Let \(F=GF(p^k)\) be a Galois field with primitive element \(\varepsilon\) and let \(F^*\) be the multiplicative group of \(F\). Let us consider all proper subgroups of \(F^*\):
\[ C_{j_1}= \langle \sigma_1 \mid \sigma_1^{j_1}=1 \rangle, \ldots , C_{j_t}= \langle \sigma_t \mid \sigma_n^{j_t}=1 \rangle, \]
where \(\sigma_i=\epsilon^{\frac{|F^*|}{j_i}}\) (\(i=1,\ldots,t\))are the generators of the corresponding cyclic groups. Denote \(T=\{ j_1, \ldots, j_t\} \). Let \(G_n = C_{k_1}\otimes \ldots \otimes C_{k_n}\) be a direct product of cyclic groups \(C_{k_i}\) (\(k_i \in T\),\(i=1,\ldots,n\)).
A discrete function of \(n\) variables over a finite field \(F\) is a mapping \(f:G_n \to C_q \), \(q \in T \), \(C_q = \langle \sigma \mid \sigma^{q}=1 \rangle \), \(\sigma = \epsilon^{\frac{|F^*|}{q}}\).
We define the function \(FSign\xi\) in the following way [Gec10]:
\[ \forall \xi \in F^* : FSign \xi = \sigma^j, \;\; \textrm{if} \;\; \frac{j|F^*|}{q}\leq \textrm{deg}\xi < \frac{(j+1)|F^*|}{q}, \]
in which \(\textrm{deg} \xi\) is obtained from the equality \(\xi=\epsilon^{\textrm{deg} \xi}\), \(j \in \{ 1, \ldots , q-1 \}\).
Let \((w_1, \ldots, w_n; T) \in F^{n+1} \). For all \(\overline{g} = (x_1,\ldots,x_n) \in G_n \) (i.e. \(x_j \in C_{k_j}\)) we define
\[ w(\overline{g}) = \sum_{i=1}^n w_i x_i + T \in F. \]
A neural element with structure vector \((w_1, \ldots, w_n; T) \in F^{n+1} \) is a logic device that realizes the function \(FSign (w(\overline{g})) \) for all \( \overline{g} \in G_n \).
Discrete function \(f\) which is relizable by a single neural element over the field \(F\) is called neurofunction.
‣ MVThresholdElement( Structure, Dimensions, Field ) | ( function ) |
For the two-element list Structure, in which the first element is a vector over the field Field, and the second element is an element of the Field, a list of positive integers Dimensions (or an integerm if all the dimensions are equal) and a Galois field Field the function MVThresholdElement returns a multi-valued threshold element with the number of inputs equal to the length of the first element of Structure list.
gap> F:=GF(13);; gap> st:=[[Z(13)^5,Z(13)^7],Z(13)^4];; gap> dim:=[2,3,3];; gap> mvte:=MVThresholdElement(st,dim,F); < multivalued threshold element over GF(13) with structure [[ 6, 11 ], 3] and dimension vector [ 2, 3, 3 ] > gap> Display(mvte); Structure vector = [[ 6, 11 ], 3] Dimension vector = [[ 2, 3, 3 ]] Field: GF(13) Multi-Valued Threshold Element realizes the function f : [ 1, 1 ] || 9 [ 1, 3 ] || 3 [ 1, 9 ] || 1 [ 12, 1 ] || 1 [ 12, 3 ] || 1 [ 12, 9 ] || 9 gap> ## If all dimensions in dimension vector are equal, the user can enter just this number. gap> F:=GF(5);; gap> st:=[[Z(5)^0,Z(5)^0],Z(5)^2];; gap> dim:=2;; gap> mvte:=MVThresholdElement(st,dim,F); < multivalued threshold element over GF(5) with structure [[ 1, 1 ], 4] and dimension vector [ 2, 2, 2 ] > gap> Display(mvte); Structure vector = [[ 1, 1 ], 4] Dimension vector = [[ 2, 2, 2 ]] Field: GF(5) Multi-Valued Threshold Element realizes the function f : [ 1, 1 ] || 1 [ 1, 4 ] || 4 [ 4, 1 ] || 4 [ 4, 4 ] || 1
The function Display outputs the stucture of the given threshold element mvte and the truth table in given alphabet. In the previous example mvte realizes a three-valued function in \(\{ 1, 3, 9 \}\) alphabet over GF(13).
‣ IsMVThresholdElement( Obj ) | ( function ) |
For the object Obj the function IsMVThresholdElement returns true if Obj is a multi-valued threshold element (see MVThresholdElement (5.1-1)), and false otherwise.
gap> IsMVThresholdElement(mvte); true
‣ OutputOfMVThresholdElement( MVThrEl ) | ( function ) |
For the multi-valued threshold element MVThrEl the function OutputOfMVThresholdElement returns the truth vector of the multi-valued Boolean function, realized by MVThrEl.
gap> F:=GF(13);; gap> st:=[[Z(13)^5,Z(13)^7],Z(13)^4];; gap> dim:=[2,3,3];; gap> mvte:=MVThresholdElement(st,dim,F); < multivalued threshold element over GF(13) with structure [[ 6, 11 ], 3] and dimension vector [ 2, 3, 3 ] > gap> f:=OutputOfMVThresholdElement(mvte); < logic function of 2 variables and dimension vector [ 2, 3, 3 ]> gap> Display(f); [ 0, 0 ] || 2 [ 0, 1 ] || 1 [ 0, 2 ] || 0 [ 1, 0 ] || 0 [ 1, 1 ] || 0 [ 1, 2 ] || 2
‣ StructureOfMVThresholdElement( MVThrEl ) | ( function ) |
For the multi-valued threshold element MVThrEl the function StructureOfMVThresholdElement returns the structure vector of MVThrEl (see MVThresholdElement (5.1-1)).
gap> StructureOfMVThresholdElement(mvte); [ [ Z(13)^5, Z(13)^7 ], Z(13)^4 ]
‣ MVBooleanFunctionBySTE( Func, Dim, F ) | ( function ) |
For the given multi-valued function Func and the prime field F the function MVBooleanFunctionBySTE returns the multi-valued threshold element over F if Func is realizable and an empty list otherwise. The algorithm realizing this function was proposed in [Gec10].
gap> f:=LogicFunction(2,2,[0,1,1,0]); < Boolean function of 2 variables > gap> mvte:=MVBooleanFunctionBySTE(f,GF(3)); [ ] gap> mvte:=MVBooleanFunctionBySTE(f,GF(5)); < multivalued threshold element over GF(5) with structure [[ 1, 1 ], 4] and dimension vector [ 2, 2, 2 ] > gap> Display(last); Structure vector = [[ 1, 1 ], 4] Dimension vector = [[ 2, 2, 2 ]] Field: GF(5) Multi-Valued Threshold Element realizes the function f : [ 1, 1 ] || 1 [ 1, 4 ] || 4 [ 4, 1 ] || 4 [ 4, 4 ] || 1 gap> ## Consider an example if dimensions are presented as a list. gap> f:=LogicFunction(2,[2,3,3],[0,0,1,1,2,2]); < logic function of 2 variables and dimension vector [ 2, 3, 3 ]> gap> mvte:=MVBooleanFunctionBySTE(f,GF(13)); < multivalued threshold element over GF(13) with structure [[ 12, 10 ], 5] and dimension vector [ 2, 3, 3 ] > gap> Display(last); Structure vector = [[ 12, 10 ], 5] Dimension vector = [[ 2, 3, 3 ]] Field: GF(13) Multi-Valued Threshold Element realizes the function f : [ 1, 1 ] || 1 [ 1, 3 ] || 1 [ 1, 9 ] || 3 [ 12, 1 ] || 3 [ 12, 3 ] || 9 [ 12, 9 ] || 9
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