2 Preliminaries

We consider propositional theories $\mathcal{T}$ over variables $\mathbf{X}$ . For a set of variables $\mathbf{Y}$ , we denote by $lit(\mathbf{Y})$ the set of literals over $\mathbf{Y}$ , by ${int}(\mathbf{Y})$ the set of assignments $\mathbf{y}$ to $\mathbf{Y}$ and by $mod(\mathcal{T})$ those assignments that satisfy $\mathcal{T}$ . Here, an assignment is a subset of $lit(\mathbf{Y})$ , which contains exactly one of a and $\neg a$ for each variable $a \in \mathbf{Y}$ . Given a partial assignment $\mathbf{y} \in {int}(\mathbf{Y})$ for a theory $\mathcal{T}$ over $\mathbf{X}$ , we denote by $\mathcal{T}\mid_{\mathbf{y}}$ the theory over $\mathbf{X}\setminus \mathbf{Y}$ obtained by conditioning $\mathcal{T}$ on $\mathbf{y}$ . We use $\models$ for the usual entailment relation of propositional logic.

Algebraic Model Counting (AMC) is a general framework for quantitative reasoning over models that generalizes weighted model counting to the semiring setting (Kimmig et al. Reference Kimmig, Van den Broeck and De Raedt2017).

Some examples of well-known commutative semirings are

• $\mathcal{P} = ([0,1], +, \cdot, 0, 1)$ , the probability semiring,

• $\mathcal{S}_{\max, \cdot} = (\mathbb{R}_{\geq 0}, \max, \cdot, 0, 1)$ , the max-times semiring,

• $\mathcal{S}_{\max, +} = (\mathbb{R}\cup\{-\infty\}, \max, +, -\infty, 0)$ , the max-plus semiring,

• EU $ = (\{(p, eu) \mid p \in [0,1], eu \in \mathbb{R}\}, +, \otimes, (0,0), (1,0))$ , the expected utility semiring, where addition is coordinate-wise and $ (a_1, b_1) \otimes (a_2, b_2) = (a_1 \cdot a_2, a_2\cdot b_1\,+ a_1\!\cdot\!b_2).$

An AMC instance $A = (\mathcal{T}, \mathcal{S}, \alpha)$ consists of a theory $\mathcal{T}$ over variables $\mathbf{X}$ , a commutative semiring $\mathcal{S}$ and a labeling function $\alpha: lit(\mathbf{X}) \rightarrow S$ . The value of A is

\begin{equation*}\mathit{AMC}(A)= {{\textstyle\bigoplus}}_{M\in mod(\mathcal{T})}{{\textstyle\bigotimes}}_{l\in M}\alpha(l).\end{equation*}

A prominent example of AMC is inference in probabilistic logic programming, which uses the probability semiring $\mathcal{P}$ and a theory for a probabilistic logic program (PLP). A PLP $\mathcal{L}=\mathbf{F}\cup R$ is a set $\mathbf{F}$ of probabilistic facts $p\!::\! f$ and a set R of rules $h \leftarrow b_1,\dots,b_{n}, {not\,} c_1, \dots, {not\,} c_{m}$ such that each assignment (world) $\mathbf{f}$ to $\mathbf{F}$ leads to exactly one model. By abuse of notation, we use $\mathcal{L}$ also for a propositional theory with the same models. We denote the Herbrand base, that is, the set of all ground atoms in $\mathcal{L}$ , by $\mathbf{H}$ . The success probability of a query q is ${SUCC(q)} = \sum_{\mathbf{h}\in mod(\mathcal{L}\mid_{q})} P(\mathbf{h})$ , the sum of the probabilities $P(\mathbf{h}) = \prod_{l \in \mathbf{h}} \alpha(l)$ of the models $\mathbf{h}$ where q succeeds. Here, $\alpha(f) = p, \alpha(\lnot f) = 1-p$ for $p\!::\! f \in \mathbf{F}$ and $\alpha(l) = 1$ , otherwise.

Example 1 (Running Example) We use the following probabilistic logic program $\mathcal{L}_{ex}$ throughout the paper.

\begin{equation*}0.4\!::\! a \hspace{50pt} c \leftarrow a \hspace{50pt} 0.6\!::\! b \hspace{50pt} d \leftarrow b \end{equation*}

Its four worlds are $\mathbf{f}_1=\{a,b\}, \mathbf{f}_2=\{a,\lnot b\}, \mathbf{f}_3=\{\lnot a, b\}, \mathbf{f}_4 = \{\lnot a, \lnot b\}$ , and the probabilities of their models $\mathbf{h}_1, \mathbf{h}_2, \mathbf{h}_3, \mathbf{h}_4$ are $P(\mathbf{h}_1) = 0.24, P(\mathbf{h}_2) = 0.16, P(\mathbf{h}_3) = 0.36$ and $P(\mathbf{h}_4) = 0.24$ . The query c succeeds in the models $\mathbf{h}_1$ and $\mathbf{h}_2$ , thus $SUCC(c) = P(\mathbf{h}_1)+P(\mathbf{h}_2)=0.4$ .

Kimmig et al. (Reference Kimmig, Van den Broeck and De Raedt2017) showed that Knowledge Compilation (KC) to sd-DNNFs solves any AMC problem. sd-DNNFs are special negation normal forms (NNFs). An NNF (Darwiche Reference Darwiche2004) is a rooted directed acyclic graph in which each leaf node is labeled with a literal, true or false, and each internal node is labeled with a conjunction $\wedge$ or disjunction $\vee$ . For any node n in an NNF graph, $\mathit{Vars}(n)$ denotes all variables in the subgraph rooted at n. By abuse of notation, we use n also to refer to the formula represented by the graph n. sd-DNNFs are NNFs that satisfy the following additional properties:

1. Decomposability (D): $\mathit{Vars}(n_i)\!\cap\!\mathit{Vars}(n_j) = \emptyset$ for any two children $n_i$ and $n_j$ of an and-node.

2. Determinism (d): $n_i \wedge n_j$ is logically inconsistent for any two children $n_i$ and $n_j$ of an or-node.

3. Smoothness (s): $\mathit{Vars}(n_i) = \mathit{Vars}(n_j)$ for any two children $n_i$ and $n_j$ of an or-node.

In order to solve second level problems, we need Constrained KC (CKC), that is, an additional property on NNFs, apart from s, d and D, that restricts the order in which variables occur.

Note that $\mathbf{X}$ -first NNFs contain a node $n_{\mathbf{x}}$ equivalent to $n\mid_{\mathbf{x}}$ for each $\mathbf{x} \in {int}(\mathbf{X})$ .

Fig. 1. Two sd-DNNFs for $\mathcal{L}_{ex}$ . Mixed nodes for partition $\{a,b\},\{c,d\}$ are circled red and pure nodes are boxed blue or boxed green, when they are from $\{a,b\}$ or $\{c,d\}$ , respectively.

3 Second Level Algebraic Model Counting

We now introduce Second Level Algebraic Model Counting (2AMC), a generalization of AMC that provides a unified view on second level problems.

• $\mathcal{T}$ is a propositional theory,

• $(\mathbf{X}_I, \mathbf{X}_O)$ is a partition of the variables in $\mathcal{T}$ ,

• $\mathcal{S}_j = (S_j, {{\oplus}}^{j}, {{\otimes}}^{j}, e_{{{\oplus}}^{j}}, e_{{{\otimes}}^{j}})$ for $j \in\{ I,O\}$ is a commutative semiring,

• $\alpha_j : lit(\mathbf{X}_j) \rightarrow S_j$ for $j \in\{ I, O\}$ is a labeling function for literals, and

• $t : S_I \rightarrow S_O$ is a weight transformation function that respects $t(e_{{{\oplus}}^{I}}) = e_{{{\oplus}}^{O}}$ .

The value of A, denoted by $\mathrm{2AMC}(A)$ , is defined as

\begin{align*}\mathrm{2AMC}(A) = {{\textstyle\bigoplus}}^{O}_{\mathbf{x}_O \in {int}(\mathbf{X}_O)} {{\textstyle\bigotimes}}^{O}_{x \in \mathbf{x}_O} \alpha_O(x) {{\otimes}}^{O} t\left({{\textstyle\bigoplus}}^{I}_{\mathbf{x}_I \in mod(\mathcal{T}\mid_{\mathbf{x}_O})} {{\textstyle\bigotimes}}^{I}_{y \in \mathbf{x}_I} \alpha_I(y)\right).\end{align*}

An AMC instance is a 2AMC instance, where $\mathbf{X}_O = \emptyset$ and t is the identity function, meaning we only sum up weights over $\mathcal{S}_I$ . Intuitively, the idea behind 2AMC is that we solve an inner AMC instance over the variables in $\mathbf{X}_I$ for each assignment to $\mathbf{X}_O$ . Then we apply the transformation function to the result, thus replacing the inner summation by a corresponding element from the outer semiring, and solve a second outer AMC instance over the variables in $\mathbf{X}_O$ .

Example 3 (cont.) Consider the question whether it is more likely for c to be true or false in $\mathcal{L}_{ex}$ from Example 1, that is, we want to find $\arg\max_{\mathbf{c}\in{int}(c)}\mathit{SUCC(\mathbf{c})}$ . To keep notation simple, we consider $\max$ rather than $\arg\max$ (see the discussion below Definition 5 for full details). Denoting the label of literal l by $\alpha(l)$ as in the definition of $\mathit{SUCC}$ , the task corresponds to

\begin{equation*}\max_{\mathbf{c}\in{int}(c)} \alpha(\mathbf{c})\sum_{\mathbf{h}\in mod(\mathcal{L}_{ex}\mid_{\mathbf{c}})} \prod_{l \in \mathbf{h}}\alpha(l).\end{equation*}

We thus have a 2AMC task with outer variables $\{c\}$ , inner variables $\{a,b,d\}$ and the probability semiring and max-times semiring as inner and outer semiring, respectively, and both kinds of labels given by $\alpha$ . The formal definition of this 2AMC instance is $A_{ex} = (\mathcal{L}_{ex}, \{a,b,d\},\{c\}, \alpha_I, \alpha_O, \mathcal{P}, \mathcal{S}_{\max,\cdot}, id)$ , where $\alpha_I(l)$ is the probability of l if $l \in lit(\{a,b\})$ and 1 if $l \in lit(\{d\})$ , and $\alpha_O(l) = 1$ , $l \in lit(\{c\})$ . We can further evaluate the value as follows:

\begin{align*} \mathrm{2AMC}(A_{ex})&= \max_{\mathbf{c}\in{int}(c)} 1 \cdot \sum_{\substack{\mathbf{a}\in{int}(\{a\}),\mathbf{b}\in{int}(b), \mathbf{d}\in{int}(d)\\ \mathbf{a}=\mathbf{c}, \mathbf{b}=\mathbf{d}}} 1 \cdot \alpha_I(\mathbf{a})\cdot \alpha_I(\mathbf{b})\\ & = \textstyle\max\{\alpha_{I}(a)\alpha_{I}(b) + \alpha_{I}(a)\alpha_{I}(\lnot b), \alpha_{I}(\lnot a)\alpha_{I}(b) + \alpha_{I}(\lnot a)\alpha_{I}(\lnot b)\} \\ & = \textstyle\max\{0.4\cdot 0.6 + 0.4\cdot 0.4,0.6\cdot 0.6 + 0.6\cdot 0.4\} = \textstyle\max\{0.4, 0.6\} = 0.6\end{align*}

that is, the most likely value is $0.6$ and corresponds to $\neg c$ .

Before we further illustrate this formalism with tasks from the literature, we prove that 2AMC can be solved in polynomial time on $\mathbf{X}_O$ -first sd-DNNFs. A similar result is already known for DTProbLog (Derkinderen and De Raedt Reference Derkinderen and De Raedt2020) and SC-ProbLog (Latour et al. Reference Latour, Babaki, Dries, Kimmig, Van den Broeck and Nijssen2017).

We illustrate 2AMC with three tasks from quantitative logic programming.

Maximum a Posteriori Inference A typical second level probabilistic inference task is maximum a posteriori inference, which involves maximizing over one set of variables while summing over another, as in Example 3.

Find the most probable assignment $\mathbf{q}$ to $\mathbf{Q}$ given the evidence $\mathbf{e}$ , with $\mathbf{R} = \mathbf{H}\setminus (\mathbf{Q}\cup\mathbf{E})$ :

\begin{equation*}\mathit{MAP}(\mathbf{Q}|\mathbf{e})= \arg\max_{\mathbf{q}} P(\mathbf{Q}=\mathbf{q}|\mathbf{e}) = \arg\max_{\mathbf{q}} \sum_{\mathbf{r}}P(\mathbf{Q}=\mathbf{q},\mathbf{e},\mathbf{R}=\mathbf{r} )\end{equation*}

MAP as 2AMC. Solving MAP requires (1) summing probabilities over the truth values of the atoms in $\mathbf{R}$ (with fixed truth values for atoms in $\mathbf{E}\cup\mathbf{Q}$ ), and (2) determining truth values of the atoms in $\mathbf{Q}$ that maximize this inner sum. Thus, we have $\mathbf{X}_O=\mathbf{Q}$ and $\mathbf{X}_I=\mathbf{E}\cup\mathbf{R}$ .

The inner problem corresponds to usual probabilistic inference, that is, $\mathcal{S}_I=\mathcal{P}$ and $\alpha_I$ assigns 1 to the literals in $\mathbf{e}$ and 0 to their negations, p and $1-p$ to the positive and negative literals for probabilistic facts $p\!::\! f$ that are not part of $\mathbf{E}$ , and 1 to both literals for all other variables in $\mathbf{X}_I$ .

We choose $\mathcal{S}_O = (\mathbb{R}_{\geq 0} \times 2^{lit(\mathbf{Q})}, {{\oplus}}, {{\otimes}}, (0, \emptyset), (1, \emptyset))$ as the max-times semiring combined with the subsets of the query literals $lit(\mathbf{Q})$ to remember the assignment that was used. Here, $(r_1, S_1) {{\oplus}} (r_2, S_2)$ is $(r_1, S_1)$ if $r_1 > r_2$ and $(r_2, S_2)$ if $r_1 < r_2$ . To ensure commutativity, if $r_1=r_2$ , the sum is $(r_1, \min_{>}(S_1,S_2))$ where $>$ is some arbitrary but fixed total order on $2^{lit(\mathbf{Q})}$ . The product $(r_1, S_1) {{\otimes}} (r_2, S_2)$ is defined as $(r_1 \cdot r_2, S_1 \cup S_2)$ . The weight function is given by $\alpha_O(l)= (p, \{l\})$ if $l=f$ for probabilistic fact $p\!::\! f$ , $\alpha_O(l) = (1-p, \{l\})$ if $l= \lnot f$ for probabilistic fact $p\!::\! f$ , and $\alpha_O(l)=(1, \{l\})$ otherwise. The transformation function is the function $t(p)=(p, \emptyset)$ .

Maximizing Expected Utility Another second level probabilistic task is maximum expected utility (Van den Broeck et al. Reference Van den Broeck, Thon, van Otterlo and De Raedt2010; Derkinderen and De Raedt Reference Derkinderen and De Raedt2020), which introduces an additional set of variables $\mathbf{D}$ whose truth value can be arbitrarily chosen by a strategy $\sigma(\mathbf{D})=\mathbf{d}, \mathbf{d}\in int(\mathbf{D})$ , and is neither governed by probability nor logical rules. A utility function u maps each literal l to a reward $u(l)\in \mathbb{R}$ for l being true.

Find a strategy $\sigma^*$ that maximizes the expected utility:

\begin{equation*}\sigma^*= \arg \max_{\sigma} \sum_{M\in mod(\mathbf{F}\cup R\cup\sigma(\mathbf{D}))} (\prod_{l\in M} \alpha(l)) (\sum_{l\in M} u(l)),\end{equation*}

where the label $\alpha(l)$ of literal l is as defined for $\mathit{SUCC}$ .

MEU as 2AMC. Solving MEU involves (1) summing expected utilities of models over the non-decision variables (with fixed truth values for $\mathbf{D}$ ), and (2) determining truth values of the atoms in $\mathbf{D}$ that maximize this inner sum. Thus, we have $\mathbf{X}_O=\mathbf{D}$ and $\mathbf{X}_I=\mathbf{H}\setminus\mathbf{D}$ .

The inner problem is solved by the expected utility semiring $\mathcal{S}_I=EU$ , with $\alpha_I$ mapping literal l to $(p_l, p_l\cdot u(l))$ if l is a probabilistic literal with probability $p_l$ , and to (1, u(l)) otherwise.

The basis for solving the outer problem is the max-plus semiring $\mathcal{S}_{\max, +}$ , with $\alpha_O$ the utility function u, and the transformation function $t((p, pu)) = pu$ , if $p\neq 0$ and $t((0, pu)) = -\infty$ . This is extended to argmax using the same idea as for MAP.

Probabilistic Inference with Stable Models A more recent second level probabilistic task is probabilistic inference with stable model semantics (Totis et al. Reference Totis, Kimmig and De Raedt2021; Skryagin et al. Reference Skryagin, Stammer, Ochs, Dhami and Kersting2021). Inference of success probabilities reduces to a variant of weighted model counting, where the weight of a (stable) model of a program $\mathcal{L}=R\cup\mathbf{F}$ is normalized with the number of models sharing the same assignment $\mathbf{f}$ to the probabilistic facts $\mathbf{F}$ :

\begin{equation*} SUCC^{sm}(q)= \sum_{\mathbf{h} \in mod(\mathcal{L} \mid_{\mathbf{f}}), \mathbf{f}\in{int}(\mathbf{F})} \frac{|mod(\mathcal{L} \mid_{\mathbf{f}\cup\{q\}})|}{|mod(\mathcal{L} \mid_{\mathbf{f}})|}\cdot\prod_{l\in \mathbf{h}}\alpha(l),\end{equation*}

where the label $\alpha(l)$ of literal l is as defined for $\mathit{SUCC}$ .

SUCC ^sm as 2AMC. Computing SUCC^sm requires (1) counting the number of models for a given total choice and those that satisfy the query q, and (2) summing the normalized probabilities. Thus, we have $\mathbf{X}_O=\mathbf{F}$ and $\mathbf{X}_I=\mathbf{H}\setminus\mathbf{F}$ . $\mathcal{S}_I$ is the semiring over pairs of natural numbers, $\mathcal{S}_I=(\mathbb{N}^2, +, \cdot, (0,0), (1,1))$ , where operations are component-wise. $\alpha_I$ maps $\lnot q$ to (0,1) and all other literals to (1,1). The first component thus only counts the models where q is true ( $|mod(\mathcal{L} \mid_{\mathbf{f}\cup\{q\}})|$ ), whereas the second component counts all models ( $|mod(\mathcal{L} \mid_{\mathbf{f}})|$ ). The transformation function is given by $t((n_1,n_2))=\frac{n_1}{n_2}$ . The outer problem then corresponds to usual probabilistic inference, that is, $\mathcal{S}_O=\mathcal{P}$ , and $\alpha_O$ assigns p and $1-p$ to the positive and negative literals for probabilistic facts $p\!::\! f$ , respectively, and 1 to all other literals.

4 Weakening X-firstness

While any 2AMC problem can be solved in polynomial time on an $\mathbf{X}_O$ -first sd-DNNF representing the logical theory $\mathcal{T}$ , such an sd-DNNF can be much bigger than the smallest (ordinary) sd-DNNF for $\mathcal{T}$ , as the $\mathbf{X}$ -firstness may severely restrict the order in which variables are decided. In the following, we show that for a wide class of transformation functions, we can exploit the logical structure of the theory to relax the $\mathbf{X}_O$ -first property.

Recall that a 2AMC task includes an AMC task for every assignment $\mathbf{x}_O$ to the outer variables, which sums over all assignments $\mathbf{x}_I$ to the inner variables that extend $\mathbf{x}_O$ to a model of the theory. Consider the CNF $\mathcal{T} = \{y \vee \neg x, \neg y \vee x\}$ and let $\mathbf{X}_O = \{y\}$ and $\mathbf{X}_I = \{x\}$ . The value of the outer variable y already determines the value of the inner variable x. Distributivity allows us to pull x out of every inner sum, as each such sum only involves one of the literals for x. If it does not matter whether we first apply the transformation function and then multiply or the other way around, we have a choice between keeping x in the inner semiring, or pushing its transformed version to the outer semiring. Thus, we can decide between an $\mathbf{X}_O$ -first or an $\mathbf{X}_O\cup\{x\}$ -first sd-DNNF. Naturally, the more such variables we have, the more freedom we gain.

This situation might also occur after we have already decided some of the variables. Consider the CNF $\mathcal{T}' = \{z \vee y \vee \neg x, z \vee \neg y \vee x\}$ and let $\mathbf{X}_O = \{z,y\}$ and $\mathbf{X}_I = \{x\}$ . If we set z to true, both y and x can take any value, therefore the value of x is not determined by z and y. However, if we set z to false, we are in the same situation as above and can move x to $\mathbf{X}_O$ on a local level.

We formalize this, starting with definability to capture when a variable is determined by others.

The intuition here is that we can decide variables from $\mathbf{Y}$ earlier, if they are defined by the variables in $\mathbf{X}$ in terms of the theory $\mathcal{T}$ conditioned on the decisions we have already made. Thus, we only decide the variables in $\mathbf{X}$ -first modulo definability.

The following lemma generalizes this example from 2 to n pairs of equivalent variables.

We see that $\mathbf{X}/\mathbf{D}$ -first sd-DNNFs can be much smaller than $\mathbf{X}$ -first sd-DNNFs, even on very simple propositional theories. It remains to show that we maintain tractability. As in the beginning of this section, we want to regard defined inner variables as outer variables. For this to work it must not matter whether we first multiply and then apply the transform t or the other way around, that is, t must be a homomorphism for the multiplications of the semirings.

Definition 10 (Monoid Homomorphism, Generated Monoid) Let $\mathcal{M}_i = (M_i, \odot^i, e_{\odot^i})$ for $i = 1,2$ be monoids. Then a monoid homomorphism from $\mathcal{M}_1$ to $\mathcal{M}_2$ is a function $f: M_1 \rightarrow M_2$ such that

\begin{align*} \text{for all $m, m' \in M_1$ } & & f(m \odot^1 m') = f(m) \odot^2 f(m') & & \text{ and } & & f(e_{\odot^1}) = e_{\odot^2}.\end{align*}

Furthermore, for a subset $M' \subseteq M$ of a monoid $\mathcal{M} = (M, \odot, e_{\odot})$ the monoid generated by M′, denoted $\langle M' \rangle_{\mathcal{M}}$ , is $\mathcal{M}^* = (M^*, \odot, e_{\odot})$ , where $M' \subseteq M^*$ and $M^* $ is the subset minimal set such that $\mathcal{M}^*$ is a monoid.

Example 8 (cont.) Consider again the 2AMC instance $A_{ex}$ from Example 3. Since a is defined in terms of c, we want to argue that the following equality holds, allowing us to see a as an outer variable:

\begin{align*} &\textstyle\max_{\mathbf{x_O} \in {int}(\{c\})} \alpha_{O}(\mathbf{x_O})\cdot id\!\left(\sum_{\mathbf{x_I} \in mod(\mathcal{L}_{ex}\mid_{\mathbf{x}_O})} \prod_{y \in \mathbf{x_I}} \alpha_I(y)\!\right)\\ = & \textstyle\max_{\mathbf{x_O} \in {int}(\{c\}),\mathbf{x_{O'}} \in {int}(\{a\})}\alpha_{O}(\mathbf{x_O})\cdot id(\alpha_{I}(\mathbf{x_{O'}})\!)\cdot id\!\left(\!\sum_{\mathbf{x_I} \in mod(\mathcal{L}_{ex}\mid_{\mathbf{x}_O \cup \mathbf{x}_{O'}})} \prod_{y \in \mathbf{x_I}} \alpha_I(y)\!\right)\end{align*}

Here, this is easy to see since id is a homomorphism between the monoid $([0,1], \cdot)$ of the inner probability semiring and the monoid $(\mathbb{R}_{\geq 0}, \cdot)$ of the outer max-times semiring, as $id: [0,1]\rightarrow \mathbb{R}_{\geq 0}$ , for any $p,q\in[0,1]$ , $id(p \cdot q) = p\cdot q = id(p) \cdot id(q)$ , and $id(1)=1$ .

In general, instead of applying the transform to a sum of products of literal labels for a set of variables, we want to apply it independently to (1) the literal labels of a defined variable and (2) the inner sum restricted to the remaining variables. It is therefore sufficient if the equality is valid for the monoid generated by the values we encounter in these situations, rather than for all values from the inner monoid′s domain. As we will illustrate for MEU at the end of this section, the transform of some 2AMC tasks only satisfies this weaker but sufficient condition. The following definition captures this idea, where the two subsets of O(A) correspond to the values observed in cases (1) and (2), respectively:

Definition 11 (Observable Values) Let $A = (\mathcal{T}, \mathbf{X}_I, \mathbf{X}_O, \alpha_I, \alpha_O, \mathcal{S}_I, \mathcal{S}_O, t)$ be a 2AMC instance. Then the set of observable values of A, denoted O(A) is

\begin{align*} &\{ \alpha_I(l) \mid \mathbf{x}_O \in {int}(\mathbf{X}_O), y \in \mathbf{D}(\mathcal{T}\cup \mathbf{x}_O, \mathbf{X}_O), l \in \{y, \neg y\}\}\\ \cup &\{{{\textstyle\bigoplus}}^{I}_{\mathbf{x}_I \in mod(\mathcal{T} \mid_{\mathbf{x}_O})} {{\textstyle\bigotimes}}^{I}_{y \in \mathbf{x}_I} \alpha_I(y) \mid \mathbf{x}_O \in {int}(\mathbf{X}_O \cup \mathbf{D}^*), \mathbf{D}^* \subseteq \mathbf{D}(\mathcal{T}\cup \mathbf{x}_O, \mathbf{X}_O)\}.\end{align*}

With this in mind, we can state our main result.

• $\mathcal{T}$ is an $\mathbf{X}/\mathbf{D}$ -first sd-DNNF

• t is a homomorphism from the monoid $\langle O(A) \rangle_{(R_I, {{\otimes}}^{I}, e_{{{\otimes}}^{I}})}$ generated by the observable values to $(R_O, {{\otimes}}^{O}, e_{{{\otimes}}^{O}})$ .

Despite the fact that checking which variables are defined for each partial assignment $\mathbf{x}$ is not feasible, as checking definability is co-NP-complete and there are more than $2^{|\mathbf{X}_O|}$ partial assignments, this result does have implications for constrained KC in practice.

Firstly, we can check which variables are defined by $\mathbf{X}_O$ in terms of the whole theory $\mathcal{T}$ , and use this to generate a variable order for compilation that leads to an $\mathbf{X}/\mathbf{D}$ -first sd-DNNF with a priori guarantees on its size. We discuss this in Section 5.

Secondly, as entailment is a special case of definability, Theorem 12 justifies the use of unit propagation during compilation, which dynamically adapts the variable order when variables are entailed by the already decided ones, and thus may violate $\mathbf{X}$ -firstness.

We still need to verify that our three example tasks satisfy the preconditions of Theorem 12, that is, their transformation functions are monoid homomorphisms on the observable values. For MAP and SUCC^sm this is easy to prove, as the transformation is already a homomorphism on all values. For MEU, however, the restriction to the monoid generated by the observable values is crucial, as the transformation function is only a homomorphism for tuples (p, pu) with $p \in \{0,1\}$ , which may not be the case in general. In the DTProbLog setting, however, assignments to decision facts and probabilistic facts are independent, and every assignment to both sets extends to a single model of the theory. Together with the (1,u)-labels of the remaining atoms, this ensures that the result of the inner sum is of the form (1,x), and MEU thus meets the criteria.

5 Implementation

The general pipeline of PLP solvers takes a program, grounds it and optionally simplifies it or breaks its cycles. Using standard KC tools, this program is then compiled into a tractable circuit either directly or via conversion to CNF. We extended this pipeline in ${\small\textsf{aspmc}}$ (Eiter et al. Reference Eiter, Hecher and Kiesel2021) and smProbLog (Totis et al. Reference Totis, Kimmig and De Raedt2021) to compile programs, via CNF, into $\mathbf{X}/\mathbf{D}$ -first circuits.

To obtain $\mathbf{X}/\mathbf{D}$ -first circuits we need to specify a variable order in which all variables in $\mathbf{X}$ are decided first modulo definedness. Preferably, the chosen order should also result in efficient compilation and thus a small circuit. Korhonen and Järvisalo (Reference Korhonen and Järvisalo2021) have shown how to generate variable orders from tree decompositions of the primal graph of the CNF that result in (unconstrained) sd-DNNFs whose size is bounded by the width of the decomposition. We adapt this result to our constrained setting, where we need to ensure that the tree decomposition satisfies an additional property that allows compilation to essentially consider the non-defined inner variables and the outer variables independently. We first define the necessary concepts.

A tree decomposition (TD) for a graph G is a pair $(T, \chi)$ , where T is a tree and $\chi$ is a labeling of V(T) by subsets of V(G) s.t. (i) for all nodes $v \in V(G)$ there is $t \in V(T)$ s.t. $v \in \chi(t)$ ; (ii) for every $(v_1, v_2) \in V(E)$ there exists $t \in V(T)$ s.t. $v_1, v_2 \in \chi(t)$ and (iii) for all nodes $v \in V(G)$ the set of nodes $\{t \in V(T) \mid v \in \chi(t)\}$ forms a connected subtree of T. The width of $(T, \chi)$ is $\max_{t \in V'} |\chi(t)| - 1$ .

Next, we show that restricted TDs allow $\mathbf{X}/\mathbf{D}$ -first compilation with performance guarantees:

To find a TD of small width for which the lemma applies, we proceed as follows. Given a CNF $\mathcal{T}$ and partition $\mathbf{X}_I, \mathbf{X}_O$ of the variables in inner and outer variables, we first compute $\mathbf{D}(\mathcal{T}, \mathbf{X}_O)$ by using an NP-oracle. Lagniez et al. (Reference Lagniez, Lonca and Marquis2016) showed that this is possible. As the width of a suitable TD is at least the size of the separator minus one, we first approximate a minimum size separator $\mathbf{S} \subseteq \mathbf{X} \cup \mathbf{D}(\mathcal{T}, \mathbf{X})$ using clingo (Gebser et al. Reference Gebser, Kaminski, Kaufmann and Schaub2014) with a timeout of 30 s. To ensure that the TD contains a node t′ with $\mathbf{S}\subseteq \chi(t')$ , we add the clique over the vertices in $\mathbf{S}$ to the primal graph, that is, we generate a tree decomposition $(T, \chi)$ of $\mathrm{PRIM}(\mathcal{T}) \cup \text{Clique}(\mathbf{S})$ . For this, we use flow-cutter (Dell et al. Reference Dell, Komusiewicz, Talmon and Weller2017) with a timeout of 5 s. The resulting TD either already contains a node with $\mathbf{S}= \chi(t')$ and thus satisfies the preconditions of Lemma 14, or can be modified locally to one that does by splitting the node with $\mathbf{S}\subset \chi(t')$ .

${\small\textsf{aspmc}}$ : ^{Footnote 1} We use the above strategy to generate a variable order, which is then given to c2d (Darwiche Reference Darwiche2004) or miniC2D (Oztok and Darwiche Reference Oztok and Darwiche2015) together with the CNF to compile an $\mathbf{X}/\mathbf{D}$ -first circuit, on which we evaluate the 2AMC task. We stress that c2d – contrary to miniC2D – always uses unit propagation during compilation, and thus can only be used due to Theorem 12.

Currently, ${\small\textsf{aspmc}}$ supports DTProbLog, MAP and smProbLog programs as inputs.

sm ProbLog: ^{Footnote 2} The smProbLog implementation of Totis et al. (Reference Totis, Kimmig and De Raedt2021) uses dsharp (Aziz et al. Reference Aziz, Chu, Muise and Stuckey2015) to immediately compile the logic program to a d-DNNF, which prevents a direct application of our new techniques. Our adapted version obtains a CNF $\mathcal{T}$ and variable ordering as in ${\small\textsf{aspmc}}$ , which it then compiles to SDD (Darwiche Reference Darwiche2011) using the PySDD library ^{Footnote 3} as in standard ProbLog. SDDs can be seen as a special case of sd-DNNFs. The main difference is that for each branch the variables are decided in the same order.

6 Experimental evaluation

Our experimental evaluation addresses the following questions:

1. Q1: How does exploiting definedness influence the efficiency of 2AMC solving?

2. Q2: How does ${\small\textsf{aspmc}}$ compare to task-specific solvers from the PLP literature?

3. Q3: How does our second level approach compare to the first level approach when definedness reduces 2AMC to AMC?

6.2 Results

Q1: How does exploiting definedness influence the efficiency of 2AMC solving? To answer the first question, we use all proper 2AMC benchmarks, and focus on the 2AMC task itself, that is, we start from the labeled CNF corresponding to the instance. We consider four different settings obtained by independently varying two dimensions: constraining compilation to either $\mathbf{X}$ -first or $\mathbf{X}/\mathbf{D}$ -first, and compiling to either sd-DNNF using c2d or to SDD using miniC2D.

On the left of Figure 2, we plot the width of the tree decompositions the solver uses to determine the variable order in the $\mathbf{X}$ -first or $\mathbf{X}/\mathbf{D}$ -first case, respectively. Recall from Lemma 14 that this width appears in the exponent of the compilation time bound. The optimal tree decomposition’s width in the $\mathbf{X}/\mathbf{D}$ -first case is at most that of the $\mathbf{X}$ -first case, and would thus result in points on or below the black diagonal only. In practice, we observe many points close to the diagonal, with two notable exceptions. MAP instances with high width tend to be slightly above the diagonal, whereas MAP grids are mostly clearly below the diagonal. These results can be explained by the shape of the problems and the fact that we only approximate the optimal decomposition, as this is a hard task. We note that for many of the benchmarks, the amount of variables defined in terms of the outer variables (decision variables for MEU, query variables for MAP, probabilistic facts for SUCCSM) is limited. The exception are the MAP grids, where the choice of queries entails definedness. We plot the same data restricted to the instances solved within the time limit on the right of Figure 2, along with summary statistics on the number of instances solved for three ranges of $\mathbf{X}/\mathbf{D}$ -width in the caption. We observe that almost no instances with $\mathbf{X/D}$ -width above 40 are solved. At the same time, almost all instances with $\mathbf{X/D}$ -width below 20 are solved, including many cases with $\mathbf{X}$ -width above 40, where we thus see a clear benefit from exploiting definedness.

Fig. 2. Q1: Comparison of tree decomposition width for $\mathbf{X}$ -first and $\mathbf{X}/\mathbf{D}$ -first variable order, across all 2AMC instances (left) and for solved 2AMC instances only (right). We solve 822 of the 825 instances with $\mathbf{X}/\mathbf{D}$ -width at most 20, 118 of the 219 instances with $\mathbf{X}/\mathbf{D}$ -width between 21 and 40, and 7 of the 353 instances with $\mathbf{X}/\mathbf{D}$ -width above 40.

In Figure 3, we plot the running times per instance for the different solvers. Given the width results, we distinguish between MAP grids and the remaining cases. On the grids, taking into account definedness results in clear performance gains when compiling SDDs (using miniC2D). On the other hand, compiling sd-DNNFs with c2d shows only a marginal difference between $\mathbf{X}$ and $\mathbf{X}/\mathbf{D}$ variable orders. The reason is that c2d implicitly exploits definedness even when given the $\mathbf{X}$ -first order through its use of unit propagation. SDD compilation, on the other hand, cannot deviate from the given order, and only benefits from definedness if it is reflected in the variable order. On the other benchmarks, with fewer defined variables, the variable order has little effect within the same circuit class, but sd-DNNFs outperform SDDs, likely because unit propagation can also exploit context-dependent definedness. In the following we thus use c2d.

Fig. 3. Q1: Running times per instance for different configurations on MAP grids (left) and all other 2AMC instances (right).

Q2: How does ${\small\textsf{aspmc}}$ compare to task-specific solvers from the PLP literature? We first consider the efficiency of the whole pipeline from instance to solution on the MAP and MEU tasks, which are addressed by both ProbLog and PITA. From the plots of running times in Figure 4, we observe that all solvers outperform clingo’s model enumeration. ProbLog is slower than both ${\small\textsf{aspmc}}$ and PITA except on the MAP graphs. Among ${\small\textsf{aspmc}}$ and PITA, ${\small\textsf{aspmc}}$ outperforms PITA on MEU, and vice versa on MAP. This can be explained by the different overall approach to knowledge compilation taken in the various solvers. ${\small\textsf{aspmc}}$ always compiles a CNF encoding of the whole ground program, including all ground atoms, ProbLog compiles a similar encoding, but restricted to the part of the formula that is relevant to the task at hand, while PITA directly compiles a circuit for the truth of atoms of interest in terms of the relevant ground choice atoms. As MAP queries are limited to probabilistic facts, this allows PITA to compile a circuit for the truth of the evidence in terms of relevant probabilistic facts only, which especially for the graph setting can be significantly smaller than a complete encoding. For MEU, PITA needs one circuit per atom with a utility, which additionally need to be combined, putting PITA at a disadvantage.

Fig. 4. Q2: Running times of different solvers on MAP problem sets (top), indicated above each plot, MEU problems (bottom left) and SUCC^SM (bottom right).

For SUCC^sm, we plot running times on the modified smokers setting for the clingo baseline, ${\small\textsf{aspmc}}$ and three variants of the dedicated ProbLog implementation, namely the original implementation of Totis et al. (Reference Totis, Kimmig and De Raedt2021) that compiles to d-DNNF as well as our modified implementation compiling to $\mathbf{X}$ -first and $\mathbf{X}/\mathbf{D}$ -first SDDs. These problems appear to be hard in general, but we observe a clear benefit from the constrained compilation enabled in our approach.

Q3: How does our second level approach compare to the first level approach when definedness reduces 2AMC to AMC?

On the regular smokers benchmark, smProbLog and ProbLog semantics coincide, that is, all inner variables of the 2AMC task are defined, and it thus reduces to AMC. In Figure 5, we plot running times for the AMC and 2AMC variants of both ${\small\textsf{aspmc}}$ and ProbLog. For ProbLog, there is a clear gap between the two approaches, which is at least in part due to the fact that ProbLog only compiles the relevant part of the program, whereas smProbLog compiles the full theory. ${\small\textsf{aspmc}}$ outperforms ProbLog on the harder instances, with limited overhead for the second level task.

Fig. 5. Q3: Running times of different solvers on SUCC programs.

Supplementary material

To view supplementary material for this article, please visit http://doi.org/10.1017/S147106842200014X.