Prolog Representation of ZCQ Matrices

[Code--ZCQ Matrix]
In Prolog, we represent the Zero-Counting-Quadrant (ZCQ) Matrices discussed above using the data structure presented in this section.

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zcm(A,B,D,C) is the basic data structure. Each argument in zcm/4 stands for a quadrant in the matrix. The correspondence is shown in Figure 3.4. This is a recursive structure which means that each one of A, B, C, or D is itself a zcm/4 or any of the other data structures presented below.

A	B
D	C

Figure 3.4: Matrix quadrants corresponding to arguments in zcm(A,B,D,C)

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The number 0 stands for a sparse matrix of 0s.

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zcu(A,B,C) represents an upper-triangular matrix as shown in Figure 3.5. Because it is an upper-triangular matrix, we already know that D is 0. The values of A and C need only be either zcu/3 themselves or 1. The value of B can be zcm/4 or any of the base cases mentioned below.

A	B
0	C

Figure 3.5: An upper triangular matrix corresponding to zcu(A,B,C)

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Base Cases:

• The number 1 stands for a $1\times1$ matrix of 1.
• zc12(A,B) is a $1\times 2$ matrix shown in Figure 3.6

  A

  B

  Figure 3.6: A $1\times 2$ matrix corresponding to zc12(A,B)

• zc21(A,D) stands for a $2\times 1$ matrix shown in Figure 3.7

  A

  D

  Figure 3.7: A $2\times 1$ matrix corresponding to zc21(A,D)