Symbolic ExecutionΒΆ

Grackle executes program using symbolic execution. Symbolic execution, like ordinary execution, executes the program one step at a time, but with two differences: (1) As discussed in the previous section, the variables range over symbolic values rather than ordinary expressions; and (2) when conditional branches are encountered, such as if statements or while loops, the simulator will need to consider multiple execution paths if the branch may be either true or false.

As an example, consider the code below:

x      = symbolic('x', 'real');
target = symbolic('target', 'real');
diff = 3;

if (x > target)
  y = x - diff;
  y = x + diff;

[r,c] = check_sat(abs(x-target) > diff & abs(y-target) > abs(x-target))

In this case, the condition x > target can be either true or false depending on the values assigned to x and target. Accordingly, Grackle will execute the script up to the if statement, and then execute both execution paths over the state prior to the if statement. When both paths reach the end of their respective paths in the if statement, Grackle will merge the two execution paths into a single state. In this case, after the merge, the assignment will map y to the expression if (x > target) (x - 3) (x + 3), which denotes the value x - 3 if x > target and x + 3 otherwise.

The post-dominator based merging stategy is particularly important for programs with loops. For example, in the following program, Grackle will execute 20 paths without merging, but would need to consider 2:sup:10 execution paths if it did not do path merging.

p = symbolic('x', 10, 1, 'logical');
true_count = 0;
false_count = 0;
for i = 1:10
  if (p(i))
    true_count = true_count + 1;
    false_count = false_count + 1;

In this case, the condition p(i) is indeterminate, and so each time it is encountered, two execution paths are formed. Due to our use of path-based merging however, a single execution path is formed after execution completes.

Internally, our path-based merging strategy operates by translating the source program into a control flow graph. For each branch, Grackle computes the immediate post-dominator of that branch, and uses that location as the merge point. This approach allows Grackle to easily deal with a variety of source language constructs, including not just if statements, but also while and for loops, even in the presence of continue and break statements. In this encoding each loop will effectively introduce two potential merge points: the first is at the entry of the loop, and used when continue is called or the loop body terminates. The second is at the loop exit, and is targetted by break.

One significant limitation of our approach is that the symbolic simulator does not effectively deal with loops that may execute an unbounded number of times. In this case, the post-dominator-based merging strategy will get stuck executing the loop body. To deal with this and other programs where merging is inappropriate, we have introduced a built-in function symbolic_fork which allows one to create a path that bypasses merging.