Interval-based Capacity Logic Programs

• Interval-based Capacity Logic Programs are an extension of probabilistic logic programming that replaces single probabilities with interval-valued belief and plausibility functions to capture epistemic uncertainty.
• They are grounded in Choquet capacity theory, using belief domains and non-additive measures to represent evidence from diverse sources such as hierarchical classifications or sensor data.
• The framework integrates probabilistic facts with belief domains, enabling interval-based query evaluation and scalable inference through methods like knowledge compilation and Monte Carlo simulation.

Searching arXiv for the specified paper and closely related work to ground the article with current references. Interval-based Capacity Logic Programs are an extension of Probabilistic Logic Programming under the Distribution Semantics in which uncertainty is represented not only by point probabilities but also by non-additive, interval-valued capacities, particularly belief and plausibility functions. The framework is introduced in "Integrating Belief Domains into Probabilistic Logic Programs" (Azzolini et al., 23 Jul 2025) to address epistemic uncertainty that is difficult to express in standard Distribution Semantics, including uncertainty arising from hierarchical classifications from computer vision models. In this formulation, a logic program combines ordinary probabilistic facts with belief domains equipped with mass functions over finite frames of discernment, so that query answers become intervals $[Bel_P(a), Pl_P(a)]$ rather than single probabilities.

1. Position within probabilistic logic programming

Probabilistic Logic Programming under the Distribution Semantics is presented as a leading approach to practical reasoning under uncertainty, with implementations available as Prolog or Python libraries, notably ProbLog and cplint/PITA (Azzolini et al., 23 Jul 2025). In its standard form, a ProbLog program $P=(R,F)$ consists of rules $R$ and probabilistic facts $F=\{p_i::f_i\}$. A world $w \subseteq F$ specifies which probabilistic facts hold, and its probability is

$$Pr(w)=\prod_{f_i\in w} p_i \cdot \prod_{f_i\notin w}(1-p_i).$$

For a query atom $a$, the usual semantics is

$P(a)=\sum_{w \models a} Pr(w).$

The limitation identified for this point-probability setting is that it makes epistemic uncertainty difficult to represent. The interval-based extension replaces single-valued probabilistic assessments with capacities that can encode uncertainty ranges. This shift is not a minor syntactic variation: it changes the semantics of worlds, query evaluation, and inference, while retaining compatibility with the programmatic style of Distribution Semantics (Azzolini et al., 23 Jul 2025).

A common misconception is to treat the resulting intervals as ordinary additive probabilities with upper and lower error bars. The framework instead uses belief functions as non-additive capacities. The interval $[Bel,Pl]$ is therefore semantically tied to evidential support and admissible plausibility, not merely to numerical imprecision.

2. Capacity-theoretic foundations

The formal basis of the framework is Choquet capacity theory. Let $\Omega$ be a finite frame of discernment and $P=(R,F)$0 its powerset. A normalized capacity is a function

$P=(R,F)$1

such that $P=(R,F)$2, $P=(R,F)$3, and, for all $P=(R,F)$4, $P=(R,F)$5 (Azzolini et al., 23 Jul 2025). Unlike a probability measure, $P=(R,F)$6 need not satisfy additivity; in general,

$P=(R,F)$7

Belief and plausibility functions arise from a basic probability assignment, or mass function, $P=(R,F)$8, satisfying $P=(R,F)$9 and $R$0. Every set $R$1 with $R$2 is a focal element. The induced capacities are

$R$3

and

$R$4

Hence $R$5, and both functions are monotone and non-additive (Azzolini et al., 23 Jul 2025). The conjugacy relation $R$6 is one of the core structural identities preserved in the logic-programming extension.

The Möbius transform provides the connection between a capacity and its mass representation. If $R$7 is induced by $R$8, then

$R$9

while conversely

$F=\{p_i::f_i\}$0

This representation matters computationally because $F=\{p_i::f_i\}$1 stores basic assignments on focal sets, whereas $F=\{p_i::f_i\}$2 aggregates them by inclusion. In the logic-programming setting, this distinction supports the move from event-level evidential assignments to world-level interval weights.

3. Semantics of belief domains and worlds

A Capacity Logic Program adds to an ordinary probabilistic program a finite set of belief domains $F=\{p_i::f_i\}$3, where each domain has its own frame $F=\{p_i::f_i\}$4 and mass function $F=\{p_i::f_i\}$5 (Azzolini et al., 23 Jul 2025). The primitive evidential atom is a ground belief fact of the form

$F=\{p_i::f_i\}$6

with $F=\{p_i::f_i\}$7, intended to mean that the evidence mass for domain $F=\{p_i::f_i\}$8 includes at least set $F=\{p_i::f_i\}$9.

The semantics introduces belief worlds. A belief world extends an ordinary world by choosing exactly one belief fact for each domain $w \subseteq F$0, and then completing it by closure under superset so that rules can succeed on weak evidence. This completion mechanism is essential because the evidential content of a chosen set propagates upward by inclusion. As a result, truth in a world is not determined solely by the explicitly selected belief facts but also by their closure properties.

Each belief world $w \subseteq F$1 is assigned an interval-valued weight $w \subseteq F$2 through the product of an ordinary probabilistic component and a belief component:

$w \subseteq F$3

and

$w \subseteq F$4

The total weight is

$w \subseteq F$5

where interval multiplication is coordinate-wise (Azzolini et al., 23 Jul 2025).

Because different belief worlds may share the same probabilistic facts but differ in their belief facts, the capacity of a set of worlds $w \subseteq F$6 is not defined by naive summation over all worlds. Instead, the worlds are partitioned into cells having identical ordinary facts. Within each cell, belief-events are unioned per domain by an upper-bound rule, collapsed to one interval weight, and then the cell weights are combined by interval addition. Formally,

$w \subseteq F$7

This cell-based construction is the mechanism by which the framework preserves evidential dependence internal to a probabilistic configuration. A plausible implication is that it prevents overcounting when multiple belief alternatives coexist over the same ordinary probabilistic support.

4. Query semantics and interval entailment

For a query atom $w \subseteq F$8, the semantics is defined through an indicator $w \subseteq F$9 that takes value $$Pr(w)=\prod_{f_i\in w} p_i \cdot \prod_{f_i\notin w}(1-p_i).$$0 when $$Pr(w)=\prod_{f_i\in w} p_i \cdot \prod_{f_i\notin w}(1-p_i).$$1 under completion and $$Pr(w)=\prod_{f_i\in w} p_i \cdot \prod_{f_i\notin w}(1-p_i).$$2 otherwise. The belief-plausibility interval of $$Pr(w)=\prod_{f_i\in w} p_i \cdot \prod_{f_i\notin w}(1-p_i).$$3 is then

$$Pr(w)=\prod_{f_i\in w} p_i \cdot \prod_{f_i\notin w}(1-p_i).$$4

This is the central semantic object of a Capacity Logic Program (Azzolini et al., 23 Jul 2025). It generalizes the scalar query probability of Distribution Semantics to an interval that bounds the support and admissibility of the query under the evidential model.

The relation to the standard probabilistic case is explicit:

$$Pr(w)=\prod_{f_i\in w} p_i \cdot \prod_{f_i\notin w}(1-p_i).$$5

and point probability is recovered when all domain mass functions are concentrated on singletons. In that limiting case, the non-additive structure collapses to the additive one. This shows that interval-based Capacity Logic Programs are conservative over standard Distribution Semantics rather than disjoint from it.

The distinction between belief and plausibility also clarifies the status of incomplete evidence. A query with low belief but high plausibility is not contradicted; it is weakly supported yet broadly compatible with the available evidence. This interpretation follows directly from the interval semantics and the conjugacy relation between belief and plausibility.

5. Inference, knowledge compilation, and complexity

The inferential machinery adapts explanation-based methods from probabilistic logic programming. Just as Poole’s explanations are composite choices of atomic facts, Capacity Logic Programs use capacity-composite choices $$Pr(w)=\prod_{f_i\in w} p_i \cdot \prod_{f_i\notin w}(1-p_i).$$6 that mix probabilistic and belief choices (Azzolini et al., 23 Jul 2025). A covering set $$Pr(w)=\prod_{f_i\in w} p_i \cdot \prod_{f_i\notin w}(1-p_i).$$7 of such choices, all of whose worlds entail a query $$Pr(w)=\prod_{f_i\in w} p_i \cdot \prod_{f_i\notin w}(1-p_i).$$8, is transformed into a pairwise-incompatible form by splitting. By Theorem 3.10, the sum of $$Pr(w)=\prod_{f_i\in w} p_i \cdot \prod_{f_i\notin w}(1-p_i).$$9 over the split set is independent of the splitting order. This order-independence is a structural property needed to make explanation aggregation well-defined.

In practical inference, one computes a propositional representation of all explanations for a query and compiles it into a deterministic decomposable circuit, such as d-DNNF or BDD. Weighted model counting on that compiled structure is generalized from scalar to interval weights by replacing each leaf probability $a$0 with either the degenerate interval $a$1 or a belief interval $a$2. Bottom-up evaluation then yields $a$3 in time linear in the circuit size (Azzolini et al., 23 Jul 2025).

The complexity profile remains demanding. Exact inference is stated to be $a$4-hard in the point cases, and deciding whether $a$5 is PP-hard. The framework therefore includes approximate schemes: $a$6-best explanations via A$a$7 or branch-and-bound; Monte Carlo simulation sampling both probabilistic and belief choices, including random selection of belief-events, to estimate bounds; and variational or loopy compilation into smaller circuits (Azzolini et al., 23 Jul 2025). These approximations place the framework in continuity with established scalable inference strategies while preserving interval-valued semantics.

6. Structural properties, worked example, and implementation

Several formal properties are highlighted. Belief is non-additive: $a$8 may exceed $a$9. It is also monotone, since $P(a)=\sum_{w \models a} Pr(w).$0, and 2-monotone, with

$P(a)=\sum_{w \models a} Pr(w).$1

Plausibility is the conjugate capacity defined by $P(a)=\sum_{w \models a} Pr(w).$2 (Azzolini et al., 23 Jul 2025). These properties distinguish the framework from additive probability calculi and are the basis for its treatment of epistemic uncertainty.

The worked example uses two belief domains, $P(a)=\sum_{w \models a} Pr(w).$3 and $P(a)=\sum_{w \models a} Pr(w).$4. For $P(a)=\sum_{w \models a} Pr(w).$5, the frame is $P(a)=\sum_{w \models a} Pr(w).$6 with masses $P(a)=\sum_{w \models a} Pr(w).$7, $P(a)=\sum_{w \models a} Pr(w).$8, and $P(a)=\sum_{w \models a} Pr(w).$9. For $[Bel,Pl]$0, the frame is $[Bel,Pl]$1 with masses $[Bel,Pl]$2, $[Bel,Pl]$3, and $[Bel,Pl]$4 (Azzolini et al., 23 Jul 2025).

In a CaLP with no probabilistic facts and rules

$\Omega$7

the explanations are capacity-composite choices $[Bel,Pl]$5, where $[Bel,Pl]$6 and $[Bel,Pl]$7. Their weights are

$[Bel,Pl]$8

and

$[Bel,Pl]$9

Because the two choices belong to independent domains, the covering set is pairwise-incompatible, yielding

$\Omega$0

and

$\Omega$1

If instead both rules use the same domain,

$\Omega$8

then their union of events is $\Omega$2, and

$\Omega$3

while

$\Omega$4

The example demonstrates that equal belief-plausibility endpoints can arise from different dependency structures. What changes is not the final interval in this case, but the route by which event unions are handled within a domain.

The same paper sketches a PITA-style prototype implementation. A CaLP is transformed into a normal Prolog program with three BDDs per atom: probability, belief, and plausibility (Azzolini et al., 23 Jul 2025). Each probabilistic fact introduces a multi-valued variable with weight $\Omega$5. Each belief fact introduces a variable whose values index focal sets and whose leaf weights are $\Omega$6. Standard BDD operations, including and, or, and not, are lifted to intervals, and a final weighted-model-count mc/2 computes the integer counts of satisfying assignments, which are then mapped to interval sums via the known leaf weights.

The applications identified for the framework are epistemic uncertainty in hierarchical classification, including a vision system that maps softmax scores to Dempster-Shafer masses over ontology nodes; sensor fusion where some sensors provide interval or evidential output; and robust decision-making under partial knowledge through credal inference (Azzolini et al., 23 Jul 2025). These use cases are all consistent with the framework’s central design goal: to preserve the operational style of probabilistic logic programming while replacing scalar uncertainty with evidential intervals.