Verification of Parameterised FPGA Circuit Descriptions with Layout ...

CHAPTER 4. VERIFYING CIRCUIT LAYOUTS 105
Once again, an alternative to this would perhaps be to attempt a deep embedding of Quartz
in Isabelle and to prove the correctness of layouts in terms of the compilation function. A
deep embedding, particularly one which encompassed a full compilation procedure, would be
an even more substantial undertaking than this shallow embedding. Layout verification based
on a deep embedding of compilation (for intersection proofs) and the size inference algorithm
(for validity and containment proofs) would, for each individual design, probably require the
proof of significantly fewer theorems but each theorem is likely to be more complex. The
benefit would be an increased level of formal assurance in the result but quite how much of
a benefit this is remains an open question.
Another aspect of our system that could be changed is the use of containment proofs. We
have based layout verification around the assumption that each block can be contained within
a rectangle and this rectangle can then be used as an abstraction for the block layout in later
proofs. However, for irregular shaped blocks this may not be optimal - for example consider
the case of two triangular circuits, which could be laid out inter-locking to form a rectangle
- but only if their size functions are not rectangular. Introducing block boundaries described
by arbitrary functions would massively complicate reasoning and the approach we would
advocate using our system is to describe shapes such as these as new combinators as and
when they are required - these layouts can then be proved as normal. An alternative is to
relax the requirement on containment proofs and verify the containment of a set of blocks.
This means verifying the layout at a level of a set of blocks rather than each individual block
in the set.
4.9 Summary
In this chapter we have described a system for verifying Quartz circuit layouts in a shallow
embedding of Quartz in Higher-Order Logic using the generic theorem prover Isabelle. We
give Quartz a formal semantics in HOL.
We have described new features of the Quartz compiler which support automatic conversion
of Quartz descriptions into Isabelle/HOL definitions and the automatic generation of proof
obligations to verify their layouts. Our modified compiler is also reasonably effective at gen-