Verification of Parameterised FPGA Circuit Descriptions with Layout ...

CHAPTER 4. VERIFYING CIRCUIT LAYOUTS 70
(x21, y21)
(x21, y21)
(x11, y11)
(x22, y22)
(x11, y11)
(a) x22 ≤ x11
(x22, y22)
(x12, y12)
(c) y12 ≤ y21
(x12, y12)
(x11, y11)
(x11, y11)
(x21, y21)
(x12, y12)
(x21, y21)
(b) x12 ≤ x21
(x12, y12)
(x22, y22)
(d) y22 ≤ y11
(x22, y22)
Figure 4.2: Situations under which two rectangles will not overlap
possibly being allocated to both. This must obviously be avoided in correct circuit descrip-
tions.
Ensuring this correctness criteria is the common target of work on relative placement. The
problem of ensuring non-overlapping of n rectangles is a well understood objective in con-
straint programming [1, 6] and is a problem in placement algorithms, however in our case
we are interested not in finding a solution to the problem but rather in checking one. Exist-
ing work on constraint solving applied to this problem is not suitable for adaptation to the
Quartz system because we permit a much ricer language of expressions than simple arithmetic
operations to define block sizes.
Two objects placed on a n-dimensional surface will not overlap if is one dimension in which
they can not intersect. For rectangles on a two-dimensional surface this means that one
block must be either below or above the other, or to the left or right of it. This means that
there are four possible situations in which a layout can be correct, illustrated graphically
in Figure 4.2. When the rectangle A is described by corners (x11, y11) and (x12, y12), and
rectangle B is described by corners (x21, y21) and (x22, y22) this can be represented by the
logical disjunction:
(x22 ≤ x11) ∨ (x12 ≤ x21) ∨ (y12 ≤ y21) ∨ (y22 ≤ y11)