COMS W4117
Compilers and Translators:
Software Verification Tools
Lecture 19: Binary Decision Diagrams
November 8, 2007

Lecture Outline

In C, pointer-alias analysis is complicated because C allows arbitrary computations on pointers.
In Java, virtual methods can cause method invocations to be indirect.
Flow-insensitive analysis ignores control flow. Its advantage is that it reduces the size of the representation of the results and it converges faster than a flow-sensitive analysis. Its disadvantage is that the points-to information can overestimate what variables can point to. See Example 12.22, p. 936.
For a simple example of a flow-insensitive pointer-alias analysis formulated in Datalog see Section 12.4.4, pp. 937-940.

A binary decision diagram (BDD) represents a boolean function f by a rooted directed acyclic graph (DAG).

Each interior node is labeled by a variable of the function.
At the bottom of the DAG are two leaves, one labeled 0 the other 1.
Each interior node v has two edges to children, one labeled "low" (corresponding to the case where v = 0) the other labeled "high" (corresponding to the case where v = 1).
Given a truth assignment for the variables in f, we can determine the value of f for that truth assignment by starting at the root of the BDD and following the edge from each interior node corresponding to the value of the variable representing by that node until we are at a leaf. If it is the 0 leaf, then f has the value 0 for that truth assignment; if it is the 1 leaf, then f has the value 1.

An ordered BDD is one in which there is a linear order on the variables represented by the interior nodes. Whenever there is an edge from a parent node v to a child labeled w, then v comes before w in that linear order.

If an interior node v has both its high and low edges go to the same node w, then we may shortcircuit v by having the edges entering v go directly to w.
If two nodes v and w have low edges that go to the same node x and high edges that go to the same node y, then we may merge v and w. Edges entering v and w now enter the merged node.

We can obtain a canonical BDD for a boolean formula f by applying these two transformations until they cannot be applied any more.
Note that the canonical BDD is dependent on the linear ordering given to the variables.
Finding a linear ordering that minimizes the canonical BDD is an NP-complete problem. Various heuristics have been developed that find small canonical forms for many BDDs arising in practice.
Given two BDDs a and b for boolean functions f and g, we can construct a BDD for the logical OR of f and g by combining the two BDDs a and b. See Algorithm 12.29, p. 956, for details.