of Products:[ 59, 155, 185, 187, 251, 315, 379, 411, 427, 441, 443, 507, 5\
71, 667, 697, 699, 763, 827, 891, 923, 939, 953, 955, 1019 ]
]]>
</Example>
</Description> </ManSection>

<ManSection>
<Func Name="IsThresholdElement" Arg="Obj"/>
<Description>


For the object <C>Obj</C> the function <C>IsThresholdElement</C> returns <C>true</C> if
<C>Obj</C> is a threshold element (see <Ref Func="ThresholdElement" />), and <C>false</C> otherwise.

<Example>
<![CDATA[
gap> te:=ThresholdElement([1,2],3);
< threshold element with weight vector [ 1, 2 ] and threshold 3 >
gap> IsThresholdElement(te);
true
gap> IsThresholdElement([[1,2],3]);
false
]]>
</Example>
</Description> </ManSection>

<ManSection>
<Func Name="OutputOfThresholdElement" Arg="ThrEl"/>
<Description>

For the threshold element <C>ThrEl</C> the function <C>OutputOfThresholdElement</C> returns the
Boolean function, realized by <C>ThrEl</C>.

<Example>
<![CDATA[
gap> te:=ThresholdElement([1,2],3);
< threshold element with weight vector [ 1, 2 ] and threshold 3 >
gap> f:=OutputOfThresholdElement(te);
< Boolean function of 2 variables >
gap> Display(f);
Boolean function of 2 variables.
[ 0, 0 ] || 0
[ 0, 1 ] || 0
[ 1, 0 ] || 0
[ 1, 1 ] || 1
]]>
</Example>
</Description> </ManSection>

<ManSection>
<Func Name="StructureOfThresholdElement" Arg="ThrEl"/>
<Description>


For the threshold element <C>ThrEl</C> the function <C>StructureOfThresholdElement</C> returns the
structure vector [<C>Weights</C>,<C>Threshold</C>] (see <Ref Func="ThresholdElement" />).

<Example>
<![CDATA[
gap> te:=ThresholdElement([1,2],3);
< threshold element with weight vector [ 1, 2 ] and threshold 3 >
gap> sv:=StructureOfThresholdElement(te);
[ [ 1, 2 ], 3 ]
]]>
</Example>
</Description> </ManSection>

<ManSection>
<Func Name="RandomThresholdElement" Arg="NumVar, Lo, Hi"/>
<Description>


For the integers <C>NumVar</C>, <C>Lo</C>, and <C>Hi</C>, the function <C>RandomThresholdElement</C> returns
a threshold element of <C>NumVar</C> variables with a pseudo random integer weight vector and an integer threshold,
where both the weights and the threshold are chosen from the interval [<C>Lo</C>, <C>Hi</C>].

<Example>
<![CDATA[
gap> te:=RandomThresholdElement(4,-10,10);
< threshold element with weight vector [ 7, -8, -6, 10 ] and threshold 2 >
]]>
</Example>
</Description> </ManSection>

<ManSection>
<Func Name="Comparison of Threshold Elements" Arg="ThrEl1, ThrEl2"/>
<Description>


Let <C>ThrEl1</C> and <C>ThrEl2</C> be two threshold elements of the same number of variables,
which realize the following Boolean functions (see <Ref Func="ThresholdElement"/>) <M>f_1</M> and <M>f_2</M>,
resprectively.
By comparison of two threshold elements we mean the comparison of the truth vectors of <M>f_1</M> and <M>f_2</M>
(see <Ref Func="OutputOfThresholdElement"/>).

<Example>
<![CDATA[
gap> te1:=ThresholdElement([1,2],3);;
gap> Print(OutputOfThresholdElement(te1),"\n");
[2, 2,[ 0, 0, 0, 1 ]]
gap> te2:=ThresholdElement([1,2],0);;
gap> Print(OutputOfThresholdElement(te2),"\n");
[2, 2,[ 1, 1, 1, 1 ]]
gap> te3:=ThresholdElement([1,1],2);;
gap> Print(OutputOfThresholdElement(te3),"\n");
[2, 2,[ 0, 0, 0, 1 ]]
gap> te1<te2;
true
gap> te1>te2;
false
gap> te1=te3;
true
]]>
</Example>
</Description> </ManSection>
</Section>

<Section Label="ste_real">
<Heading>Single Threshold Element Realizability</Heading>

One of the most important questions is whether a Boolean function can be realized by a single threshold element (STE). A Boolean function which is realizable by a STE is called a <C>Threshold Function</C>. This section is dedicated to verification of STE-realizability.<P/>

<ManSection>
<Func Name="CharacteristicVectorOfFunction" Arg="Func"/>
<Description>

Let <M>f(x_1,\ldots,x_n)</M> be a Boolean function. We can switch from the {0,1}-base to {-1,1}-base using the following transformation:
<Display>
y_i = 2x_i-1, \quad (i = 1,2,\ldots,n)
</Display>
<Display>
g(y_1,\ldots,y_n) = 2f(x_1,\ldots,x_n)-1.
</Display>
For each <M>i \in \{1,2,\ldots,n\}</M> the <M>i</M>-th column of the truth table of the function <M>g(y_1,\ldots,y_n)</M>
(in {-1,1}-base) we denote by <M>Y_i</M>, and the truth vector of <M>g</M> we denote by <M>G</M>.<P/>

Define the following vector:
<Display>
b = \big(\;Y_1 \cdot G,\; \ldots, \; Y_n \cdot G, \; \textstyle \sum_{i=0}^{2^n-1} g(i) \; \big) \in \mathbb{R}^{n+1},
</Display>
where <M>Y_k \cdot G</M> is the classical inner (scalar) product for each <M>k \in \{1,\ldots,n\}</M>. <P/>
Vector <M>b</M> is called the <A>characteristic vector</A> of the Boolean function <M>f</M>  <Cite Key="Dertouzos65"/>.
Comparing the characteristic vector of the function <M>f</M> with the lists of characteristic vectors of all STE-realizable functions
we obtain the answer wheter <M>f</M> is realizable by STE or not. In <Package>Thelma</Package> package we have a database of all
such vectors for STE-realizable functions of <M>n \leq 6</M> variables obtained from <Cite Key="Dertouzos65"/>.

For the Boolean function <C>Func</C> the function <C>CharacteristicVectorOfFunction</C> returns a characteristic vector.
There are no limitations on the cardinality of <C>Func</C>, but the database of STE-realizable functions is given only for <M>n \leq 6</M> variables.

<Example>
<![CDATA[
gap> f:=LogicFunction(2,2,[0,0,0,1]);
< Boolean function of 2 variables >
gap> CharacteristicVectorOfFunction(f);
[ 2, 2, 2 ]
]]>
</Example>

</Description> </ManSection>

<ManSection>
<Func Name="IsCharacteristicVectorOfSTE" Arg="ChVect"/>
<Description>


For the characteristic vector <C>ChVect</C> (see <Ref Func="CharacteristicVectorOfFunction"/>) the function <C>IsCharacteristicVectorOfSTE</C>
returns <C>true</C> if <C>ChVect</C> is a characteristic vector of some STE-realizable Boolean function, and <C>false</C> otherwise.
Note, that this function is implemented only for characteristic vectors of length not bigger than 7.

<Example>
<![CDATA[
gap> f:=LogicFunction(2,2,[0,0,0,1]);
< Boolean function of 2 variables >
gap> c:=CharacteristicVectorOfFunction(f);
[ 2, 2, 2 ]
gap> IsCharacteristicVectorOfSTE(c);
true
]]>
</Example>
</Description> </ManSection>

<ManSection>
<Func Name="IsInverseInKernel" Arg="Func"/>
<Description>


Let <M>f(x_1,\ldots,x_n)</M> be a Boolean function with the kernel <M>K(f)</M>. The function <C>IsInverseInKernel</C> returns <C>true</C> if there is a pair of additive inverse vectors in <M>K(f)</M> (this means that <M>f</M> is not STE-realizable, see <Cite Key="GecheRobotyshyn83"/>) or <C>false</C> otherwise.
Note that this function also accepts the kernel of the Boolean function <C>Func</C> as an input.

A vector <M>b \in \mathbb{Z}_2^n</M> is called an additive inverse to <M>a \in \mathbb{Z}_2^n</M> if <M>a \oplus b = 0</M>. <P/>

<Example>
<![CDATA[
gap> f:=LogicFunction(3,2,[0,1,0,1,0,1,1,0]);;
gap> k:=KernelOfBooleanFunction(f);;
gap> Display(k[1]);
. . 1
. 1 1
1 . 1
1 1 .
gap> IsInverseInKernel(f);
true
]]>
</Example>
</Description> </ManSection>

<ManSection>
<Func Name="IsKernelContainingPrecedingVectors" Arg="Func"/>
<Description>


A vector <M>a=(\alpha_1,\ldots,\alpha_n) \in \mathbb{Z}_2^n</M> precedes a vector <M>b=(\beta_1,\ldots,\beta_n) \in \mathbb{Z}_2^n</M>  (we denote it as <M>a \prec b</M>) if <M>\alpha_i \leq \beta_i \textrm{ for each } i=1,\ldots, n</M>. <P/>
For a given vector <M>c \in \mathbb{Z}_2^n</M> denote <M>M_c=\{\;a\in \mathbb{Z}_2^n \;\mid\; a \prec c \;\}</M>. <P/>
Let <M>f(x_1,\ldots,x_n)</M> be a Boolean function with reduced kernel <M>T(f)=\{K(f)_j \mid j=1,2,\ldots, m \}</M>. If <M>f</M> is implemented by a single threshold element (STE), then there exists  <M>j \in \{1,\ldots, m \}</M> such that
<Display>
\forall a \in K(f)_j \qquad \textrm{holds} \qquad M_a \subseteq K(f)_j.
</Display>

The function <C>IsKernelContainingPrecedingVectors</C> returns <C>false</C> for a given function <C>Func</C> if <M>Func</M> is not realizable by a single threshold element (see <Cite Key="GecheMulesa2017"/>).
Note that this function also accepts the kernel of the Boolean function <C>Func</C> as an input.

<Example>
<![CDATA[
gap> ##Continuation of the previous example
gap> IsKernelContainingPrecedingVectors(f);
false
]]>
</Example>
</Description> </ManSection>

<ManSection>

<Func Name="IsRKernelBiggerOfCombSum" Arg="Func"/>
<Description>


Let <M>f(x_1,\ldots,x_n)</M> be a Boolean function with reduced kernel <M>T(f)</M>.
Denote
<Display>
  k_i^* = \max\big\{ \; \|a\| = \textstyle \sum_{j=1}^m a_j \; \mid \; a = (a_1, \ldots, a_m) \in T(f) \; \big\}, \quad (i=1,\ldots,n)
</Display>
and
<Display>
  k_A^*=\min\big\{\;k_i^* \; \mid \; i=1, 2, \ldots,n \; \big\}.
</Display>

If <M>f</M> is implemented by a single threshold element (STE), then the following condition holds:
<Display>
|A| \geq \sum_{i=0}^{k_A^*} {{k_A^*}\choose{i}},
</Display>
where <M>{{k_A^*}\choose{i}}</M> is the classical binomial coefficient and <M>|A|</M> is the cardinality of <M>A</M>. <P/>

For a given Boolean function <C>Func</C> the function <C>IsRKernelBiggerOfCombSum</C> returns <C>false</C>
if this function is not STE-realizable (see <Cite Key="GecheMulesa2017"/>). Note that this function also accepts
the reduced kernel of the Boolean function <C>Func</C> as an input.

<Example>
<![CDATA[
gap> f:=LogicFunction(2,2,[0,1,1,0]);
< Boolean function of 2 variables >
gap> IsRKernelBiggerOfCombSum(f);
false
]]>
</Example>
</Description> </ManSection>

<ManSection>

<Func Name="BooleanFunctionBySTE" Arg="Func"/>
<Description>


For a given Boolean function <C>Func</C> the function <C>BooleanFunctionBySTE</C> determines whether <C>Func</C> is realizable by a single threshold element (STE).
The function returns a threshold element with integer weights and integer threshold.
If <C>Func</C> is not realizable by STE, it returns an empty list [].
The realization of the function <C>BooleanFunctionBySTE</C> is based on algorithms, proposed in <Cite Key="Geche2010"/>.

<Example>
<![CDATA[
gap> f:=LogicFunction(3,2,[1,1,0,0,1,0,0,0]);
< Boolean function of 3 variables >
gap> te:=BooleanFunctionBySTE(f);
< threshold element with weight vector [ -1, -4, -2 ] and threshold -2 >
gap> Display(te);
Weight vector = [ -1, -4, -2 ], Threshold = -2.
Threshold Element realizes the function f :
Boolean function of 3 variables.
[ 0, 0, 0 ] || 1
[ 0, 0, 1 ] || 1
[ 0, 1, 0 ] || 0
[ 0, 1, 1 ] || 0
[ 1, 0, 0 ] || 1
[ 1, 0, 1 ] || 0
[ 1, 1, 0 ] || 0
[ 1, 1, 1 ] || 0
Sum of Products:[ 0, 1, 4 ]
gap> f:=LogicFunction(2,2,[0,1,1,0]);
< Boolean function of 2 variables >
gap> te:=BooleanFunctionBySTE(f);
[  ]
]]>
</Example>
</Description> </ManSection>

<ManSection>

<Func Name="PDBooleanFunctionBySTE" Arg="Func"/>
<Description>


Let <M>f(x_1,\ldots,x_n)</M> be a partially defined Boolean function.
We denote by <C>x</C> the positions in truth vector, where <M>f</M> is undefined.
Then <M>f^{-1}</M><C>(x)</C> is the set of Boolean vectors of <M>n</M> variables on which the function is undefined.
The sets <M>f^{-1}(0)</M> and <M>f^{-1}(1)</M> are defined in <Ref Func="KernelOfBooleanFunction" />.
The function <M>f</M> is called a <A>threshold function</A> if there is an <M>n</M>-dimensional real vector
<M>w=(w_1,\ldots,w_n)</M> and a real threshold <M>T</M> such that
<Display>
a \in f^{-1}(1) \quad \Longrightarrow \quad a\cdot w^T \geq T,
</Display>
<Display>
a \in f^{-1}(0)\quad \Longrightarrow \quad a\cdot w^T < T,
</Display>
where <M>a\cdot w^T</M> is the classical inner (scalar) product. <P/>
For the partially defined Boolean function <C>Func</C> (presented as a string, where <C>x</C> presents the undefined values)
the function <C>PDBooleanFunctionBySTE</C> returns a threshold element if <C>Func</C> can be realized by STE and
empty list otherwise.
The realization of the function <C>PDBooleanFunctionBySTE</C> is based on the algorithm, proposed in <Cite Key="GecheRobotyshyn83"/>.
<Example>
<![CDATA[
gap> f:="1x001x0x";
"1x001x0x"
gap> te:=PDBooleanFunctionBySTE(f);
< threshold element with weight vector [ -1, -2, -3 ] and threshold -1 >
gap> Display(te);
Weight vector = [ -1, -2, -3 ], Threshold = -1.
Threshold Element realizes the function f :
Boolean function of 3 variables.
[ 0, 0, 0 ] || 1
[ 0, 0, 1 ] || 0
[ 0, 1, 0 ] || 0
[ 0, 1, 1 ] || 0
[ 1, 0, 0 ] || 1
[ 1, 0, 1 ] || 0
[ 1, 1, 0 ] || 0
[ 1, 1, 1 ] || 0
Sum of Products:[ 0, 4 ]
]]>
</Example>
</Description> </ManSection>

</Section>

<Section Label="thr_iter">
<Heading>Iterative Training Methods</Heading>

<Package>Thelma</Package> also provides a few iterative methods for threshold element training.

<ManSection>
<Func Name="ThresholdElementTraining" Arg="ThrEl, Step, Func, Max_Iter"/>
<Description>


This is a basic iterative method for the perceptron training <Cite Key="Rosenblatt58"/>.
For the threshold element <C>ThrEl</C> (which is an arbitrary threshold
element for the first iteration), the positive integer <C>Step</C> (the value on which we change parameters
while training the threshold element), the Boolean function <C>Func</C> and the positive integer <C>Max_Iter</C> -
the maximal number of iterations, the function <C>ThresholdElementTraining</C> returns a threshold element, realizing <C>Func</C>
(if such threshold element exists).

<Example>
<![CDATA[
gap> f:=LogicFunction(2,2,[0,0,0,1]);
< Boolean function of 2 variables >
gap> te1:=RandomThresholdElement(2,-2,2);
< threshold element with weight vector [ 0, -1 ] and threshold 0 >
gap> Display(OutputOfThresholdElement(te1));
Boolean function of 2 variables.
[ 0, 0 ] || 1
[ 0, 1 ] || 0
[ 1, 0 ] || 1
[ 1, 1 ] || 0
gap> te2:=ThresholdElementTraining(te1,1,f,100);
< threshold element with weight vector [ 2, 1 ] and threshold 3 >
gap> Display(OutputOfThresholdElement(te2));
Boolean function of 2 variables.
[ 0, 0 ] || 0
[ 0, 1 ] || 0
[ 1, 0 ] || 0
[ 1, 1 ] || 1
]]>
</Example>
</Description> </ManSection>

<ManSection>
<Func Name="ThresholdElementBatchTraining" Arg="ThrEl, Step, Func, Max_Iter"/>
<Description>

For the threshold element <C>ThrEl</C> (which is an arbitrary threshold
element for the first iteration), the positive integer <C>Step</C> (the value on which we change parameters
while training the threshold element), the Boolean function <C>Func</C>,
and the positive integer <C>Max_Iter</C> - the maximal number of iterations,
the function <C>ThresholdElementTraining</C> returns a threshold element, realizing <C>Func</C>
(if such threshold element exists) via batch training.

<Example>
<![CDATA[
gap> f:=LogicFunction(2,2,[0,0,0,1]);
< Boolean function of 2 variables >
gap> te1:=RandomThresholdElement(2,-2,2);
< threshold element with weight vector [ 0, 2 ] and threshold 2 >
gap> Display(OutputOfThresholdElement(te1));
Boolean function of 2 variables.
[ 0, 0 ] || 0
[ 0, 1 ] || 1
[ 1, 0 ] || 0
[ 1, 1 ] || 1
gap> te2:=ThresholdElementBatchTraining(te1,1,f,100);
< threshold element with weight vector [ 2, 2 ] and threshold 3 >
gap> Display(OutputOfThresholdElement(te2));
Boolean function of 2 variables.
[ 0, 0 ] || 0
[ 0, 1 ] || 0
[ 1, 0 ] || 0
[ 1, 1 ] || 1
]]>
</Example>
</Description> </ManSection>

<ManSection>
<Func Name="WinnowAlgorithm" Arg="Func, Step, Max_Iter"/>
<Description>

A Boolean function <M>f:\mathbb{Z}_2^n \to \mathbb{Z}_2</M> which can be presented in the following form:
<Display>
f(x_1,\ldots,x_n)=x_{i_1} \vee \cdots \vee x_{i_k}, \qquad (k \leq n)
</Display>
is called a <A>monotone disjunction</A>, i.e. it is a disjunction in which no variable
appears negated. <P/>

If the given Boolean function <M>f</M> is a monotone disjunction, the <A>Winnow algorithm</A>
is more efficient than the classical Perceptron training algorithm <Cite Key="Littlestone88"/>. <P/>

For the Boolean function <C>Func</C>, which is a monotone disjunction, <C>WinnowAlgorithm</C> returns either a threshold element
realizing <C>Func</C> or [] if <C>Func</C> is not trainable by <C>WinnowAlgorithm</C>. The positive ingetger <C>Step</C> which is not equal to 1 defines the value on which we change parameters
while running the algorithm and the positive integer <C>Max_Iter</C> defines the maximal number of iterations.

<Example>
<![CDATA[
gap> x:=Indeterminate(GF(2),"x");;
gap> y:=Indeterminate(GF(2),"y");;
gap> pol:=x*y+x+y;;
gap> f:=PolynomialToBooleanFunction(pol,2);
< Boolean function of 2 variables >
gap> Display(f);
Boolean function of 2 variables.
[ 0, 0 ] || 0
[ 0, 1 ] || 1
[ 1, 0 ] || 1
[ 1, 1 ] || 1
gap> te:=WinnowAlgorithm(f,2,100);
< threshold element with weight vector [ 1, 1 ] and threshold 1 >
gap> Display(OutputOfThresholdElement(te));
Boolean function of 2 variables.
[ 0, 0 ] || 0
[ 0, 1 ] || 1
[ 1, 0 ] || 1
[ 1, 1 ] || 1
]]>
</Example>
</Description> </ManSection>

<ManSection>
<Func Name="Winnow2Algorithm" Arg="Func, Step, Max_Iter"/>
<Description>


For any <M>X \subseteq \mathbb{Z}_2^n</M> and for any <M>\delta</M> satisfying
<M>0 < \delta \leq 1</M> let <M>F(X,\delta)</M> be the class of functions from
<M>X</M> to <M>\mathbb{Z}_2^n</M>. Assume that <M>F(X,\delta)</M> satisfies the following condition:<P/>
for each <M>f \in F(X,\delta)</M> there exist <M>\mu_1,\ldots,\mu_n \geq 0</M>
such that for all <M>(x_1,\ldots,x_n) \in X</M>
<Display>
\textstyle  \sum_{i=1}^n \mu_i x_i \geq 1, \quad \textrm{if} \quad f(x_1,\ldots,x_n)=1
</Display>
and
<Display>
\textstyle  \sum_{i=1}^n \mu_i x_i \leq 1, \quad \textrm{if} \quad f(x_1,\ldots,x_n)=0.
</Display>
In other words, the inverse images of 0 and 1 are linearly separable with a minimum separation
that depends on <M>\delta</M>. <C>Winnow2</C> algorithm is designed for training this class of
the Boolean functions <Cite Key="Littlestone88"/>.<P/>

For the Boolean function <C>Func</C> from the class of Boolean functions which is described above,
the function <C>Winnow2Algorithm</C> returns either a threshold element which realizes <C>Func</C> or [] if <C>Func</C> is not trainable by <C>Winnow2Algorithm</C>.
The positive integer <C>Step</C> which is not equal to 1
defines the value on which we change parameters while running the algorithm.

<Example>
<![CDATA[
gap> ##Conjunction can not be trained by Winnow algorithm.
gap> x:=Indeterminate(GF(2),"x");;
gap> y:=Indeterminate(GF(2),"y");;
gap> pol:=x*y;;
gap> f:=PolynomialToBooleanFunction(pol,2);
< Boolean function of 2 variables >
gap> te:=WinnowAlgorithm(f,2,100);
[  ]
gap> ## But in the case of Winnow2 we can obtain the desirable result.
gap> te:=Winnow2Algorithm(f,2,100);
< threshold element with weight vector [ 1/2, 1/2 ] and threshold 1 >
gap> Display(te);
Weight vector = [ 1/2, 1/2 ], Threshold = 1.
Threshold Element realizes the function f :
Boolean function of 2 variables.
[ 0, 0 ] || 0
[ 0, 1 ] || 0
[ 1, 0 ] || 0
[ 1, 1 ] || 1
Sum of Products:[ 3 ]
]]>
</Example>
</Description> </ManSection>

<ManSection>
<Func Name="STESynthesis" Arg="Func"/>
<Description>


The function <C>STESynthesis</C> is based on the algorithm proposed in <Cite Key="Dertouzos65"/>.
In each iteration we perturb an <M>n+1</M>-dimensional weight-threshold vector in such manner that
the distance between the given vector and a desired weight-threshold vector, if such vector exists, is reduced.
So if the Boolean function <C>Func</C> is STE-realizable, then this procedure will eventually yield an acceptable
weight-threshold vector. Otherwise iteration process will eventually enter a limit cycle and the execution
of <C>STE_Synthesis</C> will be stopped. <P/>

For the Boolean function <C>Func</C> the function <C>STESynthesis</C> returns a threshold element if
<C>Func</C> is STE-realizable or an empty list otherwise.<P/>

<Example>
<![CDATA[
gap> f:=x*y+x+y;;
gap> x:=Indeterminate(GF(2),"x");;
gap> y:=Indeterminate(GF(2),"y");;
gap> pol:=x*y+x+y;;
gap> f:=PolynomialToBooleanFunction(pol,2);;
gap> te:=STESynthesis(f);
< threshold element with weight vector [ 2, 2 ] and threshold 1 >
gap> Display(te);
Weight vector = [ 2, 2 ], Threshold = 1.
Threshold Element realizes the function f :
Boolean function of 2 variables.
[ 0, 0 ] || 0
[ 0, 1 ] || 1
[ 1, 0 ] || 1
[ 1, 1 ] || 1
Product of Sums:[ 0 ]
]]>
</Example>
</Description> </ManSection>

</Section>


</Chapter>

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