Verification of Parameterised FPGA Circuit Descriptions with Layout ...

CHAPTER 3. GENERATING PARAMETERISED LIBRARIES WITH LAYOUT 44
Where a single block is instantiated the width and height functions are extracted from the σ
environment and are returned after the application of curried parameters p1 . . . pn. The size of
a parallel composition of blocks is calculated as though the blocks are arranged vertically, with
the overall height being the sum of the height of each individual block and the overall width
being the maximum width of any block in the composition. The size of a series composition
of blocks is calculated as though the blocks are arranged horizontally in a similar manner.
Series composition is represented as a fully associative binary operator (an n-size composition
can be represented as nested series compositions) for convenience. The series composition
size function internal signal s is defined in terms of the ι definite description operator. This
operator, in the form ιx. P(x) can be read as “the x such that P(x) holds”. In this case
the internal signal s is the one such that it obeys the predicates produced by the application
of function B ′ to the two blocks. This is made necessary by the relational nature of Quartz
blocks which means it is not possible to state whether block b1 or block b2 generates the value
s (or some combination of the two, if it is a tuple of values). This logical nicety is practically
irrelevant for us in this chapter since we will convert relational Quartz descriptions into
functional ones during compilation while evaluating any size functions.
The grey boxes with dotted outlines in Figure 3.5 illustrate the sizes calculated for series and
parallel compositions diagrammatically.
3.4.3 Size Inference
As noted in the previous section each type of statement has a particular layout interpretation
and we can exploit that to describe the size of each type of statement recursively. The size
of an entire block can then be described in terms of the sizes of its constituent statements.
The function SB, shown in Figure 3.6, returns a pair of expressions describing the height
and width of a block in terms of its domain, range and internal signals.
Unlike the SI function we described previously this function is directly implementable since
it does not use the definite description operator or refer explicitly to internal signals. Note
also that this function does not infer size functions for the block, it infers size expressions
in terms of the block’s local environment. These size expressions can be converted into