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词条 Draft:Bit-Vector Matrix
释义

  1. The Boolean algebra [2] [3] based on the Bit-Vector Matrix

      The Commutative property based on the Bit-Vector Matrix    The Associative property based on the Bit-Vector Matrix    The Distributive property based on the Bit-Vector Matrix    The De-Morgan theorem based on the Bit-Vector Matrix  

  2. Verifying the De-Morgan of 3D Bit-matrices by VHDL

  3. Applications [4] of the Bit-Vector Matrix on the Images

  4. References

  5. External links

{{AFC submission|d|v|u=Dr assem Badr|ns=118|decliner=Jovanmilic97|declinets=20181213191401|ts=20181115072755}} {{AFC comment|1=It may be, or not be notable, but it is improperly sourced. Most of the draft is without any references, making the draft look like OR which is not allowed on Wikipedia. Jovanmilic97 (talk) 19:14, 13 December 2018 (UTC)}}

In mathematics and computer science, the Bit-Vector Matrix [1] (or Bit-Array Matrix) is two-dimension mathematical matrix of Bit arrays (or Bit vectors).

Each Bit-Vector has one-dimension array of Boolean elements ('0' or '1'). Therefore, the Bit-Vector Matrix consider as 3D Bit-Matrix with three parameters (i, j and k). The two parameters “i x j” define the 2D matrix that include i-rows and j-columns of Bit Vectors. Besides, the parameter “k” indicates the total number of Boolean elements (Bits) in each Bit Vector.

Therefore, the Bit-Vector Matrix consider as 3D dimension matrix of Bits.

The Bit-Vector Matrix is more effectively during exploiting the bit-level parallelism in the hardware designing, testing and implementation.

The formula (1) is 2D matrix "Qij" includes "ixj" Bit-Vectors. The formula (2) is Bit-Vector includes k-number of bits.

(2)

By composing the formula (2) into the formula (1), will get the formula (3) of mathematical three-dimensional Bit-Matrix (3-D Bit-Matrix) with three parameters (i,j and k). This type of matrix can be used to analysis the different electronic digital circuits mathematically. In addition, it can be utilized in synthesizing process from mathematical models to real electronic module.

(3)

The suffix number that attended with each bit has 3 parameters. The left hand side parameter represent its row number in the 3D matrix, the middle parameter represents its column number. Besides, the right hand side parameter (the third dimension) represents its depth in the 3D Bit-Matrix.

The Boolean algebra [2] [3] based on the Bit-Vector Matrix

Assume three Bit-Vector-Matrices (A, B and C) have similar dimensions (same 3 parameters) as illustrate in the formulas (4), (5) and (6).

(4)

(5)

(6)

Whereas "Vaij", "Vbij", "Vcij" are bit-vectors. They include “k” number of bits as following formulas (7), (8) and (9).

(7)

(8)

(9)

The Commutative property based on the Bit-Vector Matrix

Assume conjunction operation (˄) between the two mentioned bit-vectors "Vaij" and "Vbij", so all bits in the first bit-vector will conjunct individually with their corresponding bits in the second bit-vector as formula (10).

[Va]ij ˄ [Vb]ij =

= [Va ˄ Vb]ij (10)

By applying the standard commutative algebra Boolean expression {a ˄ b = b ˄ a} for all the corresponding bits of the formula (10), will get formula (11).

[Va ˄ Vb]ij =

= [Vb ˄ Va]ij (11)

After applying the exclusive operation between the two equal terms of formula (11), will get the formula (12).

[Va ˄ Vb]ij ⊕ [Vb ˄ Va]ij = [0]ij (12)

By composing ij-number of the bit-vectors of the formula (12) in the general Bit-Vector Matrix (Q) of formula (1), will get the formula (13).

(13)

By reducing the formula (13) using the two formulas (4) and (5), will get the Boolean algebra commutative conjunction expression for the two mentioned Bit-Vector-Matrices “A” and “B” in the formula (14).

[A ˄ B]ijk ⊕ [B ˄ A]ijk = [0]ijk (14)

Similarly, the modified Boolean algebra commutative disjunction expression as the formula (15)

[A ˅ B]ijk ⊕ [B ˅ A]ijk = [0]ijk (15)

The Associative property based on the Bit-Vector Matrix

Assume the conjunction operation (˄) among the three mentioned bit-vectors "Vaij", "Vbij" and "Vcij" as illustrate in formula (16).

[Va ˄ Vb]ij ˄ [Vc]ij =

(16)

By applying the basic associative Boolean algebra expression {(a ˄ b)˄c = a˄(b ˄ c)} for all corresponding bits of the expression (16), will get formula (17).

[Va ˄ Vb]ij ˄ [Vc]ij =

= [Va ]ij ˄ [Vb ˄ Vc]ij (17)

By applying the exclusive operation between the two equal terms of formula (17), will get the formula (18).

( [Va ˄ Vb]ij ˄ [Vc] )ij ⊕ ( [Va]ij ˄ [Vb ˄ Vc] )ij = [0]ij (18)

By composing ij-number of bit-vectors of the formula (18) in the bit-vector matrix (Q) of formula (1), will get the formulas (19).

(19)

By reducing the formula (19) using the formulas (4), (5) and (6), will get the Boolean algebra associative conjunction expression as in the formula (20).

( [A ˄ B]ijk ˄ Cijk ) ⊕ ( Aijk ˄ [B ˄ C]ijk) = [0]ijk (20)

Similarly, the general associative disjunction expression can be reduced as the modified Boolean expression as in formula (21).

( [A ˅ B]ijk ˅ Cijk ) ⊕ ( Aijk ˅ [B ˅ C]ijk) = [0]ijk (21)

The Distributive property based on the Bit-Vector Matrix

Assume disjunction operation (˅) between the two mentioned bit-vectors "Vbij" and "Vcij" besides conjunction operation (˄) with the third bit-vector"Vaij" as shown in the formula (22).

[Va]ij ˄ [Vb ˅ Vc]ij =

(22)

By applying the distributed property of the Boolean expression

{a ˄(b ˅ c) = (a ˄ b)˅(a ˄ c)}, Will get the formula (23).

[Va]ij ˄ [Vb ˄ Vc]ij =

= [Va ˄ Vb]ij ˄ [Va ˄ Vc]ij (23)

By applying exclusive operation between the two equal terms of formula (23), will get the formula (24).

( [Va]ij ˄ [Vb ˅ Vc] )ij ⊕ ( [Va ˄ Vb]ij ˅ [Va ˄ Vc] )ij = [0]ij (24)

By composing ij-number of bit-vectors of the formula (24) into the bit-vector matrix (Q), will get the formulas (25).

(25)

The formula (25) can be simplified in the modified Boolean distributive expression as formula (26).

( Aijk ˄ [B ˅ C]ijk ) ⊕ ( [A ˄ B]ijk ˅ [A ˄ C]ijk) = [0]ijk (26)

Similarly, the modified Boolean distributive expression illustrates as in the formula (27).

( Aijk ˅ [B ˄ C]ijk ) ⊕ ( [A ˅ B]ijk ˄ [A ˅ C]ijk) = [0]ijk (27)

The De-Morgan theorem based on the Bit-Vector Matrix

Assume the disjunction (˅) operation between the two complement bit-vectors "Vaij" and "Vbij" as shown in the formula (28).

¬[Va]ij ˅ ¬[Vb]ij =

(28)

By Applying the De-Morgan's theorem ¬a ˅ ¬b = ¬(a˄b) on formula (28), will get the formula (29).

¬[Va]ij ˅ ¬[Vb]ij =

= ¬( [Va]ij ˄ [Vb] )ij (29)

After making the exclusive operation between the two equal terms of formula (29), will get the formula (30).

( [¬Va]ij ˅ [¬Vb]ij ) ⊕ ¬([Va]ij ˄ [Vb]ij) = [0]ij (30)

By composing "ixj" of bit-vectors of the formula (30) in the general formula of the bit-vector matrix (Q), will get the formula (31).

(31)

The modified expression of the De-Morgan's theorem for two Bit-Vector-Matrices in formula (31) can be reduced as the formula (32).

( ¬Aijk ˅ ¬Bijk ) ⊕ ¬[A ˄ B]ijk = [0]ijk (32)

Similarly, the modified expression of the De-Morgan's theorem for three Bit-Vector-Matrices shows in the formula (33).

( ¬Aijk ˅ ¬Bijk ˅ ¬Cijk ) ⊕ ¬[A ˄ B ˄ C]ijk = [0]ijk (33)

Verifying the De-Morgan of 3D Bit-matrices by VHDL

The attached VHDL code for realizing the De-Morgan theory based on the Bit-Vector Matrices (3-bit matrices). The VHDL code built using Xilinx editor ISE. It declares the Bit-Vector Matrix as "Bit_V_Matrix". It has 3 input Bit-Vector-Matrices (Matrix1, Matrix2 and Matrix3) besides the output matrix (Matrix_OUT). All matrices in the code have three dimensions "i", "j" and "k". This meaning that each bit in any mentioned matrix has 3D location in its Bit-Vector Matrix.

The code has 2 main paths for handling the logical data in the 3 Bit-Vector-Matrices. The first path performs the "AND" and "NOT" operations for the Bit-Vector-Matrices. In addition, the second path carry out the "NOT" and "OR" operations. The results of the 2 paths are performing by "XOR" operation. All logical operations are carried out concurrently.

The code can be modified easily to increasing or decreasing the number of rows, columns, bits for the manipulated Bit-Vector-Matrices.

Moreover, the code can modify to either increasing or decreasing the number of input Bit-Vector-Matrices related to the capacity of bits in the different applications.

According to the attached formulas of the Bit-Vector-Matrices De-Morgan theory, the results from the XOR-operation is allows zero-bits. The VHDL code is verified using a reliable simulator.

Applications [4] of the Bit-Vector Matrix on the Images

The images can be represented mathematically using the Bit-Vector-Matrices by the concept of the Bit Per Pixel (Bpp). The total amount of its height pixels can represent by the total number of rows "i" in a Bit-Vector Matrix, the total amount of width pixels can represent by the total number of columns "j" in this matrix. In addition, the BPP value (depth) can be represented as the parameter "k" of the same matrix. In other words, the image can be represented in one Bit-Vector Matrix. This issue can define as image in single matrix (IISM).

Assume an image with resolution 1024 x 1024 with high color (BPP = 16-bit). This image can be represented mathematically in single Bit-Vector Matrix with parameters "i", "j" and "k" equal "1024", "1024" and "16" respectively. The size of this image will be as the following formula

The capacity of image bit in the matrix = i * j * K = 1024 * 1024 * 16 = 16777216 bits = 2 MB

This mathematical representation consider as mathematical model. It helps in image processing by carrying out the suitable logical and mathematical operation on it. This mathematical model can be converted to a computational model using any computer language to run this images on either the operating systems or display it on GLCD from memory.

Many instructions can be produced through handling operations based on the Bit-Vector-Matrices inside the microprocessor especially the graphic processors.

References

1. ^{{Cite journal|title= Modifying the logic gate symbols to enrich the designing of the computer systems by 3-D bit-matrices|journal= Ain Shams Engineering Journal|volume= 9|issue= 4|pages= 3207–3216|language=en|doi= 10.1016/j.asej.2018.06.002|year= 2018|last1= Badr|first1= Assem}}
2. ^{{Cite book|url=https://books.google.com/?id=NJzgAAAAMAAJ&q=mathematical+matrix&dq=mathematical+matrix|title= The Mathematics of Matrices|language=en|isbn= 9780898747560|last1= Davis|first1= Philip J.|year= 1965}}
3. ^{{Cite book|url=https://books.google.com/?id=QwqrGXVpiY4C&pg=PA260&dq=mathematical+array#v=onepage&q=mathematical%20array&f=false|title= Mathematical and computational Models|language=en|isbn= 9788177645545|last1= Arulmozhi|first1= G.|year= 2003}}
4. ^{{Cite book|url=https://books.google.com/?id=cwEs5AkGp0MC&pg=PA64&dq=mathematical+array#v=onepage&q=mathematical%20array&f=false|title= Modern Mathematical Methods for Physicists and Engineers|language=en|isbn= 9780521598279|last1= Cantrell|first1= C. D.|date= 2000-10-09}}

External links

  • [https://www.sciencedirect.com/science/article/pii/S2090447918300480?via%3Dihub] Modifying the logic gate symbols to enrich the designing of the computer systems by 3-D bit-matrices
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