2.1 Probabilistic answer set programming

We consider the Credal Semantics (CS) (Lukasiewicz, Reference Lukasiewicz and Godo2005; Cozman and Mauá, Reference Cozman, Mauá, Hommersom and Abdallah2016; Mauá and Cozman, Reference Mauá and Cozman2020) that associates a meaning to Answer Set Programs extended with probabilistic facts (De Raedt et al., Reference De Raedt, Kimmig, Toivonen and Veloso2007) of the form $p::a$ where $p$ is the probability associated with the atom $a$ . Intuitively, such notation means that the fact $a$ is present in the program with probability $p$ and absent with probability $1-p$ . These programs are called Probabilistic Answer Set Programs (PASP, and we use the same acronym to also indicate Probabilistic Answer Set Programming – the meaning will be clear from the context). A selection for the presence or absence of each probabilistic fact defines a world $w$ whose probability is $P(w) = \prod _{a \in w} p \cdot \prod _{a \not \in w} (1-p)$ , where with $a \in w$ we indicate that $a$ is present in $w$ and with $a \not \in w$ that $a$ is absent in $w$ . A program with $n$ probabilistic facts has $2^n$ worlds. Let us indicate with $W$ the set of all possible worlds. Each world is an answer set program and it may have zero or more answer sets but the CS requires at least one. If this holds, the probability of a query $q$ (a conjunction of ground literals) is defined by a lower and an upper bound. A world $w$ contributes to both the lower and upper probability if each of its answer sets contains the query, that is it is a cautious consequence. If only some answer sets contain the query, that is it is a brave consequence, $w$ only contributes to the upper probability. If the query is not present, we have no contribution to the probability bounds from $w$ . In formulas,

(1) \begin{align} P(q) = [\underline{P}(q),\overline{P}(q)] = [\sum _{w_i \in W \mid \forall m \in AS(w_i), \ m \models q} P(w_i), \sum _{w_i \in W \mid \exists m \in AS(w_i), \ m \models q} P(w_i)]. \end{align}

The conditional probability of a query $q$ given evidence $e$ , also in the form of conjunction of ground literals, is (Cozman and Mauá, Reference Cozman and Mauá2020):

(2) \begin{align} \underline{P}(q \mid e) = \frac{\underline{P}(q, e)}{\underline{P}(q, e) + \overline{P}(not \ q, e)}, \ \overline{P}(q \mid e) = \frac{\overline{P}(q, e)}{\overline{P}(q, e) + \underline{P}(not \ q, e)}. \end{align}

For the lower conditional probability $\underline{P}(q \mid e)$ , if $\underline{P}(q, e) + \overline{P}(not \ q, e) = 0$ and $\overline{P}(q, e) \gt 0$ , then $\underline{P}(q \mid e) = 1$ . Similarly, for the upper conditional probability $\overline{P}(q \mid e)$ , if $\overline{P}(q, e) + \underline{P}(not \ q, e) = 0$ and $\overline{P}(not \ q, e) \gt 0$ , then $\overline{P}(q \mid e) = 0$ . Both formulas are undefined if $\overline{P}(q, e)$ and $\overline{P}(not \ q, e)$ are 0.

To clarify, consider the following example.

Example 1. The following PASP encodes a simple graph reachability problem.

The first three facts are probabilistic. The rules state that there is a path between $X$ and $Y$ if they are directly connected or if there is a path between $Z$ and $Y$ and $X$ and $Z$ are connected. Two nodes may or may not be connected if there is an edge between them. There are $2^3 = 8$ worlds to consider, listed in Table 1. If we want to compute the probability of the query $q = path(1,4)$ , only $w_6$ and $w_7$ contribute to the upper bound (no contribution to the lower bound), obtaining $P(q) = [0,0.06]$ . If we observe $e = edge(2,4)$ , we get $P(q,e) = [0,0.06]$ , $P(not \ q,e) = [0.24,0.3]$ , thus $\underline{P}(q \mid e) = 0$ and $\overline{P}(q \mid e) = 0.2$ .

Table 1. Worlds and their probabilities for Example1. The second, third, and fourth columns contain 0 or 1 if the corresponding probabilistic fact is respectively false or true in the considered world. The LP/UP column indicates whether the considered world contributes to the lower (LP) or upper (UP) bound (or does not contribute, marked with a dash) for the probability of the query $path(1,4)$

Inference in PASP can be expressed as a Second Level Algebraic Model Counting Problem (2AMC) (Kiesel et al., Reference Kiesel, Totis and Kimmig2022). Given a propositional theory $T$ where its variables are grouped into two disjoint sets, $X_o$ and $X_i$ , two commutative semirings $R^{i} = (D^i, \oplus ^i, \otimes ^i, n_{\oplus ^i}, n_{\otimes ^i})$ and $R^{o} = (D^o, \oplus ^o, \otimes ^o, n_{\oplus ^o}, n_{\otimes ^o})$ , two weight functions, $w_i : lit(X_i) \to D^i$ and $w_o : lit(X_o) \to D^o$ , and a transformation function $f : D^i \to D^o$ , 2AMC is encoded as:

\begin{equation*} 2AMC(T) = \bigoplus \nolimits _{I_{o} \in \mu (X_{o})}^{o} \bigotimes \nolimits ^{o}_{a \in I_{o}} w_{o}(a) \otimes ^{o} f( \bigoplus \nolimits _{I_{i} \in \varphi (T \mid I_{o})}^{i} \bigotimes \nolimits ^{i}_{b \in I_{i}} w_{i}(b) ) \end{equation*}

where $\varphi (T \mid I_{o})$ is the set of assignments to the variables in $X_i$ such that each assignment, together with $I_o$ , satisfies $T$ and $\mu (X_{o})$ is the set of possible assignments to the variables in $X_{o}$ . In other words, 2AMC requires to solve two Algebraic Model Counting (AMC) (Kimmig et al., Reference Kimmig, Van den Broeck and De Raedt2017) tasks. The outer task focuses on the variables $X_o$ , and for each possible assignment to these variables, an inner AMC task is performed by considering $X_i$ . These two tasks are connected via a transformation function that turns values from the inner task into values for the outer task. Different instantiations of the components allow different tasks to be represented. To perform inference in PASP, Azzolini and Riguzzi (Reference Azzolini, Riguzzi, Basili, Lembo, Limongelli and Orlandini2023a) proposed to consider as innermost semiring $\mathcal{R}^{i} = (\mathbb{N}^2, +, \cdot, (0,0), (1,1))$ with $X_i$ containing the atoms of the Herbrand base except the probabilistic facts and $w_i$ mapping $not \ q$ to $(0, 1)$ and all other literals to $(1, 1)$ , as outer semiring the two-dimensional probability semiring, that is, $\mathcal{R}^{o} = ([0, 1]^2, +, \cdot, (0, 0),(1, 1))$ , with $X_o$ containing the atoms of the probabilistic facts and $w_o$ associating $(p, p)$ and $(1 - p, 1 - p)$ to $a$ and $not \ a$ , respectively, for every probabilistic fact $p :: a$ and $(1, 1)$ to all the remaining literals, and as transformation function $f((n_1,n_2))$ returning the pair $(v_{lp},v_{up})$ where $v_{lp} = 1$ if $n_1 = n_2$ , 0 otherwise, and $v_{up} = 1$ if $n_1 \gt 0$ , 0 otherwise.

2AMC can be solved via knowledge compilation (Darwiche and Marquis, Reference Darwiche and Marquis2002), often adopted in probabilistic logical settings, since it allows us to compactly represent the input theory with a tree or graph and then computing the probability of a query by traversing it. aspmc (Eiter et al., Reference Eiter, Hecher and Kiesel2021, Reference Eiter, Hecher and Kiesel2024) is one such tool, that has been proven more effective than other tools based on alternative techniques such as projected answer set enumeration (Azzolini et al., Reference Azzolini, Bellodi and Riguzzi2022; Azzolini and Riguzzi, Reference Azzolini, Riguzzi, Basili, Lembo, Limongelli and Orlandini2023a). aspmc converts the input theory into a negation normal form (NNF) formula, a tree where each internal node is labeled with either a conjunction (and-node) or a disjunction (or-node), and leaves are associated with the literals of the theory. More precisely, aspmc targets sd-DNNFs which are NNFs with three additional properties: (i) the variables of children of and-nodes are disjoint (decomposability property); (ii) the conjunction of any pair of children of or-nodes is logically inconsistent (determinism property); and (iii) children of an or-node consider the same variables (smoothness property). Furthermore, aspmc also requires $X$ -firstness (Kiesel et al., Reference Kiesel, Totis and Kimmig2022), a property that constraints the order of appearance of variables: given two disjoint partitions $X$ and $Y$ of the variables in an NNF $n$ , a node is termed pure if all variables departing from it are members of either $X$ or $Y$ . If this does not hold, the node is called mixed. An NNF has the $X$ -firstness property if for each and-node, all of its children are pure nodes or if one child is mixed and all the other nodes are pure with variables belonging to $X$ .

2.2 Parameter learning in probabilistic answer set programs

We adopt the same Learning from Interpretations framework of (Azzolini et al., Reference Azzolini, Bellodi and Riguzzi2024), that we recall here for clarity. We denote a PASP with $\mathcal{P}(\Pi )$ , where $\Pi$ is the set of parameters that should be learnt. The parameters are the probabilities associated to (a subset of) probabilistic facts. We call such facts as learnable facts. Note that the probabilities of some probabilistic facts can be fixed, that is there can be some probabilistic facts that are not learnable facts. A partial interpretation $I = \langle I^+,I^- \rangle$ is composed by two sets $I^+$ and $I^-$ that respectively represent the set of true and false atoms. It is called partial since it may specify the truth value of some atoms only. Given a partial interpretation $I$ , we call the interpretation query $q_I = \bigwedge _{i^+ \in I^+} i^+ \bigwedge _{i^- \in I^-} not \ i^-$ . The probability of an interpretation $I$ , $P(I)$ , is defined as the probability of its interpretation query, which is associated with a probability range since we interpret the program under the CS. Given a PASP $\mathcal{P}(\Pi )$ , the lower and upper probability for an interpretation $I$ are defined as

\begin{equation*} \overline {P}(I \mid \mathcal {P}(\Pi )) = \sum _{w \in \mathcal {P}(\Pi ) \ \mid \ \exists m \in AS(w), \ m \models I} P(w), \end{equation*}

\begin{equation*} \underline {P}(I \mid \mathcal {P}(\Pi )) = \sum _{w \in \mathcal {P}(\Pi ) \ \mid \ \forall m \in AS(w), \ m \models I} P(w). \end{equation*}

Definition 1 (Parameter Learning in probabilistic answer set programs) Given a PASP $\mathcal{P}(\Pi )$ and a set of (partial) interpretations $\mathcal{I}$ , the goal of the parameter learning task is to find a probability assignment to the probabilistic facts such that the product of the lower (or upper) probabilities of the partial interpretations is maximized, that is solve:

\begin{equation*} \Pi ^* = \mathrm {arg\,max}_\Pi \underline {P}(\mathcal {I} \mid \mathcal {P}(\Pi )) = \mathrm {arg\,max}_\Pi \prod _{I \in \mathcal {I}} \underline {P}(I \mid \mathcal {P}(\Pi )) \end{equation*}

which can be equivalently expressed as

(3) \begin{align} \Pi ^* = \mathrm{arg\,max}_\Pi \mathrm{log}(\underline{P}(\mathcal{I} \mid \mathcal{P}(\Pi ))) = \mathrm{arg\,max}_\Pi \sum _{I \in \mathcal{I}} \ \mathrm{log}(\underline{P}(I \mid \mathcal{P}(\Pi ))) \end{align}

also known as log-likelihood (LL). The maximum value of the LL is 0, obtained when all the interpretations have probability 1. The use of log probabilities is often preferred since summations instead of products are considered, thus possibly preventing numerical issues, especially when many terms are close to 0.

Note that, since the probability of a query (interpretation) is described by a range, we need to select whether we maximize the lower or upper probability. A solution that maximizes one of the two bounds may not be a solution that also maximizes the other bound. To see this, consider the program $\{\{q \,{:\!-}\ a, b, not\ nq\}, \{nq \,{:\!-}\ a, b, not\ q\}\}$ where $a$ and $b$ are both probabilistic with probability $p_a$ and $p_b$ , respectively. Suppose we have the interpretation $I = \langle \{q\}, \{\} \rangle$ . Here, $\underline{P}(I) = 0$ while $\overline{P}(I) = p_a \cdot p_b$ . Thus, any probability assignment to $p_a$ and $p_b$ maximizes the lower probability (which is always 0) but only the assignment $p_a = 1$ and $p_b = 1$ maximizes $\overline{P}(I)$ .

Azzolini et al. (Reference Azzolini, Bellodi and Riguzzi2024) focused on ground probabilistic facts whose probabilities should be learnt and proposed an algorithm based on Expectation Maximization (EM) to solve the task. Suppose that the target is the upper probability. The treatment for the lower probability is analogous and only differs in the considered bound. The EM algorithm alternates an expectation phase and a maximization phase, until a certain criterion is met (usually, the difference between two consecutive iterations is less than a given threshold). This involves computing, in the expectation phase, for each probabilistic fact $a_i$ whose probability should be learnt:

(4) \begin{align} \overline{E}[a_{i0}] = \sum _{I \in \mathcal{I}} \overline{P}(not \ a_{i} \mid I), \ \ \ \overline{E}[a_{i1}] = \sum _{I \in \mathcal{I}} \overline{P}(a_{i} \mid I). \end{align}

These values are used in the maximization step to update each parameter $\Pi _i$ as:

(5) \begin{align} \Pi _i = \frac{\overline{E}[a_{i1}]}{\overline{E}[a_{i0}] + \overline{E}[a_{i1}]} = \frac{\sum _{I \in \mathcal{I}} \overline{P}(a_{i} \mid I)}{\sum _{I \in \mathcal{I}}\overline{P}(not \ a_{i} \mid I) + \overline{P}(a_{i} \mid I)}. \end{align}

3.1 Extracting equations from a NNF

The upper probability of a query $q$ is computed as a sum of products (see equation 1). If, instead of using the probabilities of the facts, we keep them symbolic (i.e. with their name), we can extract a nonlinear symbolic equation for the query, where the variables are the parameters associated with the learnable facts. Call this equation $f_{up}(\Pi )$ where $\Pi$ is the set of parameters. Its general form is $f_{up}(\Pi ) = \sum _{w_i}\prod _{a_j \in w_i} p_j \prod _{a_j \not \in w_i} (1-p_j) \cdot k_i$ where $k_i$ is the contribution of the probabilistic facts with fixed probability for world $w_i$ . We can cast the task of extracting an equation for a query as a 2AMC problem. To do so, we can consider as inner semiring and as transformation function the ones proposed by Azzolini and Riguzzi (Reference Azzolini, Riguzzi, Basili, Lembo, Limongelli and Orlandini2023a) and described in Section 2.1. From this inner semiring we obtain two values, one for the lower and one for the upper probability. The sensitivity semiring by Kimmig et al. (Reference Kimmig, Van den Broeck and De Raedt2017) allows the extraction of an equation from an AMC task. We have two values to consider, so we extend that semiring to $R^o = (\mathbb{R}[X], +, -, (0, 0), (1, 1))$ with

\begin{equation*} w_o(l) = \begin {cases} (p,p) & \text {for a p.f. } p::a \text { with fixed probability and } l = a \\ (1-p,1-p) & \text {for a p.f. } p::a \text { with fixed probability and } l = not \ a \\ (\pi _a,\pi _a) & \text {for a learnable fact } \pi _a::a \text { and } l = a \\ (1 - \pi _a, 1 - \pi _a) & \text {for a learnable fact } \pi _a::a \text { and } l = not \ a \\ (1,1) & \text {otherwise} \end {cases} \end{equation*}

where p.f. stands for probabilistic fact and $\mathbb{R}[X]$ is the set of real valued functions parameterized by $X$ . Variable $\pi _a$ indicates the symbolic probability of the learnable fact $a$ . In this way, we obtain a nonlinear equation that represents the probability of a query. When evaluated by replacing variables with actual numerical values, we obtain the probability of the query when the learnable facts have such values. Simplifying the obtained equation is also a crucial step since it might significantly reduce the number of operations needed to evaluate it.

We now show how to adopt symbolic equations to solve the parameter learning task.

3.2 Solving parameter learning with constrained optimization

The learning task requires maximizing the sum of the log probabilities of the interpretations equation (3) where the tunable parameters are the facts whose probability should be learnt. Algorithm1 sketches the involved steps. We can extract the equation for the probability of each interpretation from the NNF and consider each parameter of a learnable fact as a variable (function GetEquationFromNNF). To reduce the size of the equation we simplify it (function Simplify). Then, we set up a constrained nonlinear optimization problem where the target is the maximization of the sum of the equations representing the log probabilities of the interpretations (function SolveOptimizationProblem). We need to also impose that the parameters of the learnable facts are between 0 and 1. In this way, we can easily adopt off-the-shelf solvers and do not need to write a specialized one.

Algorithm 1 Function LearningOPT: solving the parameter learning task targeting the upper probability with constrained optimization in a PASP P with learnable probabilistic facts II and interpretations I.

3.3 Solving parameter learning with expectation maximization

We propose another algorithm that instead performs Expectation Maximization as per equation (4) and (5). It is sketched in Algorithm2. For each interpretation $I$ , we add its interpretation query $q_I$ into the program. To compute $\overline{P}{(a_j \mid I_k)}$ for each learnable probabilistic fact $a_j$ and each interpretation $I_k$ we proceed as follows: first, we compute $\overline{P}{(a_j, I_k)}$ and $\underline{P}{(not \ a_j, I_k)}$ . Then, equation (2) allows us to compute $\overline{P}(a_j \mid I_k)$ . Similarly for $\underline{P}{(a_j \mid I_k)}$ . We extract an equation for each of these queries (function GetEquationsFromNNF) and iteratively evaluate them until the convergence of the EM algorithm (lines 5–10 that alternates the expectation phase with function Expectation, the maximization phase with function Maximization, and the computation of the log-likelihood with function ComputeLL). We consider as default convergence criterion a variation of the log-likelihood less than $5 \cdot 10^{-4}$ between two subsequent iterations. However, this parameter can be set by the user. If we denote with $n_p$ the number of probabilistic facts whose probabilities should be learnt and $n_i$ the number of interpretations, we need to extract equations for $2 \cdot n_p \cdot n_i$ queries. However, this is possible with only one pass of the NNF: across different iterations, the structure of the program is the same, only the probabilities, and thus the result of the queries, will change. Thus, we can store the performed operations in memory and use only those to reevaluate the probabilities, without having to rebuild the NNF at each iteration.

Algorithm 2 Function LearningEM: solving the parameter learning task targeting the upper probability with Expectation Maximization in a PASP $P$ with learnable probabilistic facts $\Pi$ and with interpretations $I$ .

5.1 Datasets descriptions

We considered four datasets. Where not explicitly specified, all the initial probabilities of the learnable facts are set to 0.5. For all the instances, we generated configurations with 5, 10, 15, and 20 interpretations. The atoms to be included in the interpretations are taken uniformly at random from the Herbrand base of each program and some of them are considered as negated, again uniformly at random.

The coloring dataset models the graph coloring problem, where each node is associated with a color and configurations are considered valid only if nodes associated with the same color are not connected. We do not explicitly remove invalid solutions but only mark them as not valid. This makes the following dataset crucial to test our proposal when there are an increasing number of answer sets for each world. All the instances have the following rules:

The goal is to learn the probabilities of the $edge/2$ facts given an increasing number of interpretations containing the observed colors for some edges and whether the configuration is $valid$ or not. We considered complete graphs of size 4 (coloring4) and 5 (coloring5). Here the interpretations have a random length between 3 and 4.

The path dataset models a path reachability problem and every instance has the rules described in Example1. We considered this dataset since it represents a graph structure that can model many problems. The goal is to learn the probabilities of the $edge/2$ facts given that some paths are observed. We considered random connected graphs with 10 (path10) and 15 (path15) edges. Here the interpretations have a random length between 1 and 3.

The shop dataset models the shopping behavior of some people, with different products available where some of them cannot be bought together. An example of instance (with 2 people) is:

The goal is to learn the probabilities of the $shops/1$ learnable facts given an increasing number of products that are observed being bought (or not bought) and an increasing number of people. We considered instances with 4 (shop4), 8 (shop8), 10 (shop10), and 12 (shop12) people. Here the interpretations have a random length between 1 and 10. With this dataset we assess the learning algorithms in programs when constraints prune some solutions.

The smoke dataset, adapted from (Totis et al., Reference Totis, De Raedt and Kimmig2023), models a network where some people smoke and others are influenced by the smoking behavior of their friends. This is a well-known dataset often used to benchmark probabilistic logic programming systems. An example of instance with two people is:

Here, the learnable facts have signature $\mathit{influences}/2$ and the goal is to learn their probabilities given that some $ill/1$ facts are observed. We consider instances with 3 (smoke2), 4 (smoke4), and 6 (smoke6) people. Here the interpretations have a random length between 1 and 3.

5.2 Results

Fig. 1. Execution times for EM, constrained optimization solved with COBYLA and SLSQP, and PASTA, by increasing the number of interpretations. For path15 and coloring5 the line for EM is missing since it cannot solve any instance. The initial probabilities for learnable facts are set to 0.5.

Table 2. Final log-likelihood (LL) values for the tested algorithms on six selected instances with the initial probability of the learnable facts set to 0.5. The column # int. contains the number of interpretations considered, the column EM contains the results obtained with expectation maximization, columns C. COBYLA and C. SLSQP stands for constrained optimization solved with, respectively, COBYLA and SLSQP, and the column PASTA contains the results obtained with the PASTA solver

Fig. 2. Execution times of constrained optimization solved with COBYLA and SLSQP on coloring5, path15, shop12, and smoke6 with 0.1, 0.5, and 0.9 as initial values for the learnable facts.

First, we compare the execution times of the four proposed algorithms. Results are reported in Figure 1. COBYLA and SLSQP have comparable execution times while EM and PASTA are often the slowest. The only instances where PASTA is faster are coloring4 and coloring5. Given the imposed time and memory limits, EM cannot solve the instances shop12 with any number of interpretations and shop10 with 15 and 20 interpretations, and shop8 with 20 interpretations, coloring5 with any number of interpretations, and path15 with any number of interpretations (all due to memory limit) while PASTA was not able to solve smoke6 with any number of interpretations and smoke5 with 10, 15, and 20 interpretations (all due to time limit). The two algorithms based on constrained optimization are able to solve every considered instance.

We also evaluated the algorithms in terms of final log-likelihood. Table 2 shows the results on the coloring4, path10, shop4, shop8, smoke3, and smoke4 instances. SLSQP has the best performance: it seems to be able to better maximize the LL in coloring4, shop4, and shop8, while COBYLA cannot reach such maxima. For the other two, the results are equal. EM is also competitive with SLSQP. PASTA has the worst performance for all the datasets and cannot reach a LL of 0 while other algorithms, such as the ones based on constrained optimization, succeed.

Lastly, we also run experiments with the two optimization algorithms by considering different initial probability values. Figure 2 shows the results. In general, the initial probability values have little influence on the execution time for SLSQP. The impact is more evident on COBYLA: for example, for coloring5 with 10 interpretations, setting the initial values to 0.1 requires 100 s of computation while setting them to 0.5 or 0.9 requires 60 s. We extended this evaluation also to EM, but we do not report the results since the initial value makes no difference in terms of execution time. In the current version, PASTA does not allow setting an initial probability value different from 0.5.

Overall, the algorithm based on constrained optimization outperforms in terms of efficiency and in terms of final log-likelihood both the algorithm based on EM and PASTA. EM also outperforms PASTA in terms of final LL but it often requires excessive memory and cannot solve instances solvable by PASTA. Nevertheless, several improvements may be integrated within our approach to possibly increase the scalability: (i) considering different alternatives for the representation of symbolic equations, possibly more compact, also considering symmetries (Azzolini and Riguzzi, Reference Azzolini and Riguzzi2023b); (ii) developing ad-hoc simplification algorithms to reduce the size of the symbolic equations since they are only composed by summations of products. Furthermore, considering higher runtimes could be beneficial for enumeration-based techniques.