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A Directed acyclic graph representing a Boolean function. A decision node tests a Boolean variable and has zero and one successors; evaluation follows the corresponding arrows until a terminal node returns a Boolean value. Sharing subgraphs can make a binary decision diagram much smaller than a binary decision tree.

An internal vertex that tests a condition and selects an outgoing branch. In a binary decision diagram, a decision node tests a Boolean variable and has one zero arrow and one one arrow.

Solutions to the paper whose cover dates the examination to 6 June 2015, 2:30–4:00 pm. Source: released examination archive, member 2015-cafv.pdf. All three questions are covered, although candidates were asked to attempt at most two. In the binary decision diagrams, dashed arrows mean zero and solid arrows mean one. In the linear temporal logic calculations, $F=◊$, $G=\square$ and $X=◯$.

Follow the binary decision diagram from its top decision node to a terminal node, choosing the outgoing arrow specified by each tested Boolean variable. If $x=0$, the left $y$ decision node returns $y$, independently of $z$. If $x=1$, the right $y$ decision node returns one when $y=1$ and otherwise returns $z$. Thus the truth table, with rows in the order $(x,y,z)$, is:

Thus the output column of the truth table, in the displayed order, is $\boxed{(0,0,1,1,0,1,1,1)}$. In particular, the skipped $z$ test on the left does not constrain $z$: both values give the same result.

A reduced ordered binary decision diagram is obtained by sharing equal terminal nodes, removing a decision node whose two children are equal, and sharing decision nodes with the same tested Boolean variable and the same ordered pair of children. Apply these rules to the binary decision tree in part (c) as follows.

The left panel below is therefore the resulting reduced ordered binary decision diagram. Its full description is

There are $\boxed{4+2=6}$ vertices: four decision nodes and two terminal nodes.

The original binary decision diagram also has four decision nodes and two terminal nodes, and is already reduced for $x<y<z$: the two $y$ decision nodes have different children and no decision node has two equal children. Thus reversing the order has not made this example smaller. Indeed $y\vee (x\wedge z)$ is unchanged by interchanging $x$ and $z$, and the new binary decision diagram is the original one with those two labels exchanged. The uniqueness of a reduced ordered binary decision diagram applies to a fixed variable ordering of a binary decision diagram; it does not assert that every ordering has the same size.

Choose the variable ordering of a binary decision diagram $y<x<z$. The Boolean cofactor with $y=1$ is the constant one, while the Boolean cofactor with $y=0$ is $x\wedge z$. Consequently the right panel in part (d) has the following arrows:Here an entry such as $0\to x$ denotes the zero arrow to the $x$ decision node. There are $\boxed{3+2=5}$ vertices: three decision nodes and two terminal nodes, saving one decision node relative to either of the previous orderings. Testing $y$ first is effective because its one branch settles the entire Boolean function immediately. The ordering $y<z<x$ works equally well.

This is also minimal among ordinary binary decision diagrams for this Boolean function. Each Boolean variable is essential: setting $y=0,z=1$ leaves $f=x$; setting $y=0,x=1$ leaves $f=z$; and setting $x=z=0$ leaves $f=y$. If a Boolean variable were never tested, the represented Boolean function could not depend on it. Thus at least three decision nodes are necessary, and the construction attains that bound.

An ordered binary decision diagram with shared equal terminal nodes, no decision node having identical zero and one children, and no two decision nodes having the same tested Boolean variable and the same ordered pair of children. For a fixed variable ordering of a binary decision diagram, it is unique up to labelled graph isomorphism: each nonconstant remaining Boolean cofactor has a uniquely determined first essential Boolean variable, and its two children are recursively determined by its zero and one Boolean cofactors. This uniqueness does not compare different orderings.

A vertex that returns a final output rather than performing another test. In a binary decision diagram the two possible outputs are the Boolean constants zero and one.

A fixed order in which Boolean variables may be tested along each branch of an ordered binary decision diagram. Branches may skip a test, but cannot reverse the order. The order can change the number of distinct remaining Boolean cofactors and therefore the size of the representation.