Verification of Parameterised FPGA Circuit Descriptions with Layout ...

CHAPTER 3. GENERATING PARAMETERISED LIBRARIES WITH LAYOUT 59
If the generated size expression depends on internal signals hidden within the series
composition then these must be explicitly instantiated. This is already the process
that occurs when Quartz series compositions are compiled into Pebble and if the series
composition exactly matches an instantiation carried out anyway then the same newly
declared signals can be used. However, if the composition is not otherwise present
in the block new variables must be declared and code that sets their values must be
lifted into the local context. This implements the semantics of the definite description
operator used in function SIσ β.
3.7.4 Recursive Size Expressions
Many Quartz blocks have both recursive and iterative definitions. The two are usually
equivalent, although the iterative versions compile more neatly into Pebble/VHDL. It is
quite common to use the recursive definition of a common combinator for formal reasoning
purposes and the iterative version in the generated hardware description.
Some blocks, such as the binary tree combinator, can be more clearly defined recursively than
iteratively. Size inference of recursively defined blocks can give recursive size expressions and
while it is sometimes possible to manually specify iterative size expressions this is not always
the case. The height and width of the recursively defined map combinator can be easily
described using the same expressions as the iterative version, however the same can not be
said for more complex combinators such as rows or grids where there are internal signals
hidden by the recursion that can not be referenced or where blocks do not have a clearly
defined size in terms of the simple sum and maxf functions (such as a tree). In any event,
automation of this transformation is non-trivial so where size inference is used alone it is
possible to be left with recursive size functions.
For example, the recursively defined map combinator can be described by:
map 0 R ⇔ id
map n R ⇔ apln−1 −1 ; [R, map n−1 R] ; apln−1
After expanding the parallel composition, apl and id this gives size expressions for map n