On the Read-Once Property of Branching Programs and CNFs of Bounded Treewidth

Abstract

In this paper we prove a space lower bound of $n^{\varOmega (k)}$ for non-deterministic (syntactic) read-once branching programs (nrobps) on functions expressible as cnfs with treewidth at most k of their primal graphs. This lower bound rules out the possibility of fixed-parameter space complexity of nrobps parameterized by k. We use lower bound for nrobps to obtain a quasi-polynomial separation between Free Binary Decision Diagrams and Decision Decomposable Negation Normal Forms, essentially matching the existing upper bound introduced by Beame et al. (Proceedings of the twenty-ninth conference on uncertainty in artificial intelligence, Bellevue, 2013) and thus proving the tightness of the latter.

Appendices

Appendix 1: Transformation of an nrobp into a Uniform One

Let Z be the nrobp being transformed and let F be the function of n variables realized by Z. Let $a_1,\dots , a_m$ be the non-leaf nodes of Z being ordered topologically. We show that there is a sequence $Z_{a_1}=Z, Z_{a_2}, \dots , Z_{a_m}$ such that each $Z_{a_i}$ for $i>1$ is an nrobp of F obtained from $Z_{a_{i-1}}$ by subdividing the in-coming edges of $a_i$ by adding at most n nodes and O(n) edges to each such an in-coming edge. Moreover, the edges of any two paths $P_1$ and $P_2$, starting the root of $Z_{a_i}$ and ending at the same node which is ether $a_i$ or topologically precedes $a_i$, are labeled with literals of the same set of variables. Observe that since each edge has only one head, say $a_j$, it is subdivided only once, namely during the construction of $Z_{a_j}$. Hence the number of new added edges of $Z_{a_m}$ is O(n) per edge of Z and hence the size of $Z_{a_m}$ is O(n) times larger than the size of Z.

Regarding $Z_{a_1}$ this existence statement is vacuously true so assume $i>1$. Denote by $AllVar(a_i)$ the set of all variables whose literals label edges of paths of $Z_{a_{i-1}}$ from the root to $a_i$.

For each in-neighbour $a'$ of $a_i$, we transform the edge $(a',a_i)$ as follows. Let P be a path from the root of $Z_{a_{i-1}}$ to $a_i$ passing through $(a',a_i)$. Let $x^1, \dots , x^q$ be the elements of $AllVar(a_i) {\setminus } Var(A(P))$. We subdivide $(a',a_i)$ as follows. We introduce new nodes $a'_1, \dots , a'_q$ and let $a'_{q+1}=a_i$. Then instead $(a',a_i)$ we introduce an edge $(a',a'_1)$ carrying the same label as $(a',a_i)$ (or no label in case $(a',a_i)$ carries no label). Then, for each $1 \le i \le q$ we introduce two edges $(a'_i,a'_{i+1})$ carrying labels $x^q$ and $\lnot x^q$, respectively.

Let us show that the edges of any two paths $P_1$ and $P_2$ from the root of $Z_{a_i}$ to $a_i$ are labelled with literals of the same set of variables. Let $a'$ be an in-neighbour of $a_i$ in $Z_{a_{i-1}}$. By the induction assumption, any two paths from the root to $a'$ are labelled with literals of the same set of variables. It follows that as a result any two paths from the root to $a_i$ passing through $a'$ are labelled by literals of the same set of variables, namely $AllVar(a_i)$. Since this is correct for an arbitrary choice of $a'$, we conclude that in $Z_{a_i}$ any two paths from the root to $a_i$ are labelled with $AllVar(a_i)$, that is with literals of the same set of variables. Observe that the paths to the nodes of Z preceding $a_i$ are not affected so the ‘uniformity’ of paths regarding them holds by the induction assumption. Regarding the new added nodes on the subdivided edge $(a',a_i)$ the uniformity clearly follows from the uniformity of paths from the root to $a'$.

To verify the read-once property of $Z_{a_i}$, let $P'$ be a path from the root to the leaf of $Z_{a_i}$. Taking into account the induction assumption, the only reason why $P'$ may contain two edges labelled by literals of the same variable is that $P'$ is obtained from a path P of $Z_{a_{i-1}}$ by subdivision of an edge $(a',a_i)$ of this path. By construction the variables of the new labels put on $(a',a_i)$ do not occur on the prefix of P ending at $a_i$. Furthermore, by definition of $AllVar(a_i)$ the variable x of each new label, in fact, occurs in some path of $Z_{a_{i-1}}$ from the root to $a_i$ and hence, by the read-once property, x does not occur on any path starting from $a_i$. It follows that the variables of the new labels do not occur on the suffix of $P'$ starting at $a_i$. Taking into account that all the new labels of $(a',a_i)$ are literals of distinct variables, the read-once property of $P'$, and hence the read-once property of $Z_{a_i}$, due to the arbitrary choice of $P'$, follow. Thus we know now that $Z_{a_i}$ is an nrobp.

It remains to verify that $Z_{a_i}$ indeed realizes F. Let $P'$ be a path of $Z_{a_i}$ from the root to the leaf. Then $A(P')$ is an extension of A(P) of some path P of $Z_{a_i}$. By the induction assumption, any extension of A(P) is a satisfying assignment of F, hence so is $A(P')$. Conversely, for each satisfying assignment A of F we can find a path P of $Z_{a_{i-1}}$ such that $A(P) \subseteq A$. If an edge of path P is subdivided then the new labels are opposite literals on multiple edges. So, for every such multiple edge we can choose one edge carrying the literal occurring in A and obtain a path $P'$ such that $A(P') \subseteq A$.

For the leaf node we do a similar transformation but this time add new labels on the in-coming edges of the leaf so that the set of labels on each path from the root to the leaf is a set of literals of Var(F). A similar argumentation to the above shows that the resulting structure is indeed a uniform nrobp realizing F. Clearly the size of the resulting nrobp remains O(n) times larger than the size of Z.

Appendix 2: Equivalence of the arosrn and the Traditional Definition of the nrobp

A (nrobp) is traditionally defined as a dag Z with one root and two leaves. Some of non-leaf nodes are labelled with variables so that no variable occurs as a label twice on a directed path of Z. A node labelled with a variable has two outgoing edges one labelled with true the other with false. Finally, the leaves are labelled with true and false.

It is convenient to see each edge e labelled with true or false being in fact respectively labelled with the positive or negative literal of the variable labelling the tail of e. With such a labelling an assignment A(P) associated with each directed path of Z is simply the set of literals labelling the edges of P. The satisfying assignments of the function computed by Z are precisely those that are extensions of A(P) for paths P from the root to the true leaf.

It is not hard to see that for any function that is not constant false, nrobp can be thought as a special case of arosrn. Indeed, with edges labelled by literals as specified in the previous paragraph, remove the labels from the vertices, remove the false leaf as well as all nodes of Z from which th true leaf is not reached and the obtained graph is an arosrn computing exactly the same function as Z.

Conversely, an arosrn can be transformed into a nrobp as follows. Denote the only leaf of the arosrn as the true leaf and introduce a new node to be the false leaf. Then for each edge (u, v) labelled with a literal x, apply the following transformation.

• Subdivide (u, v) by introducing a new node w and edges (u, w) and (w, v) instead (u, v).

• Introduce a new edge e from w to the false leaf.

• Label w by Var(x), the variable of x.

• If x is the positive literal then label (w, v) with true and e with false. Otherwise, label (w, v) with false and e with true.

The transformation of labeled edges is illustrated in Fig. 5.

Fig. 5

Transformation of a labelled edge of an arosrn

It is not hard to see that there is a bijection between root-leaf paths of the arosrn and root-true leaf paths of the resulting nrobp preserving the associated sets of literals. Therefore, we conclude that this transformation is valid.