The verb (dyad ,"1 0) has left rank 1, and the left operand has rank 2, so the operation will be applied to 1-cells of the left operand. The verb has right rank 0, and the right operand has rank 1, so the operation will be applied to 0-cells of the right operand. The left frame is 3, the right frame is 3 .

The operation dyad , is

performed on each pair of 0 1 2 0 3 4 5 1 6 7 8 2 cells:

|0 1 2 comma 0|3 4 5 comma 1|6 7 8 comma 2| +-------------+-------------+-------------+

(i. 3 3) ,"1 0 i. 3 0 1 2 0 3 4 5 1 6 7 8 2

When Dyad Frames Differ: Operand Agreement

The processing of dyads has an extra step not present for monads, namely the pairing of corresponding cells of the left and right operands. As long as the frames lf and rf are the same, as in the examples so far, this is straightforward. If the frames are different, J may still be able to pair left and right cells, using another level of implicit looping, one that provides considerable additional programming power. The formal description that

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follows is not easy to follow—you might want to skim over it and read it in detail after you have studied the examples that follow.

J requires that one of the frames be a prefix of the other (if the frames are identical, each is a prefix of the other and all the following reduces to the simple case we have studied). The common frame cf is the part of the frames that is identical, namely the shorter of the two frames; its length is designated rcf. If we look at the cells of the operands relative to this common frame (i. e. the (-rcf)-cells), we see that for the operand with the shorter frame, these cells are exactly the rank that will be operated on, while for the operand with the longer frame, each (-rcf)-cell contains multiple operand cells.

First, the (-rcf)-cells of the two operands are paired one-to-one (because they have the same frame), leaving each shorter-frame operand cell paired with a longer-frame (-rcf)-cell. Then, the longer-frame (-rcf)-cells are broken up into operand cells, with each operand cell being paired with a copy of the shorter-frame operand cell that was paired with the (-rcf)-cell (an equivalent statement is that the cells of the shorter-frame operand are replicated to match the surplus frame of the longer-frame operand). This completes the pairing of operand cells, and the operation is then performed on the paired operand cells, and collected using the longer frame. Maybe some examples will help.

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100 200 + i. 2 3

100 101 102

203 204 205

The simplest and most common case of different-length frames is when the shorter frame is of zero length; in other words, when one of the operands has only one cell. In that case, the single cell is replicated to match every cell of the longer operand. An easy way to force an operand to be viewed as a single cell is to make the verb have infinite rank for that operand. This is not a special case—the behavior follows from the rules already given—but it's worth an example:

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'abc' ,"_ 0 'defg' abcd

abce abcf abcg

The verb (dyad ,"_ 0) has left rank _, and the left operand has rank 1, so the operation will be applied to 1-cells of the left operand. The verb has right rank 0, and the right operand has rank 1, so the operation will be applied to 0-cells of the right operand. The left frame is (empty), the right frame is 4 .

+-----------+-----------+-----------+-----------+ |abc comma d|abc comma e|abc comma f|abc comma g| +-----------+-----------+-----------+-----------+

You must thoroughly understand this example, where one operand has only one cell, because it occurs frequently. The handling of the general case of dissimilar frames is uncommon enough that you do not need to understand it perfectly right now—you'll

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