Verification of Parameterised FPGA Circuit Descriptions with Layout ...

CHAPTER 6. LAYOUT CASE STUDIES 153
block cube (int x, int y, int z, block R (‘a, ‘b, ‘c) ∼ (‘c, ‘b, ‘a))
(‘a x d[z][y], ‘b y d[z][x], ‘c z d[y][x]) ∼
(‘c z r[y][x], ‘b y r[z][x], ‘a x r[z][y]) {
‘a xs[x+1][z][y]. ‘b ys[y+1][z][x]. ‘c zs[z+1][y][x].
int ix, iy, iz.
}
xs[0] = x d. ys[y] = y d. zs[z] = z d.
x r = xs[x]. y r = ys[0]. z r = zs[0].
for ix = 0..x−1 {
for iy = 0..y−1 {
for iz = 0..z−1 {
(xs[ix ][ iz ][ iy ], ys[iy+1][iz ][ ix ], zs[ iz+1][iy ][ ix ]) ;
R ;
(zs[ iz ][ iy ][ ix ], ys[iy ][ iz ][ ix ], xs[ix+1][iz ][ iy]).
} .
} .
} .
Figure 6.17: cube combinator defined iteratively with explicit internal signals
generalises easy to the n-dimensional case where extra dimensions can simply be added.
We can describe a cube combinator iteratively as shown in Figure 6.17 in terms of explicit
internal signals. Multiple copies of the R block are instantiated and connected to internal
signals xs, ys and zs that hold values flowing along the x-axis, y-axis and z-axis respectively.
This definition uses three different internal signals rather than declaring a single vector of
internal signals with an extra dimension, this is because the signals flowing along each axis
can have different types - as illustrated in the type signature for the R block and the cube
block itself.
This interpretation of a cube also identifies the final important element of our convention –
that the (0, 0, 0) point is located at the front, left of a cubical structure. This is reflected in
the indexes chosen to connect to the R block in the loop. While this is a short description it
is quite complex and has no particularly obvious layout interpretation.
We can envisage a cube as a 1D array of 2D arrays. We exploit this to represent a cube as a
column of grids. This kind of description immediately gives the cubical circuit a 2D layout
interpretation – as grids laid out vertically on top of one another. Figure 6.18 illustrates the
wiring involved in this kind of arrangement. In this example two grids are placed on top of