How Rules Represent Causal Knowledge: Causal Modeling with Probabilistic Logic Programming

Technical Summary: How Rules Represent Causal Knowledge: Causal Modeling with Probabilistic Logic Programming

Problem Statement
Judea Pearl's theory of causality, which distinguishes between observational and interventional knowledge, has been foundational in artificial intelligence. However, Pearl's framework is developed exclusively within Bayesian networks and causal models, restricting it largely to acyclic causal relationships. Transferring these concepts to other formalisms, such as Probabilistic Logic Programming (PLP), risks misinterpretation or inconsistency. Specifically, existing semantics for PLP frameworks like ProbLog and LPMLN (LogLinear Models) often fail to satisfy the principle of causal irrelevance (the idea that unobserved effects should not alter beliefs about other variables) and non-interference (the idea that interventions should not affect variables outside their causal downstream). This leads to counterintuitive results when reasoning about interventions in cyclic or non-stratified programs.

Methodology
The authors bridge Pearl's interventionist approach with the philosophical foundations of causality established by Eelink et al. [10], which rely on logical explanations rather than temporal succession. The methodology proceeds as follows:

1. Theoretical Alignment: The paper adopts the view that causal explanations are composed of rules (e.g., "smoking causes carrying matches") and that scientific knowledge requires justifications that respect a causal order. This order is formalized through principles of Natural Necessity (causes necessitate effects) and Causal Sufficiency (every effect has a cause), grounded in external premises (abducibles).
2. Formal Semantics Definition: The authors define a Causal System $CS := (P, A, \Phi)$, where $P$ is an LPMLN program, $A$ is a set of abducibles, and $\Phi$ is a LogLinear model over $A$.
   • They formalize Natural Necessity via the constraint content of the program and Causal Sufficiency via the explanatory content.
   • They define an intervention as a modification of the system where clauses with heads in the intervention set are removed, and facts forcing specific values are added.
3. Critique of Existing Semantics: The authors demonstrate that the standard P-log semantics for ProbLog and the standard semantics for LPMLN do not satisfy Causal Irrelevance (Formalization 6). In non-stratified or cyclic cases, these semantics allow beliefs to change when additional unobserved effects are considered, violating the principle that interventions should not propagate "upstream" or to unrelated branches.
4. Proposed Causal Semantics: To resolve this, the authors propose a new Causal Semantics ($\pi^{causal}_{CS}$) based on a "greedy" maximization of entropy along the causal order.
   • The dependence graph is decomposed into Strongly Connected Components (SCCs).
   • The SCCs form a Directed Acyclic Graph (DAG).
   • Probabilities are computed by conditioning on parent components and maximizing entropy locally within each component, ensuring that the resulting distribution aligns with a Bayesian network structure derived from the causal dependencies.

Key Contributions

• Formal Causal Semantics for PLP: The paper transfers the formal causal semantics of Eelink et al. [10] to the probabilistic logic programming frameworks of ProbLog and LPMLN, accommodating both stratified and non-stratified (cyclic) programs.
• Identification of Flaws in Standard Semantics: Through Examples 20–22, the authors show that standard P-log and LPMLN semantics violate causal irrelevance and non-interference, leading to incorrect predictions of intervention effects in cyclic scenarios.
• Theoretical Guarantees:
  • Theorem 2: Proves that for stratified ProbLog programs, the standard P-log semantics coincides with the proposed causal semantics, thereby satisfying causal irrelevance.
  • Theorem 3: Establishes that the proposed causal semantics is a consistent generalization of the P-log semantics for stratified programs.
  • Theorem 4: Proves that the proposed causal semantics satisfies causal irrelevance for general (including cyclic) causal systems.
• Implementation: The authors provide an implementation of the proposed semantics using the PLP-BN tool suite. This system connects PLP programs with Bayesian Networks, utilizing the Clingo answer set solver for computing stable models of SCCs and factor/variable elimination for inference.

Results

• Consistency with Stratified Programs: For stratified ProbLog programs, the new semantics yields identical results to the established P-log semantics, confirming that the standard approach is sound for acyclic cases.
• Correction of Cyclic Cases: In non-stratified cases (e.g., the "Anna and Kilian" doorbell example), the standard semantics produces probabilities that violate causal irrelevance (e.g., an intervention on a downstream variable incorrectly altering the probability of an upstream variable). The proposed causal semantics corrects these probabilities, ensuring that interventions only affect downstream variables as dictated by the causal structure.
• Intervention Handling: The implementation demonstrates that the new semantics correctly models interventions by modifying the program structure and recomputing probabilities based on the causal DAG of SCCs, preventing the "backtracking" of causal influence that occurs in standard LPMLN/ProbLog interpretations.

Significance
The paper claims that by aligning probabilistic logic programming with the philosophical foundations of causal explanation (specifically the work of Eelink et al. [10] and the principles of necessity and sufficiency), it is possible to extend Pearl's causal reasoning to logic programming formalisms that handle cycles and uncertainty. The proposed semantics ensures that the representation of causal knowledge in PLP is robust against misinterpretation when performing interventions. This work provides a theoretically grounded method for reasoning about "what if" scenarios in logic programs that was previously restricted to acyclic Bayesian networks, thereby enabling more reliable causal modeling in domains requiring cyclic dependencies.