Verification of Parameterised FPGA Circuit Descriptions with Layout ...

CHAPTER 3. GENERATING PARAMETERISED LIBRARIES WITH LAYOUT 41
〈expr〉 ::= 〈expr〉 〈bop〉 〈expr〉
| 〈uop〉 〈expr〉
| 〈var〉
| 〈num〉
| true | false
| height ( 〈blkinst〉 )
| width ( 〈blkinst〉 )
| max ( 〈expr〉, 〈expr〉 (, 〈expr〉)* )
| if ( 〈expr〉 , 〈expr〉 , 〈expr〉 )
| sum ( 〈id〉 = 〈expr〉 .. 〈expr〉 , 〈expr〉 )
| maxf ( 〈id〉 = 〈expr〉 .. 〈expr〉 , 〈expr〉 )
〈bop〉 ::= and | or | nand | nor | xor | xnor
| + | - | * | / | ** | mod | == | !=
| < | | >=
〈uop〉 ::= - | abs | not
Figure 3.2: Grammar of Quartz expressions
once again returning zero if the lower bound is greater than the upper bound.
More formally, the semantics of Quartz expressions are described by the evaluation function
E. The clauses describing the semantics of the new operations are shown in Figure 3.3.
The concept of a “function” is kept clearly restricted to block height and width expressions,
while elsewhere the semantics are based on substitution (shown as {a ↦→ b}) rather than
λ-abstraction. This distinction exists because Quartz expressions must be compiled into
Pebble/VHDL expressions without functions, λ-calculus like semantics are restricted to block
size functions since these will be eliminated during Quartz compilation. We will show in
Chapter 4 how λ-calculus style functions can be used to achieve the same semantics.
Note that we have extended expressions with max and maxf operators but have not included
their duals min and minf. While this might be desirable in the interests of symmetry actu-
ally we do not require minimum-finding operations for generating layouts and in any event
expressions using these functions can be manipulated to eliminate them using the following
two relationships:
Theorem 2 ∀a b. min(a, b) = −max(−a, −b)
Theorem 3 ∀e1 e2 e3. minf(i = e1..e2, e3) = −maxf(i = e1..e2, −e3)