2 Preliminaries

Throughout this paper, we restrict our attention to discrete probability measures and capacities, both defined over finite sets, and to programs without function symbols.

2.1 Capacities and belief functions

We recall the definition of probability measures and spaces (Ash and Doléans-Dade, Reference Ash and Doléans-Dade2000).

Definition 1 (Probability Measure and Space). Let $\mathcal{X}$ be a non-empty finite set called the sample space and $\mathbb{P}({\mathcal{X}})$ the powerset of $\mathcal{X}$ called the event space. A probability measure is a function $P:\mathbb{P}({{\mathcal{X}}})\rightarrow {\mathbb{R}}$ such that i) $\forall x \in \mathbb{P}({\mathcal{X}}), P(x) \geq 0$ , ii) $P({\mathcal{X}}) = 1$ , and iii) for disjoint $S_i \in \mathbb{P}({\mathcal{X}}): P(\bigcup _i {\mathcal{S}}_i) = \sum _i P({\mathcal{S}}_i)$ . A probability space is a triple $({\mathcal{X}},\mathbb{P}({\mathcal{X}}),P)$ .Footnote ¹

All measures, including probability measures, have the additivity property. That is, the measure assigned to the union of disjoint events is equal to the sum of the measures of the sets. Belief and plausibility functions (Shafer, Reference Shafer1976) are non-additive capacities rather than measures (Choquet, Reference Choquet1954; Grabish, Reference Grabish2016).

CaLPs in Section 3 use belief and plausibility functions based on multiple capacity spaces or domains. Accordingly, we use domain identifiers in the following definitions.

Definition 2 (Capacity Space). Let $\mathcal{X}$ be a non-empty finite set called the frame of discernment. A capacity is a function $\xi :\mathbb{P}({\mathcal{X}}) \rightarrow [0,1]$ such that i) $\xi (\emptyset )= 0$ and ii) for $\mathcal{A},{\mathcal{B}} \subseteq {\mathcal{X}}$ , $\mathcal{A} \subseteq {\mathcal{B}} \Rightarrow \xi (\mathcal{A}) \leq \xi ({\mathcal{B}})$ . If $\xi ({\mathcal{X}})=1$ , $\xi$ is normalized. A capacity space is a tuple $(D,{\mathcal{X}},\mathbb{P}({\mathcal{X}}),\xi )$ where $D$ is a string called a domain identifier.

Belief and plausibility functions are capacities based on mass functions.

Definition 3 (Mass Functions). Let $\mathcal{X}$ be a non-empty finite set and $D$ a domain identifier. A mass function $\mathit{mass} : D\times \mathbb{P}({\mathcal{X}}) \rightarrow {\mathbb{R}}$ satisfies the following properties: i) $\mathit{mass}(D,\emptyset ) = 0$ , ii) $\forall x \in \mathbb{P}({\mathcal{X}}), \mathit{mass}(D,x) \geq 0$ , and iii) $\sum _{x \in \mathbb{P}({\mathcal{X}})} \mathit{mass}(D,x) = 1$ . A set $x \subseteq \mathbb{P}({\mathcal{X}})$ where $\mathit{mass}(D,x) \neq 0$ is a focal set.

Definition 4 (Belief and Plausibility Capacities). Let $\mathcal{X}$ be a non-empty finite set, $mass$ a mass function, $D$ a domain identifier, and $X \in \mathbb{P}({\mathcal{X}})$ .

• • The belief of $X$ is the sum of masses of all (not necessarily proper) subsets of $X$

  (1) \begin{equation} \mathit{Belief}(D,X) = \sum _{B \subseteq X} \mathit{mass}(D,B) \end{equation}

• • The plausibility of $X$ is the sum of the masses of all sets that are non-disjoint with $X$ , and can be stated in terms of belief. Denoting the set complement of $X$ as $\neg X$ :

  (2) \begin{equation} \mathit{Plaus}(D,X) = 1 - \mathit{Belief}(D,\neg X) \end{equation}

From the above definitions, it can be seen that $\mathit{Belief}$ and $\mathit{Plaus}$ are capacities. Also, belief and plausibility are conjuncts, as it is also true that $\mathit{Belief}(D,A) = 1 - \mathit{Plaus}(D,\neg A)$ . A capacity space $(D,{\mathcal{X}},\mathbb{P}({\mathcal{X}}),\xi )$ for which $\xi$ is a belief or plausibility function is a belief domain, and $\mathbb{P}({\mathcal{X}})$ is called the belief event space of such a space.

Example 3. We may represent a belief domain by the predicates domain/2 which represents the frame of discernment ( $\mathcal{X}$ ) and mass/3 which represents the mass function. The belief domain of Example 1, which we call urn1, can be represented as:

\begin{align*} \begin{split} & \mathit{domain}(urn1,\{blue,red,yellow\}). \\ & \mathit{mass}(urn1,\{blue\},0.1). \ \mathit{mass}(urn1,\{red\},0.3). \ \mathit{mass}(urn1,\{blue,yellow\},0.6). \end{split} \end{align*}

Here, $\mathit{Belief}(urn1,\{red,yellow\}) = 0.3$ and $\mathit{Plaus}(urn1,\{red,yellow\}) = 1 - \mathit{Belief}{}(urn1,\{blue\}) = 0.9$ .

2.2 Probabilistic logic programs (PLPs)

Among several equivalent languages for PLP under the distribution semantics, we consider ProbLog (Reference De Raedt, Kimmig, Toivonen and VelosoDe Raedt et al. 2007) for simplicity of exposition. A ProbLog program $\mathcal{P}={{({{{\mathcal{R}}},{{\mathcal{F}}}})}}$ consists of a finite set ${\mathcal{R}}$ of (certain) logic programming rules and a finite set ${\mathcal{F}}$ of probabilistic facts of the form $p_i::f_i$ where $p_i\in [0,1]$ and $f_i$ is an atom, meaning that we have evidence of the truth of each ground instantiation $f_i\theta$ of $f_i$ with probability $p_i$ and of its negation with probability $1-p_i$ . Without loss of generality, we assume that atoms in probabilistic facts do not unify with the head of any rule and that all probabilistic facts are independent (cf. (Riguzzi, Reference Riguzzi2022)). Given a ProbLog program $\mathcal{P}={{({{{\mathcal{R}}},{{\mathcal{F}}}})}}$ , its grounding ( $ground(\mathcal{P})$ ) is defined as ${({ground({{\mathcal{R}}}),ground({{\mathcal{F}}})})}$ . With a slight abuse of notation, we sometimes use ${\mathcal{F}}$ to indicate the set of atoms $f_i$ of probabilistic facts. The meaning of ${\mathcal{F}}$ will be clear from the context.

2.2.1 The distribution semantics for ProbLog programs without function symbols

For a ProbLog program $ \mathcal{P} = {{({{{\mathcal{R}}},{{\mathcal{F}}}})}}$ , a grounding ${({{{\mathcal{R}}},{{\mathcal{F}}}'})}$ , such that ${{\mathcal{F}}}' \subseteq ground({{\mathcal{F}}})$ is called a world. $W_{\mathcal{P}}$ denotes the set of all worlds. Whenever $\mathcal{P}$ contains no function symbols, $ground({{\mathcal{F}}})$ is finite, so $W_{\mathcal{P}}$ is also finite. We also assume that each ProbLog program $\mathcal{P} = {{({{{\mathcal{R}}},{{\mathcal{F}}}})}}$ is uniformly total, that is that each world has a total well-founded model (Przymusinski, Reference Przymusinski1989), so in each world every ground atom is either true or false. The condition that a program is uniformly total is the most general way to express that every world is associated with a stratified program, so that the distribution semantics is well-defined. We define a measure on a set of worlds, and how to compute the probability for a query.

Definition 5 (Measures on Worlds and Sets of Worlds). $\rho _{\mathcal{P}}:W_{\mathcal{P}}\rightarrow {\mathbb{R}}$ is a measure on worlds such that for each $w\in W_{\mathcal{P}}$

\begin{equation*} \rho _{\mathcal{P}}(w) = \prod _{p::a \in {{\mathcal{F}}} \mid a\in w} p \prod _{p::a \in {{\mathcal{F}}} \mid a\not \in w} (1-p). \end{equation*}

$\mu _{\mathcal{P}}:\mathbb{P}(W_{\mathcal{P}})\rightarrow {\mathbb{R}}$ is a measure on sets of worlds as such that for each $\omega \in \mathbb{P}(W_{\mathcal{P}})$

\begin{equation*} \mu _{\mathcal{P}}(\omega )=\sum _{w\in \omega }\rho _{\mathcal{P}}(w). \end{equation*}

From Definition5, $(W_{\mathcal{P}},\mathbb{P}(W_{\mathcal{P}}),\mu _{\mathcal{P}})$ is a probability space and $\mu _{\mathcal{P}}$ a probability measure.

Definition 6. Given a ground atom $q$ , define $Q:W_{\mathcal{P}}\to \{0,1\}$ as

(3) \begin{equation} Q(w) = \left \{ \begin{array}{ll} 1 & \mbox{if }w\models q \\ 0 & \mbox{otherwise} \end{array} \right . \end{equation}

$w\models q$ means that $q$ is true in the well-founded model of $w$ .

Since $Q^{-1}(\gamma )\in \mathbb{P}(W_{\mathcal{P}})$ for all $\gamma \subseteq \{0,1\}$ , and since the range of $Q$ is measurable, $Q$ is a random variable. The distribution of $Q$ is defined by $P(Q=1)$ ( $P(Q=0)$ is given by $1-P(Q=1)$ ) and for a ground atom $q$ we indicate $P(Q=1)$ by $P(q)$ .

Given this discussion, we compute the probability of a ground atom $q$ called query as

\begin{equation*} P(q) = \mu _{\mathcal{P}}(Q^{-1}(1)) = \mu _{\mathcal{P}}(\{w \mid w \in W_{\mathcal{P}},w\models q\}) = \sum _{w\in W_{\mathcal{P}} \mid w\models q}\rho _{\mathcal{P}}(w). \end{equation*}

That is, $P(q)$ is the sum of probabilities associated with each world in which $q$ is true.

2.2.2 Computing queries to probabilistic logic programs

Let us introduce some terminology. An atomic choice indicates whether or not a ground probabilistic fact $p_i :: f_i$ is selected and is represented by the pair $(f_i,k_i)$ where $k_i \in \{0,1\}$ : $k_i = 1$ indicates that $f_i$ is selected, $k_i = 0$ that it is not. A set of atomic choices is consistent if only one alternative is selected for any probabilistic fact, that is the set does not contain atomic choices $(f_i,0)$ and $(f_i,1)$ for any $f_i$ . A composite choice $\kappa$ is a consistent set of atomic choices.

Definition 7 (Measure of a Composite Choice). Given a composite choice $\kappa$ , we define the function $\rho _{c}$ as

\begin{equation*} \rho _{c}(\kappa )=\prod _{(f_i,1)\in \kappa }p_i\prod _{(f_i,0)\in \kappa }(1-p_i). \end{equation*}

A selection $\sigma$ (also called a total composite choice) contains one atomic choice for every probabilistic fact. From the preceding discussion, it is immediate that a selection $\sigma$ identifies a world $w_{\sigma }$ . The set of worlds $\omega _\kappa$ compatible with a composite choice $\kappa$ is $\omega _\kappa =\{w_{\sigma }\in W_{\mathcal{P}} \mid \kappa \subseteq \sigma \}$ . Therefore, a composite choice identifies a set of worlds. Given a set of composite choices $K$ , the set of worlds $\omega _K$ compatible with $K$ is $\omega _{K}=\bigcup _{\kappa \in {K}}\omega _\kappa$ . Two sets $K_1$ and $K_2$ of composite choices are equivalent if $\omega _{K_1}=\omega _{K_2}$ , that is, if they identify the same set of worlds. If the union of two composite choices $\kappa _1$ and $\kappa _2$ is not consistent, then $\kappa _1$ and $\kappa _2$ are incompatible. We define as pairwise incompatible a set $K$ of composite choices if $\forall \kappa _1\in K, \forall \kappa _2\in K$ , $\kappa _1\neq \kappa _2$ implies that $\kappa _1$ and $\kappa _2$ are incompatible. If $K$ is a pairwise incompatible set of composite choices, define $\mu _c(K)=\sum _{\kappa \in K}\rho _{c}(\kappa )$ .

Given a general set $K$ of composite choices, we can construct a pairwise incompatible equivalent set through the technique of splitting. In detail, if $f$ is a probabilistic fact and $\kappa$ is a composite choice that does not contain an atomic choice $({f},k)$ for any $k$ , the split of $\kappa$ on $f$ can be defined as the set of composite choices $S_{\kappa ,{f}}=\{\kappa \cup \{({f},0)\},\kappa \cup \{({f},1)\}\}$ . In this way, $\kappa$ and $S_{\kappa ,{f}}$ identify the same set of possible worlds, that is $\omega _\kappa =\omega _{S_{\kappa ,{f}}}$ , and $S_{\kappa ,{f}}$ is pairwise incompatible. It turns out that, given a set of composite choices, by repeatedly applying splitting it is possible to obtain an equivalent mutually incompatible set of composite choices.

Theorem 1 ((Poole, Reference Poole2000)). Let $K$ be a set of composite choices. Then there is a pairwise incompatible set of composite choices equivalent to $K$ .

Theorem 2 ((Poole, Reference Poole1993) Measure Equivalence for Composite Choices ). If $K_1$ and $K_2$ are both pairwise incompatible sets of composite choices such that they are equivalent, then $\mu _c(K_1)=\mu _c(K_2)$ .

Theorem 3 (Probability Space of a Program). Let $\mathcal{P}$ be a ProbLog program without function symbols and let $\Omega _{\mathcal{P}}$ be

\begin{equation*}\{\omega _K \mid K \mbox{ is a set of composite choices}\}.\end{equation*}

Then $\Omega _{\mathcal{P}}=\mathbb{P}(W_{\mathcal{P}})$ and $\mu _{\mathcal{P}}(\omega _K)=\mu _c(K')$ where $K'$ is a pairwise incompatible set of composite choices equivalent to $K$ .

A composite choice $\kappa$ is an explanation for a query $q$ if $ \forall w \in \omega _\kappa : w \models q$ . If the program is uniformly total, $w \models q$ is either true or false for every world $w$ . Moreover, a set $K$ of composite choices is covering with respect to a query $q$ if every world in which $q$ is true belongs to $\omega _K$ . Therefore, in order to compute the probability of a query $q$ , we can find a covering set $K$ of explanations for $q$ , then make it mutually incompatible and compute the probability from it. This is the approach followed by ProbLog and PITA for example (Reference Kimmig, Demoen, De Raedt, Costa and RochaKimmig et al. 2011; Riguzzi and Swift, Reference Riguzzi and Swift2011).

3 Capacity logic programs

We now present Capacity Logic Programs (CaLPs) that extend PLPs to include belief domains (Section 2.1).

Belief facts should be distinguished from the belief function Belief/2 of Definition4. Note that belief facts alone are sufficient to specify a program’s use of a belief domain, since plausibility and belief are conjuncts, so each can be computed from the other. We assume that all belief domains are function-free. To simplify presentation, we also assume belief facts are ground.

As notation, for a set $\mathcal{B}$ of belief facts and belief domain $D$ , $\mathit{facts}(D,{\mathcal{B}})$ designates the set of belief facts in $\mathcal{B}$ that have domain $D$ . In addition, $\mathit{doms}({\mathcal{B}})$ denotes the set of all domains that are domain arguments of belief facts in $\mathcal{B}$ .

When used in a belief world, a set of belief facts is made canonical to take into account that while belief facts from different belief domains are independent, belief facts from the same belief domain are not. Informally, if set of belief facts contains $\mathit{belief}(D_1,B_1)$ and $\mathit{belief}(D_1,B_2)$ , these belief facts are replaced by $\mathit{belief}(D_1,B_1 \cap B_2)$ , to ensure all belief facts are independent.

Definition 10 (Canonicalization of a Set of Belief Facts). The canonicalization of a set of belief facts $Bel$ is

\begin{equation*} \mathit{canon}(Bel) = \{ \mathit{belief}(D,B) \mid D \in \mathit{doms}(Bel) \mbox{ and } B = \bigcap _{\mathit{belief}(D,B_i) \in Bel} B_i \} \end{equation*}

The set $canon(Bel)$ may contain facts whose belief event is $\emptyset$ . If $canon({\mathcal{B}})$ contains such facts it is inconsistent, otherwise it is consistent.

Since $\mathcal{D}$ is a finite set of function-free belief domains, $W_{\mathcal{P}}^{{\mathcal{D}}}$ is finite. Also, for any world $({{\mathcal{R}}},{{\mathcal{F}}}',{\mathcal{D}},{\mathcal{B}}') \in W_{\mathcal{P}}^{{\mathcal{D}}}$ , $\mathcal{B}$ contains exactly one belief fact for every domain in $\mathcal{D}$ : by Definition11, ${\mathcal{B}}'$ contains at least one belief fact for a given $D_i \in {\mathcal{D}}$ , while canonicalization ensures that there is at most one belief fact for $D_i$ .

As defined in Section 2.2, CaLPs are assumed to be uniformly total: that each world has a two-valued well-founded model (i.e., that each world gives rise to a stratified program). Thus, to extend the semantics of Section 2.2 it needs to be ensured that the well-founded model can be properly computed when belief facts are present, and for this the step of completion is used. Suppose $w$ ’s belief set contained belief(urn1,{blue}) as the (unique) belief fact for the belief domain urn1 of Example3. Because $\mathit{belief}(urn1,\{blue\})$ provides evidence for $\mathit{belief}(urn1,\{blue,yellow\})$ , a positive literal $\mathit{belief}(urn1,\{blue,yellow\})$ in a rule should succeed.

Thus a belief set $\mathcal{B}$ must be canonical before being used to construct a world $w$ . On the other hand, the completion of $\mathcal{B}$ is used when computing $WFM(w)$ and ensures rules requiring weak evidence (such as that a ball is yellow or blue) will be satisfied in a belief world that provides stronger evidence (such that a ball is blue).

Belief domains can be seen as a basis for imprecise probability (Halpern and Fagin, Reference Halpern and Fagin1992), so the capacity of a belief world is an interval containing both belief and plausibility. Accordingly, circumflexed products and sums represent interval multiplication and addition. In the following definition, $\beta _{prb}$ computes the portion of the capacity due to probabilistic facts (similar to Definition5), and $\beta _{blf}$ the portion due to belief facts.

Definition 13 (Capacity of a Belief World). $\beta _{prb},\beta _{blf}, \beta :W_{\mathcal{P}}^{{\mathcal{D}}} \rightarrow {\mathbb{R}}^2$ are capacities such that for $w = ({{\mathcal{R}}},{\mathcal{F}},{\mathcal{D}},{\mathcal{B}})$

\begin{align*} \begin{split} \beta _{prb}(w)&= \widehat {\prod }_{p::a\in w}[p,p] \widehat {\prod }_{p::a\not \in w}[(1-p),(1-p)] \\ \beta _{blf}(w)&= \widehat {\prod }_{\mathit{belief}(D,B)\in {\mathcal{B}}} [\mathit{Belief}(D,B),Plaus(D,B)] \\ \beta (w)&= \beta _{prb}(w)\widehat {\times }\beta _{blf}(w) \end{split} \end{align*}

The restriction that a world’s belief set contain a belief fact for each belief domain need not affect the capacity of a world, since the fact may indicate trivial evidence for the entire discernment frame, for which the belief and plausibility are both 1. For instance, {blue, red, yellow} in the $urn1$ domain has a belief of 1.

To assign a non-additive capacity to a set of belief worlds, the super-additivity of belief and plausibility capacities must be addressed .Footnote ³ The following definition provides one step in this process.

Next, a set $\mathcal{B}$ of belief worlds is partitioned by grouping into cells worlds in $\mathcal{B}$ that have the same set of probabilistic facts. Within each cell, the probabilistic facts are factored out and the belief events for each domain are then unioned, leading to a single world and unique capacity for each partition cell. Finally the capacities of all cells are added. This process is necessary to avoid counting the mass of probabilistic facts more than once when the capacities of belief worlds are summed.

Definition 15 (Capacities of Sets of Belief Worlds). Let ${\mathcal{P}} = ({{\mathcal{R}}},{\mathcal{F}},{\mathcal{D}},{\mathcal{B}})$ . The partition function $Dom{Prtn}({\mathcal{S}}):\mathbb{P}(W_{\mathcal{P}}^{{\mathcal{D}}}) \rightarrow \mathbb{P}(\mathbb{P}(W_{\mathcal{P}}^{{\mathcal{D}}}))$ , given ${\mathcal{S}} \in \mathbb{P}(W_{\mathcal{P}}^{{\mathcal{D}}})$ produces the domain partition of $\mathcal{S}$ : the coarsest partition such that for each $S_i \in Dom{Prtn}({\mathcal{S}})$ :

\begin{equation*}({{\mathcal{R}}},{\mathcal{F}}_j,{\mathcal{D}},{\mathcal{B}}_j) \in S_i \mbox{ and }({{\mathcal{R}}},{\mathcal{F}}_k,{\mathcal{D}},{\mathcal{B}}_k) \in S_i \Rightarrow {\mathcal{F}}_j = {\mathcal{F}}_k.\end{equation*}

Using the domain partition, we define $probfacts(S_i)$ as the unique ${\mathcal{F}}'$ for any $w = ({{\mathcal{R}}},{\mathcal{F}}',{\mathcal{D}},{\mathcal{B}}')$ s.t., $w \in S_i$ . The capacity $\beta _{prtn}$ for $S_i \in Dom{Prtn}({\mathcal{S}})$ is

(4) \begin{equation} \beta _{prtn}(S_i) = \beta _{prb}(probfacts(S_i)) \widehat {\times } \widehat {\prod }_{({{\mathcal{R}}},{\mathcal{F}},{\mathcal{D}},{\mathcal{B}}) \in {\mathcal{S}}_i;D \in doms({\mathcal{B}})}BP^{\uparrow }(D,{\mathcal{B}}). \end{equation}

Finally, the capacity for $\mathcal{S}$ is

(5) \begin{equation} \xi ^{{\mathcal{B}}}({\mathcal{S}}) = \widehat {\sum }_{S_i \in Dom{Prtn}({\mathcal{S}})} \beta _{prtn}(S_i). \end{equation}

Example 5. (Computing the Capacity for a Set of Belief Worlds). Consider a new domain $urn2$ analogous to $urn1$ of Example 3:

domain(urn2,{green,orange,purple})

mass(urn2,{green},0.1) mass(urn2,{orange},0.3) mass(urn2,{green,purple},0.6)

Let CaLP ${\mathcal{P}} = ({{\mathcal{R}}},{\mathcal{F}},{\mathcal{D}},{\mathcal{B}})$ where ${\mathcal{F}} = \{p_1::f_1,p_2::f_2\}$ and ${\mathcal{D}} = \{urn1,urn2\}$ , along with a set $\mathcal{S}$ of 3 worlds of $\mathcal{P}$ with the following fact sets and belief domains.

$w_1: {\mathcal{F}}_1 = \{p_1\}, {\mathcal{B}}_1= \{\mathit{belief}(urn1,\{blue\})\}$ $w_2: {\mathcal{F}}_2 = \{p_2\}, {\mathcal{B}}_2= \{\mathit{belief}(urn2,green)\}$

$w_3: {\mathcal{F}}_3 = \{p_1\}, {\mathcal{B}}_3= \{\mathit{belief}(urn1,\{yellow\})\}$

Following Definition 15, $DomPrtn({\mathcal{S}}$ ) partitions $\mathcal{S}$ into $\{w_1,w_3\}$ and $\{w_2\}$ . To compute $\beta _{prtn}(\{w_1,w_3\})$ , $BP^{\uparrow }$ is calculated for each belief domain giving

\begin{equation*}[\mathit{Belief}(urn1,\{blue,yellow\}),\mathit{Plaus}(urn1,\{blue,yellow\}])= [0.7,0.7]\end{equation*}

for the domain urn1 and $[1,1]$ for the domain $urn2$ . Next, $\beta _{prb}(\{w_1,w_3\})$ is calculated as $[p_1,p_1]\widehat {\times }[(1-p_2),(1-p_2)]$ and multiplied with $[0.7,0.7]$ . The result is then summed with $\beta _{prtn}(\{w_2\})$ .

One might ask why Definition15 uses $BP^{\uparrow }$ to combine beliefs of the same domain rather than Dempster’s Rule of Combination. Dempster’s rule was designed to combine independent evidence to determine a precise combined belief. This approach is useful for some purposes but has been criticized for many others (cf. (Jøsang, Reference Jøsang2016)). When combining CaLP worlds, epistemic commitment for a given domain is reduced by $BP^{\uparrow }$ in a manner similar to combining CaLP explanations (Section 4).

Since the range of $\xi ^{{\mathcal{B}}}$ is $\mathbb{R}^2$ , we designate $proj_{B}, proj_{P}:\mathbb{R}^2 \rightarrow \mathbb{R}$ as projection functions for the belief and plausibility portions, respectively. The following theorem states that capacity spaces formed using belief worlds and the capacity function $\xi ^{{\mathcal{B}}}$ have belief and plausibility capacities whose value is 1 for the entire belief event space.

Proof. From Definitions2 and 15 it is clear that $\xi ^{{\mathcal{B}}}$ is a capacity. To show that $\xi ^{{\mathcal{B}}}$ is normalized, consider that $\xi ^{{\mathcal{B}}}$ is a function composition, where the first function, $DomPrtn(W_{\mathcal{P}}^{{\mathcal{D}}})$ returns the domain partition of $W_{\mathcal{P}}^{{\mathcal{D}}}$ , while $\beta _{prb}(probfacts(S_i))$ is the separate probability of $\mathcal{F}$ . Consider a cell ${\mathcal{S}}_i \in DomPrtn({\mathcal{S}})$ . From equation (4)

\begin{equation*}\beta _{prtn}(S_i) = \beta _{prb}(probfacts(S_i)) \widehat {\times } \widehat {\prod }_{({{\mathcal{R}}},{\mathcal{F}},{\mathcal{B}}) \in {\mathcal{S}}_i;D \in doms({\mathcal{B}})}BP^{\uparrow }({\mathcal{D}},B)\end{equation*}

Every world in ${\mathcal{S}}_i$ contains the same set of probabilistic facts, $(probfacts(S_i))$ , but differs in belief facts. Let $D$ be a belief domain in $\mathcal{D}$ . Since ${\mathcal{S}}_i$ is a partition of $W_{\mathcal{P}}^{{\mathcal{D}}}$ , and because all belief domains in all worlds are normalized, $S_i$ contains a world with every belief event of $D$ . Thus $BP^{\uparrow }(D,B) = [1,1]$ so that

\begin{equation*}\beta _{prtn}(S_i) = \beta _{prb}(probfacts(S_i)) \widehat {\times } \widehat {\prod }_{({{\mathcal{R}}},{\mathcal{F}},{\mathcal{B}}) \in {\mathcal{S}}_i;D \in doms({\mathcal{B}})}[1,1] = \beta _{prb}(probfacts(S_i)) \end{equation*}

Thus, the value of $\beta _{prtn}({\mathcal{S}}_i)$ relies only on the probabilistic facts that were used for the partition. Since $W_{\mathcal{P}}^{{\mathcal{D}}}$ contains all possible worlds, $\xi ^{{\mathcal{B}}}(W_{\mathcal{P}}^{{\mathcal{D}}})$ reduces to the sum of all subsets of probabilistic facts of $\mathcal{P}$ , which is 1.

Definition 16 (cf. Section 2.2, Definition6). Given a ground atom $q$ in $ground(\mathcal{P})$ we define $Q_{{\mathcal{B}}}: W_{\mathcal{P}}^{{\mathcal{D}}}\to \{0,1\}$ as

(6) \begin{equation} Q_{{\mathcal{B}}}(w)=\left \{\begin{array}{ll}1 & \mbox{if }w\vDash q\ (\mathit{Definition}\,12)\\ 0 & \mbox{otherwise} \end{array}\right . \end{equation}

By definition $Q_{{\mathcal{B}}}^{-1}(\gamma )\in \mathbb{P}(W_{\mathcal{P}}^{{\mathcal{D}}})$ for $\gamma \subseteq \{0,1\}$ , and since the range of $Q_{{\mathcal{B}}}$ is measurable, $Q_{{\mathcal{B}}}$ is a capacity-based random variable.

Given the above discussion, we compute $[\mathit{belief}(q),\mathit{plaus}(q)]$ for a ground atom $q$ as

\begin{equation*}[\mathit{belief}(q),\mathit{plaus}(q)] = \xi ^{{\mathcal{B}}}(Q_{{\mathcal{B}}}^{-1}(1))=\xi ^{{\mathcal{B}}}(\{w \mid w\in W_{\mathcal{P}}^{{\mathcal{D}}}, w\models q\}).\end{equation*}

4 Computing queries to capacity logic programs

We extend the definition of atomic choice from Section 2.2.2 to CaLPs as follows.

Atomic belief choices with different belief domains are considered independent choices, much as atomic Bayesian choices. However, atomic belief choices with the same belief domain are not independent: so fitting them into the distribution semantics requires care.

Given a set $\mathcal{S}$ of atomic choices, let $Bay({\mathcal{S}})$ be the subset of $\mathcal{S}$ that contains only the atomic Bayesian choices of $\mathcal{S}$ and let $Bel({\mathcal{S}})$ be the subset of $\mathcal{S}$ that contains only the atomic belief choices of $\mathcal{S}$ . Note that $Bel({\mathcal{S}})$ is a set of belief facts, so that definitions of Section 3 can be used directly or with minimal change. Then $\mathcal{S}$ is consistent if both $Bay({\mathcal{S}})$ and $Bel({\mathcal{S}})$ are consistent (cf. Definition10). A capacity composite choice $\kappa$ is a consistent set of atomic choices for a CaLP ${\mathcal{P}}^{{\mathcal{B}}}= ({\mathcal{P}},{{\mathcal{F}}},{\mathcal{D}},{\mathcal{B}})$ . Given a capacity composite choice $\kappa$ , let $doms(\kappa )$ be $\{D \mid \mathit{belief}(D,B)\in \kappa \}$ .

Definition 18 (Measure for a Single Capacity Composite Choice). Given a capacity composite choice $\kappa$ we define $\rho ^{Comp}_{{\mathcal{B}}}$ as

\begin{equation*} \rho ^{Comp}_{{\mathcal{B}}}(\kappa ) = \rho _{{\mathcal{P}}^{{\mathcal{B}}}}(\kappa ) \widehat {\times } \widehat {\prod }_{\mathit{belief}(D,Elt) \in canon(Bel(\kappa ));{\mathcal{D}} \in doms(\kappa )}BP^{\uparrow }(D,Elt ) \end{equation*}

where $\rho _{{\mathcal{P}}^{{\mathcal{B}}}}$ is the measure for composite choices from Definition 7.

The set of worlds $\omega _\kappa$ compatible with a capacity composite choice $\kappa$ is

\begin{equation*}\omega _\kappa =\{w_{\sigma } = ({{\mathcal{R}}},{{\mathcal{F}}},{\mathcal{D}},{\mathcal{B}}) \in W_{\mathcal{P}}^{{\mathcal{D}}} \mid Bay(\kappa )\subseteq {\mathcal{F}}\mbox{ and } canon(Bel(\kappa ))\subseteq {\mathcal{B}}\}.\end{equation*}

With this definition of worlds compatible with a capacity composite choice, the definitions of equivalence of two sets of capacity composite choices, of incompatibility of two atomic choices and of a pairwise incompatible set of capacity composite choices are the same as for probabilistic logic programs.

If $K$ is a pairwise incompatible set of capacity composite choices, define $\xi _c(K)=\sum _{\kappa \in K}\rho _{{\mathcal{B}}}^{Comp}(\kappa )$ . Given a general set $K$ of capacity composite choices, we can construct a pairwise incompatible equivalent set through the technique of splitting. In detail, (1) if $f$ is a probabilistic fact and $\kappa$ is a capacity composite choice that does not contain an atomic Bayesian choice $({f},k)$ for any $k$ , the split of $\kappa$ on $f$ , $S_{\kappa ,{f}}$ , is defined as for probabilistic logic programs; (2) if $D$ is a domain not appearing in $doms(\kappa )$ and $B$ a belief event of $D$ , the split of $\kappa$ on $\mathit{belief}(D,B)$ , is defined as the set of capacity composite choices $S_{\kappa ,D,B}=\{\kappa \cup \{\mathit{belief}(D,B)\},\kappa \cup \{\mathit{belief}(D,\neg B)\}\}$ . In this way, $\kappa$ and $S_{\kappa ,{f}}$ ( $S_{\kappa ,D,B}$ ) identify the same set of possible worlds, that is $\omega _\kappa =\omega _{S_{\kappa ,{f}}}$ ( $\omega _\kappa =\omega _{S_{\kappa ,D,B}}$ ) and $S_{\kappa ,{f}}$ ( $S_{\kappa ,D,B}$ ) are pairwise incompatible.

As for probabilistic logic programs, given a set of capacity composite choices, by repeatedly applying splitting it is possible to obtain an equivalent mutually incompatible set of composite choices (Poole, Reference Poole2000) and the analogous of Theorem1 can be proved.

The definitions of explanations and covering sets of explanations for a query $q$ are the same as for PLP. As a result, systems like ProbLog and Cplint/PITA can be extended to inference over CaLPs: first a covering set of explanations is found using a conversion to propositional logic (ProbLog (Reference Kimmig, Demoen, De Raedt, Costa and RochaKimmig et al. 2011)) or a program transformation plus tabling and answer subsumption (PITA (Riguzzi and Swift, Reference Riguzzi and Swift2011)), then knowledge compilation is used to convert the explanations into an intermediate representation (d-DNNF for ProbLog and BDD for PITA) that makes them pairwise incompatible and allows the computation of the belief/plausibility.

Example 6. To illustrate these topics, consider r_indepthat uses belief domains urn1 and urn2.

r_indep:-belief(urn1,{blue}). r_indep:- belief(urn2,{orange}).

Since the belief domains (from Examples 3 and 5) are independent, a covering and pairwise incompatible set of explanations for r_indepis

$K=\{$ $\{{belief}(urn1,\{blue\}),{belief}(urn2,\neg \{orange\})\},$

$\{{belief}(urn1,\neg \{blue\}),{belief}(urn2,\{orange\})\}$

$\{{belief}(urn1,\{blue\}),{belief}(urn2,\{orange\})\}\}$

whose capacity is:

\begin{equation*}\xi _c(K)= ([0.1,0.7]\widehat {\times }[0.7,0.7]) \ \widehat {+}\ ([0.3,0.9]\widehat {\times }[0.3,0.3]) \ \widehat {+}\ ([0.1,0.7]\widehat {\times }[0.3,0.3]) = [0.19,0.97]\end{equation*}

Next, consider r_dep, both of whose rules both use the urn1 belief domain

r_dep:- belief(urn1,{blue}). r_dep:- belief(urn1,{red})

The capacity of r_depis:

$\xi _c(\{{belief}(urn1,\{blue\}),{belief}(urn1,\{red\}) \}) = \xi _c(\{{belief}(urn1,\{blue,red\})\}) = [0.4,1].$

5 Related work

Among the possible semantics for probabilistic logic programs, such as CP-logic (Reference Meert, Struyf, Blockeel and De RaedtMeert et al. 2010), Bayesian Logic Programming (Kersting and De Raedt, Reference Kersting and De Raedt2001), CLP(BN) (Reference Costa, Page, Qazi and CussensCosta et al. 2003), Prolog Factor Language (Gomes and Costa, Reference Gomes, Costa, Riguzzi and Železný2012), Stochastic Logic Programs (Reference MuggletonMuggleton et al. 1996), and ProPPR (Reference Wang, Mazaitis, Lao and CohenWang et al. 2015), none of them, to the best of our knowledge, consider belief functions.

Other semantics, still not considering belief functions, target Answer Set Programming (Brewka et al, Reference Brewka, Eiter and Truszczyński2011), such as LPMLN (Lee and Yang, Reference Lee, Yang, Singh and Markovitch2017), P-log (Reference Baral, Gelfond and RushtonBaral et al. 2009), and the credal semantics (CS) (Cozman and Mauá, Reference Cozman and Mauá2017). The latter deserves more attention: given an answer set program extended with probabilistic facts, the probability of a query $q$ is computed as follows. First worlds are computed, as in the distribution semantics. Each world $w$ now is an answer set program which may have zero or more stable models. The atom $q$ can be present in no models of $w$ , in some models, or in every model. In the first case, $w$ makes no contribution to the probability of $q$ . If $q$ is present in some but not all models of $w$ , $P(w)$ is computed as in Equation5, and contributes to the upper probability of $q$ . If instead the query is present in every model of $w$ , $P(w)$ contributes to both the upper and lower probability. That is, a query no longer has a point-probability but is represented by an interval. Furthermore, Cozman and Mauá (Reference Cozman and Mauá2017) also proved the CS is characterized by the set of all probability measures that dominate an infinitely monotone Choquet capacity.

The approach of Wan and Kifer (Reference Wan and Kifer2009) incorporates belief functions into logic programming, but unlike ours is not based on the distribution semantics. Rather, belief and plausibility intervals are associated with rules. Within a model $M$ , the belief of an atom $A$ is a function of the belief intervals associated with $A$ and the support of $A$ in $M$ .