ProbLog

Origin. De Raedt et al. (2007). Probabilistic logic programming. Facts with probabilities. Distribution semantics. Foundation for probabilistic inference in logic programs.

Models. Probabilistic facts: P::f. Logic program over facts. Possible worlds: subsets of facts. Query probability: sum over worlds.

Formalism.

Probabilistic facts: p::f (fact f holds with probability p) Example: 0.3::earthquake.

Logic program: Definite clauses over probabilistic facts. alarm :- earthquake. alarm :- burglary.

Distribution semantics: Each probabilistic fact: independent Bernoulli. World: subset of probabilistic facts. P(world) = ∏{pf | f ∈ world} · ∏{1-pf | f ∉ world}

Query probability: P(q) = ∑{P(w) | w ⊨ q} Sum over worlds where query holds.

Example: 0.1::burglary. 0.2::earthquake. alarm :- burglary. alarm :- earthquake. P(alarm) = P(burglary ∨ earthquake) = 0.1 + 0.2 - 0.02 = 0.28

Inference: Exact: knowledge compilation (SDD, BDD). Approximate: sampling, bounds.

Learning: Parameter learning: learn probabilities. Structure learning: learn rules. LFI: learning from interpretations.

Extensions: Annotated disjunctions: p₁::a₁; ...; pₙ::aₙ. Continuous distributions (DC-ProbLog).

Symbols.

Symbol	Unicode	Name	Meaning
p::f	—	Probabilistic fact	p probability
:-	—	Clause	Definite clause
P(q)	—	Query probability	Marginal
⊨	U+22A8	Entails	World satisfies

Metatheory. Distribution semantics well-defined. Complexity: #P-complete. Exact inference via compilation. Sound learning.

Applies to. Probabilistic reasoning. Bioinformatics. NLP. Probabilistic databases. Neuro-symbolic AI.

Limitations. Scalability. Independence assumption. Ground inference. Continuous extensions complex.