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I'm trying to understand a paper (Tandem Networks of Universal Cells, Butler, 1978 1), but I can't make it past the first paragraph:

Consider the $x_l … x_{k - 1} | x_k$ partition matrix of a function $f( x_1, x_2, …, x_k)$, as shown in Fig. 1, where $C_0$ and $C_1$ represent the $x_k = 0$ and $x_k = 1$ columns, respectively. Let $0$ and $1$ represent the columns of all 0's and all 1's, respectively. Let $X$ denote a column with at least one 0 and at least one 1, and $\overline{X}$ its complement. The concatenation of two columns will represent a complete partition matrix. Thus, for example, $01$ represents $f(x_1, x_2, …, x_k) = x_k$ and $XX$ represents a function independent of $x_k$, but dependent on at least one of the remaining variables.

I'm trying to build something like a Karnaugh map, but the result I'm getting doesn't make sense.

For example, here's what I made from $f = x_1 \times x_2 \times x_3$ (with digital logic "&" for boolean times):

The first column ($C_0$) I would call $0$, from the author's definition, but the second column ($C_1$) doesn't seem to be a $0$, $1$, or $X$. What am I doing wrong?

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You're pretty close.

I don't think that's what they intended as the partition matrix. Instead, I think they intended the following matrix:

         x3
x1 x2 \  0 1
       -----
 0  0  | 0 0
 0  1  | 0 0
 1  0  | 0 0
 1  1  | 0 1

That's slightly different from what you have written. In particular, the partition matrix will have $2^{k-1}$ rows and 2 columns.

It's best to think of the first sentence as one paragraph, and the remaining sentences as a second disconnected paragraph that is trying to explain a taxonomy for classifying a column of this matrix. The author is proposing to classify every column into one of three types: all-zeros, all-ones, or contains both a 0 and a 1. The paper represents an all-zeros column by 0, an all-ones column by 1, and the third case by X. Thus, X is used to represent any column that contains at least one 0 and at least 1, i.e., any column that is not all-zeros and not all-ones.

By this classification, the partition matrix above would be of the form 0X: the first column is all-zeros (type 0), and the second column is type X.

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