On the Relative Succinctness of Sentential Decision Diagrams

Abstract

Sentential decision diagrams (SDDs) introduced by Darwiche in 2011 are a promising representation language for propositional knowledge bases. The relative succinctness of representation languages is an important subject in knowledge compilation. The aim of the paper is to identify which kind of Boolean functions can be represented by SDDs of small size with respect to the number of variables the functions are defined on. For this reason the sets of Boolean functions representable by different representation languages in polynomial size are investigated and SDDs are compared with representation languages from the classical knowledge compilation map of Darwiche and Marquis. Ordered binary decision diagrams (OBDDs) which are a popular data structure for Boolean functions are one of them. SDDs are more general than OBDDs by definition but only recently, a Boolean function was presented with polynomial SDD size but exponential OBDD size. This result is strengthened in several ways. The main result is that a function can be represented by SDDs of small size if the function and its negation have small restricted nondeterministic OBDD representations. Moreover, for important Boolean functions called storage access function polynomial-size SDDs are presented. As a side effect an open problem about the relative succinctness between SDDs and free binary decision diagrams which are more general than OBDDs is answered.

Appendices

Appendix A: Proof of Lemma 1

Proof

We give a proof by contradiction in order to show that Φ is a partition. Suppose to the contrary that there is an ∨-node u of $\mathcal {F}$ and a (partial) assignment β ∈ β(u) such that the described set of functions Φ is not a partition. So Φ has to violate at least one of the partition properties. It will be shown that the violation of at least one partition property leads to a contradiction.

Let x_i be the smallest variable of vars(u) w.r.t. π and X_≥i = {x_i,…,x_n}.

Satisfiability

Suppose there is a function φ ∈Φ with φ = ⊥. By definition of $R^{+}(\mathcal {F}, \beta )$ and $R^{+}(\overline {\mathcal {F}}, \beta )$ the nodes u₁,…,u_k,v₁,…,v_l are no sinks. Therefore, an inner node u_i or v_j of $\mathcal {F}$ or $\overline {\mathcal {F}}$, respectively, represents the constant function ⊥. This is a contradiction to the assumption of $\mathcal {F}$ and $\overline {\mathcal {F}}$ being simple.

Disjointness

Suppose there are functions φ₁,φ₂ ∈Φ with φ₁ ∧ φ₂≠⊥. For this purpose, consider the following cases.

• 1. The functions φ₁,φ₂ are represented by nodes of the same ∨₁-OBDD, i.e., either $\varphi _{1} = {\Phi }_{u_{i}}$, $\varphi _{2} = {\Phi }_{u_{j}}$ or $\varphi _{1} = {\Phi }_{v_{i}}$, $\varphi _{2} = {\Phi }_{v_{j}}$ holds for i≠j. Suppose ${\Phi }_{u_{i}} \wedge {\Phi }_{u_{j}} \neq \bot$. According to the definition of $R^{+}(\mathcal {F}, \beta )$ the assignment β can be extended (maybe differently) such that there are accepting paths for these extensions of β in $\mathcal {F}$ containing u_i and u_j. As ${\Phi }_{u_{i}} \wedge {\Phi }_{u_{j}} \neq \bot$ holds, there is an assignment β^∗ of X_≥i (variables not assigned by β) such that ${\Phi }_{u_{i}}[\beta ^{*}] = 1$ and ${\Phi }_{u_{j}}[\beta ^{*}] = 1$. However, if we extend β by β^∗ then there are accepting paths for (β,β^∗) in $\mathcal {F}$ containing u_i and u_j with i≠j. Here (β,β^∗) denotes the complete assignment to the variables in X that is consistent with the partial assignments β and β^∗. Because of the maximality of u_i and u_j w.r.t. X_≥i ($\mathcal {F}_{u_{i}}$ cannot be a subgraph of $\mathcal {F}_{u_{j}}$ or vice versa) we know that there must be two distinct accepting paths. This is a contradiction to the property of $\mathcal {F}$ being unambiguous. If ${\Phi }_{v_{i}} \wedge {\Phi }_{v_{j}} \neq \bot$ holds, the contradiction can be derived analogously.

• 2. The functions φ₁,φ₂ are represented by nodes of $\mathcal {F}$ and $\overline {\mathcal {F}}$, i.e., ${\Phi }_{u_{i}} \wedge {\Phi }_{v_{j}} \neq \bot$. Hence, there is an assignment β^∗ of X_≥i such that ${\Phi }_{u_{i}}[\beta ^{*}] = 1$ and ${\Phi }_{v_{j}}[\beta ^{*}] = 1$ leading to accepting paths for β^∗ in the subgraphs $\mathcal {F}_{u_{i}}$ and $\overline {\mathcal {F}}_{v_{j}}$. By definition of $R^{+}(\mathcal {F}, \beta )$ and $R^{+}(\overline {\mathcal {F}}, \beta )$ the assignment β can be extended such that there are accepting paths in $\mathcal {F}$ and $\overline {\mathcal {F}}$ containing u_i and v_j, respectively. Like in the former case β can be extended by β^∗ such that there are accepting paths for (β,β^∗) in $\mathcal {F}$ and $\overline {\mathcal {F}}$ leading to a contradiction to ${\Phi }_{\mathcal {F}} = \overline {{\Phi }_{\overline {\mathcal {F}}}}$.

Cover

Suppose ${\Phi }_{u_{1}} \vee {\dots } \vee {\Phi }_{u_{k}} \vee {\Phi }_{v_{1}} \vee {\dots } \vee {\Phi }_{v_{l}} \neq \top$. Then, there exists an assignment β^∗ of X_≥i such that ${\Phi }_{u_{1}}[\beta ^{*}] = {\dots } = {\Phi }_{u_{k}}[\beta ^{*}] = {\Phi }_{v_{1}}[\beta ^{*}] = {\dots } = {\Phi }_{v_{l}}[\beta ^{*}] = 0$. Hence, there is no accepting path for β^∗ in $\mathcal {F}_{u_{1}}, \dots , \mathcal {F}_{u_{k}}, \overline {\mathcal {F}}_{v_{1}}, \dots , \overline {\mathcal {F}}_{v_{l}}$. Since every accepting path for an extension of β in $\mathcal {F}$ and $\overline {\mathcal {F}}$ contains exactly one node from u₁,…,u_k,v₁,…,v_l, it is not possible to extend β by β^∗ resulting in an accepting path in $\mathcal {F}$ or $\overline {\mathcal {F}}$. This is a contradiction to ${\Phi }_{\mathcal {F}} \vee \overline {{\Phi }_{\overline {\mathcal {F}}}} = \top$.

Now, we get the claimed lemma because the violation of at least one partition property leads to a contradiction. □

Appendix B: Proof of Proposition 1

Proof

We even prove a more general result than given in Proposition 1. Let f be a Boolean function such that f and $\overline {f}$ can be represented by polynomial-size unambiguous nondeterministic OBDDs with only one nondeterministic node at the beginning and both OBDDs respect the same variable ordering. Let G_f be the given unambiguous nondeterministic OBDD representing f and $G_{\overline {f}}$ be the one for $\overline {f}$. Then f can also be represented by SDDs of polynomial size and the SDD size is bounded above by $\mathcal {O}(|G_{f}|+ |G_{\overline {f}}|)$.

A Boolean function f is equal to $(f\wedge \top )\vee (\overline {f}\wedge \bot )$. Due to our assumptions the Boolean function f can be written as f₁₁ ∨ f₁₂ ∨⋯ ∨ f_1k and the function $\overline {f}$ as $\overline {f}_{01}\vee \overline {f}_{02} \vee {\cdots } \vee \overline {f}_{0\ell }$, where f₁₁,f₁₂,…,f_1k and $\overline {f}_{01}, \overline {f}_{02}, {\ldots } , \overline {f}_{0\ell }$ can be represented by deterministic OBDDs of polynomial size with respect to a variable ordering π and k and ℓ are polynomially bounded. Moreover, f₁₁,f₁₂,…,f_1k and $\overline {f}_{01}, \overline {f}_{02}, {\ldots } , \overline {f}_{0\ell }$ are a partition. Therefore, f₁₁,f₁₂,…,f_1k and $\overline {f}_{01}, \overline {f}_{02}, {\ldots } , \overline {f}_{0\ell }$ can all be represented by SDDs of polynomial size with respect to a vtree T that corresponds to the variable ordering π.

The conjunction f ∧⊤ is equal to (f₁₁ ∧⊤) ∨ (f₁₂ ∧⊤) ∨⋯ ∨ (f_1k ∧⊤) and $\overline {f}\wedge \bot$ is equal to $(\overline {f}_{01}\wedge \bot )\vee (\overline {f}_{02}\wedge \bot ) \vee {\cdots } \vee (\overline {f}_{0\ell }\wedge \bot )$. Now, we generate a new vtree T^′ with root q. T is the left subtree of q and a leaf labeled by an auxiliary variable h is the right subtree. Altogether we obtain the result that the disjunction of (f₁₁ ∧⊤) ∨ (f₁₂ ∧⊤) ∨… ∨ (f_1k ∧⊤) and $(\overline {f}_{01}\wedge \bot )\vee (\overline {f}_{02}\wedge \bot ) \vee {\cdots } \vee (\overline {f}_{0\ell }\wedge \bot )$ can be represented by an SDD of polynomial size with respect to the vtree T^′. □

Appendix C: Proof of Lemma 4

Proof

Our aim is to prove that each function representable by a k-OBDD of polynomial size, where k is an arbitrary constant, can also be represented by an unambiguous nondeterministic OBDD of polynomial size with only one nondeterministic node at the beginning. For this reason we present a polynomial transformation from k-OBDDs into equivalent restricted unambiguous nondeterministic OBDDs. The following construction was first used in [5] proving that the satisfiability problem can be solved in polynomial time for functions represented by k-OBDDs. Later it was also used in [7] in order to prove that k-OBDDs can be polynomially transformed into OBDDs which use so-called parity nondeterminism.

Let f be the function represented by a given k-OBDD G and let k be a constant. We start with the observation that there is exactly one accepting path for each 1-input in a k-OBDD since it is a deterministic model. Now, the crucial idea is a suitable decomposition of a given k-OBDD G. For this we consider the at most s = |G|^k− 1 possibilities to switch between the layers of G. The i-th auxiliary function, 1 ≤ i ≤ s, equals 1 for the 1-inputs of f that choose the i-th possibility which means that the accepting paths for these inputs run through the layers of the given k-OBDD G in the chosen way. Such an auxiliary function can be represented by an OBDD of size |G|^k by combining parts of the k-OBDD via conjunction. Here we use the fact that in a k-OBDD all layers respect the same variable ordering. (OBDDs in general do not have nice algorithmic properties. There are examples known such that g_n and h_n are two Boolean functions which have OBDDs of linear size (for different variable orderings) but f_n = g_n ∧ h_n has even exponential nondeterministic FBDD size. The so-called permutation test function is an example of such a function f_n. If only OBDDs respecting the same variable ordering are considered, all important operations can be performed efficiently. For more details see, e.g., [32].)

Next, we describe these ideas more precisely. Let G₁,…,G_k be the layers of G. If b is a 1-input, the accepting path for b leads through some layers ℓ(1) = 1 < ℓ(2) < ⋯ < ℓ(r) ≤ k of G, where v₁ is the source of G, G_ℓ(i) is reached at some node v_i, and from some node in G_ℓ(r) the sink labeled by 1 is reached. There are at most |G|^k− 1 possibilities to choose r,ℓ(2),…,ℓ(r),v₂,…,v_r. For an arbitrary but fixed choice of these parameters we consider the layers G_ℓ(1),…,G_ℓ(r) and the sinks. We transform G_ℓ(i), i ∈{1,…,r}, into an OBDD $G^{\prime }_{\ell (i)}$ with source v_i in the following way. An edge leaving G_ℓ(i) is replaced by an edge to a 1-sink if either i < r and the edge leads to v_i+ 1 or i = r and the edge leads to the 1-sink. All other edges leaving a node in G_ℓ(i) are replaced by edges to the 0-sink. Now, $G^{\prime }_{\ell (i)}$ consists of all nodes (and corresponding edges) reachable from v_i. The function represented by G has a 1-input iff for some r,ℓ(2),…,ℓ(r),v₂,…,v_r the corresponding OBDDs $G^{\prime }_{\ell (1)}, \ldots , G^{\prime }_{\ell (r)}$ have a common 1-input. Since all these OBDDs respect the same variable ordering, Bryant’s apply algorithm [11] can be used to obtain an OBDD of size $\mathcal {O}(|G|^{k})$ for the conjunction of the functions represented by $G^{\prime }_{\ell (1)}, \ldots , G^{\prime }_{\ell (r)}$ in time $\mathcal {O}(|G|^{k})$. Considering all choices of the parameters r,ℓ(2),…,ℓ(r),v₂,…,v_r we obtain a unambiguous nondeterministic OBDD of size $\mathcal {O}(|G|^{2k-1})$ which has only one nondeterministic node at the beginning. □