Control dependency is a situation in which a program instruction executes if the previous instruction evaluates in a way that allows its execution.

An instruction B has a control dependency on a preceding instruction A if the outcome of A determines whether B should be executed or not. In the following example, the instruction ${\displaystyle S_{2))$ has a control dependency on instruction ${\displaystyle S_{1))$. However, ${\displaystyle S_{3))$ does not depend on ${\displaystyle S_{1))$ because ${\displaystyle S_{3))$ is always executed irrespective of the outcome of ${\displaystyle S_{1))$.

S1.         if (a == b)
S2.             a = a + b
S3.         b = a + b

Intuitively, there is control dependence between two statements A and B if

• B could be possibly executed after A
• The outcome of the execution of A will determine whether B will be executed or not.

A typical example is that there are control dependences between the condition part of an if statement and the statements in its true/false bodies.

A formal definition of control dependence can be presented as follows:

A statement ${\displaystyle S_{2))$ is said to be control dependent on another statement ${\displaystyle S_{1))$ iff

• there exists a path ${\displaystyle P}$ from ${\displaystyle S_{1))$ to ${\displaystyle S_{2))$ such that every statement ${\displaystyle S_{i))$ ≠ ${\displaystyle S_{1))$ within ${\displaystyle P}$ will be followed by ${\displaystyle S_{2))$ in each possible path to the end of the program and
• ${\displaystyle S_{1))$ will not necessarily be followed by ${\displaystyle S_{2))$, i.e. there is an execution path from ${\displaystyle S_{1))$ to the end of the program that does not go through ${\displaystyle S_{2))$.

Expressed with the help of (post-)dominance the two conditions are equivalent to

• ${\displaystyle S_{2))$ post-dominates all ${\displaystyle S_{i))$
• ${\displaystyle S_{2))$ does not post-dominate ${\displaystyle S_{1))$

Construction of control dependences

Control dependences are essentially the dominance frontier in the reverse graph of the control-flow graph (CFG).^[1] Thus, one way of constructing them, would be to construct the post-dominance frontier of the CFG, and then reversing it to obtain a control dependence graph.

The following is a pseudo-code for constructing the post-dominance frontier:

for each X in a bottom-up traversal of the post-dominator tree do:
    PostDominanceFrontier(X) ← ∅
    for each Y ∈ Predecessors(X) do:
        if immediatePostDominator(Y) ≠ X:
            then PostDominanceFrontier(X) ← PostDominanceFrontier(X) ∪ {Y}
    done
    for each Z ∈ Children(X) do:
        for each Y ∈ PostDominanceFrontier(Z) do:
            if immediatePostDominator(Y) ≠ X:
                then PostDominanceFrontier(X) ← PostDominanceFrontier(X) ∪ {Y}
        done
    done
done

Here, Children(X) is the set of nodes in the CFG that are immediately post-dominated by X, and Predecessors(X) are the set of nodes in the CFG that directly precede X in the CFG. Note that node X shall be processed only after all its Children have been processed. Once the post-dominance frontier map is computed, reversing it will result in a map from the nodes in the CFG to the nodes that have a control dependence on them.