From probability to causality in probabilistic logic programming

Pearl famously argues that causal knowledge enables the prediction of intervention effects. By contrast, purely descriptive knowledge supports only conclusions drawn from observations. His theory of causality, however, is developed exclusively within Bayesian networks and causal models. Consequently, it is largely restricted to acyclic causal relationships, and transferring its ideas to other formalisms risks misinterpretation or inconsistency. This paper brings Pearl's approach to causality into probabilistic logic programming (PLP). To this end, such programs are aligned with philosophical foundations established in prior work that do not rely on temporal notions; that is, all relevant events are assumed to occur simultaneously. A formal causal semantics for these programs, together with a notion of intervention and an implementation, is proposed. It is shown that this semantics coincides with the P-log semantics for stratified ProbLog programs, while the two may differ in the non-stratified case and for other PLP formalisms.

Motivated by challenging modelling issues in the life sciences, we investigate the relationship between logic programming semantics and the eventual states of causal processes compatible with those logic programs. More precisely, we show that while stable models of positive logic programs correspond to the eventual states of processes commencing from a neutral state and continuing undisturbed indefinitely, supported models describe the eventual states reachable from arbitrary starting points. This also contributes to the discussion of the appropriate semantics for logic programming as a causal rule language, adding a temporal perspective to recent interpretations of the stable and supported model semantics from an explanatory viewpoint of causality.

Pearl's causal hierarchy shows that observational, interventional, and counterfactual queries are qualitatively distinct. We ask a quantitative version of this question: how many additional bits are needed to specify higher-rung causal answers once lower-rung answers are known? We formalize this via query-class description length, the Kolmogorov complexity of the answer oracle induced by an SCM for a class of queries. Our main construction gives binary acyclic SCMs whose observational distribution has constant description length, while the single-variable interventional answer oracle has description length $Θ(n^2)$. A degree-sensitive upper bound shows that finite-gate-schema SCMs of indegree $d$ have observational-interventional gap at most $O(nd \log(en/d) + n \log n)$, making the quadratic construction order-optimal in the dense regime and a rooted-tree construction order-optimal for bounded indegree. The quadratic separation persists under $\varepsilon$-accurate total-variation descriptions for every fixed $\varepsilon < 1/4$. At the next rung, the full hard-do interventional oracle can still leave a $Θ(n)$ counterfactual description gap. A general ambiguity-to-bits theorem and Shannon analogue show that these gaps equal the logarithm of residual higher-rung ambiguity up to lower-order terms.