# simple network in sw format

One interpretation of the BKO scheme is that it is a general notation to represent networks. Indeed, it is almost trivial to define networks in BKO. Here is a simple network, but extends easily:

Here is the BKO, and we have just decided to propagate the network using the "O" operator. You can choose anything really.
```O |a1> => |a2>
O |a2> => |a3>
O |a3> => |a4>
O |a4> => |a5>
O |a5> => |a6>
O |a6> => |a7>
O |a7> => |a8>
O |a8> => |a9>
O |a9> => |a10>
O |a10> => |a1> + |b1>

O |b1> => |b2>
O |b2> => |b3>
O |b3> => |b4>
O |b4> => |b5>
O |b5> => |b6>
O |b6> => |b7>
O |b7> => |b1>```
And being a network, we have a corresponding matrix representation:
```sa: matrix[O]
[ a1  ] = [  0  0  0  0  0  0  0  0  0  1  0  0  0  0  0  0  0  ] [ a1  ]
[ a2  ]   [  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  ] [ a2  ]
[ a3  ]   [  0  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  ] [ a3  ]
[ a4  ]   [  0  0  1  0  0  0  0  0  0  0  0  0  0  0  0  0  0  ] [ a4  ]
[ a5  ]   [  0  0  0  1  0  0  0  0  0  0  0  0  0  0  0  0  0  ] [ a5  ]
[ a6  ]   [  0  0  0  0  1  0  0  0  0  0  0  0  0  0  0  0  0  ] [ a6  ]
[ a7  ]   [  0  0  0  0  0  1  0  0  0  0  0  0  0  0  0  0  0  ] [ a7  ]
[ a8  ]   [  0  0  0  0  0  0  1  0  0  0  0  0  0  0  0  0  0  ] [ a8  ]
[ a9  ]   [  0  0  0  0  0  0  0  1  0  0  0  0  0  0  0  0  0  ] [ a9  ]
[ a10 ]   [  0  0  0  0  0  0  0  0  1  0  0  0  0  0  0  0  0  ] [ a10 ]
[ b1  ]   [  0  0  0  0  0  0  0  0  0  1  0  0  0  0  0  0  1  ] [ b1  ]
[ b2  ]   [  0  0  0  0  0  0  0  0  0  0  1  0  0  0  0  0  0  ] [ b2  ]
[ b3  ]   [  0  0  0  0  0  0  0  0  0  0  0  1  0  0  0  0  0  ] [ b3  ]
[ b4  ]   [  0  0  0  0  0  0  0  0  0  0  0  0  1  0  0  0  0  ] [ b4  ]
[ b5  ]   [  0  0  0  0  0  0  0  0  0  0  0  0  0  1  0  0  0  ] [ b5  ]
[ b6  ]   [  0  0  0  0  0  0  0  0  0  0  0  0  0  0  1  0  0  ] [ b6  ]
[ b7  ]   [  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  1  0  ] [ b7  ]
```
Now we have a network, we can propagate around it by applying the O operator repeatedly, which corresponds directly to the idea of applying the O matrix to a vector, where superpositions can be considered to be "sparse representations" of vectors -- we drop/ignore all elements that have coeff = 0. Let's start with 1 seed node, and step around. OK. I picked |a5> as the starting node:
```sa: O |a5>
|a6>

sa: O O |a5>
|a7>

-- yeah. I got lazy, and dropped to the exponential notation (as you should!):
sa: O^3 |a5>
|a8>

sa: O^4 |a5>
|a9>

sa: O^5 |a5>
|a10>

sa: O^6 |a5>
|a1> + |b1>

sa: O^7 |a5>
|a2> + |b2>

sa: O^8 |a5>
|a3> + |b3>
```
And so on.
Now, let's pick more than one seed node, and add in some non {0,1} coeffs:
```sa: O (|a2> + 7|a5> + 30 |a10> + 3.14|b5>)
|a3> + 7.000|a6> + 30.000|a1> + 30.000|b1> + 3.140|b6>

sa: O^2 (|a2> + 7|a5> + 30 |a10> + 3.14|b5>)
|a4> + 7.000|a7> + 30.000|a2> + 30.000|b2> + 3.140|b7>

sa: O^3 (|a2> + 7|a5> + 30 |a10> + 3.14|b5>)
|a5> + 7.000|a8> + 30.000|a3> + 30.000|b3> + 3.140|b1>

sa: O^4 (|a2> + 7|a5> + 30 |a10> + 3.14|b5>)
|a6> + 7.000|a9> + 30.000|a4> + 30.000|b4> + 3.140|b2>

sa: O^5 (|a2> + 7|a5> + 30 |a10> + 3.14|b5>)
|a7> + 7.000|a10> + 30.000|a5> + 30.000|b5> + 3.140|b3>

sa: O^6 (|a2> + 7|a5> + 30 |a10> + 3.14|b5>)
|a8> + 7.000|a1> + 7.000|b1> + 30.000|a6> + 30.000|b6> + 3.140|b4>

sa: O^7 (|a2> + 7|a5> + 30 |a10> + 3.14|b5>)
|a9> + 7.000|a2> + 7.000|b2> + 30.000|a7> + 30.000|b7> + 3.140|b5> ```
One advantage of the BKO notation is that it is trivial to define and redefine our network (much easier than manually entering elements into a matrix). How about we change a couple of rules:
```sa: O |a4> => |a5> + 300 |b5>
sa: O |b2> => |b3> + 0.5 |a2>
```
Now, check out the new matrix:
```sa: matrix[O]
[ a1  ] = [  0  0  0  0    0  0  0  0  0  1  0  0    0  0  0  0  0  ] [ a1  ]
[ a2  ]   [  1  0  0  0    0  0  0  0  0  0  0  0.5  0  0  0  0  0  ] [ a2  ]
[ a3  ]   [  0  1  0  0    0  0  0  0  0  0  0  0    0  0  0  0  0  ] [ a3  ]
[ a4  ]   [  0  0  1  0    0  0  0  0  0  0  0  0    0  0  0  0  0  ] [ a4  ]
[ a5  ]   [  0  0  0  1    0  0  0  0  0  0  0  0    0  0  0  0  0  ] [ a5  ]
[ a6  ]   [  0  0  0  0    1  0  0  0  0  0  0  0    0  0  0  0  0  ] [ a6  ]
[ a7  ]   [  0  0  0  0    0  1  0  0  0  0  0  0    0  0  0  0  0  ] [ a7  ]
[ a8  ]   [  0  0  0  0    0  0  1  0  0  0  0  0    0  0  0  0  0  ] [ a8  ]
[ a9  ]   [  0  0  0  0    0  0  0  1  0  0  0  0    0  0  0  0  0  ] [ a9  ]
[ a10 ]   [  0  0  0  0    0  0  0  0  1  0  0  0    0  0  0  0  0  ] [ a10 ]
[ b1  ]   [  0  0  0  0    0  0  0  0  0  1  0  0    0  0  0  0  1  ] [ b1  ]
[ b2  ]   [  0  0  0  0    0  0  0  0  0  0  1  0    0  0  0  0  0  ] [ b2  ]
[ b3  ]   [  0  0  0  0    0  0  0  0  0  0  0  1    0  0  0  0  0  ] [ b3  ]
[ b4  ]   [  0  0  0  0    0  0  0  0  0  0  0  0    1  0  0  0  0  ] [ b4  ]
[ b5  ]   [  0  0  0  300  0  0  0  0  0  0  0  0    0  1  0  0  0  ] [ b5  ]
[ b6  ]   [  0  0  0  0    0  0  0  0  0  0  0  0    0  0  1  0  0  ] [ b6  ]
[ b7  ]   [  0  0  0  0    0  0  0  0  0  0  0  0    0  0  0  1  0  ] [ b7  ]
```
And our code can also spit out a matrix that has been applied a few times, eg 10 times:
```sa: merged-matrix[O,O,O,O,O,O,O,O,O,O]
[ a1  ] = [  1    0    0      0      0    0    0    0    0    0      0    0.5  0    0    0    0    0    ] [ a1  ]
[ a2  ]   [  0    1    0.5    0      0    0    0    0    0    150.5  75   0    0    0    0    0    0.5  ] [ a2  ]
[ a3  ]   [  150  0    1      0.5    0    0    0    0    0    0      0.5  75   0    0    0    0    0    ] [ a3  ]
[ a4  ]   [  0    150  0      1      0.5  0    0    0    0    0      0    0.5  0    0    0    0    0    ] [ a4  ]
[ a5  ]   [  0    0    150    0      1    0.5  0    0    0    0      0    0    0.5  0    0    0    0    ] [ a5  ]
[ a6  ]   [  0    0    0      150    0    1    0.5  0    0    0      0    0    0    0.5  0    0    0    ] [ a6  ]
[ a7  ]   [  0    0    0      0      0    0    1    0.5  0    0      0    0    0    0    0.5  0    0    ] [ a7  ]
[ a8  ]   [  0    0    0      0      0    0    0    1    0.5  0      0    0    0    0    0    0.5  0    ] [ a8  ]
[ a9  ]   [  0    0    0      0      0    0    0    0    1    0.5    0    0    0    0    0    0    0.5  ] [ a9  ]
[ a10 ]   [  0    0    0      0      0    0    0    0    0    1      0.5  0    0    0    0    0    0    ] [ a10 ]
[ b1  ]   [  1    0    0      0      0    0    0    301  150  0      0    0.5  0    0    1    150  0    ] [ b1  ]
[ b2  ]   [  0    1    0      0      0    0    0    0    301  150    0    0    0    0    0    1    150  ] [ b2  ]
[ b3  ]   [  0    0    1      0      0    0    0    0    0    301    150  0    0    0    0    0    1    ] [ b3  ]
[ b4  ]   [  300  0    0      1      0    0    0    0    0    0      1    150  0    0    0    0    0    ] [ b4  ]
[ b5  ]   [  0    300  45000  0      301  150  0    0    0    0      0    1    150  0    0    0    0    ] [ b5  ]
[ b6  ]   [  0    0    300    45000  0    301  150  0    0    0      0    0    1    150  0    0    0    ] [ b6  ]
[ b7  ]   [  0    0    0      300    0    0    301  150  0    0      0    0    0    1    150  0    0    ] [ b7  ]
```
Now, it would be nice if we could just do: merged-matrix[O^10] or something, but my code can't currently do that. Shouldn't be too hard to add if I get the inclination.

That's it for this post. I think some more matrices in the next post or two.

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updated: 19/12/2016
by Garry Morrison
email: garry -at- semantic-db.org