Probabilistic Logic

Probabilistic Neurosymbolic Learning (PNL) combines neural predictions with symbolic reasoning, enabling end-to-end learning from final-output supervision without labels for intermediate concepts. A central challenge is probabilistic inference: state-of-the-art approaches often rely on materializing the logical provenance of a query, which can itself become a major computational bottleneck. We introduce Dynamic Probabilistic Neurosymbolic Learning (DPNL), an oracle-guided framework that avoids requiring complete provenance materialization before inference. DPNL lazily explores the space of intermediate assignments, while oracles resolve entire regions that can already be certified to produce or exclude the target output. We establish conditions ensuring soundness and termination. ApproxDPNL extends the same search with early termination while maintaining certified bounds on the exact output probability, providing controlled approximation guarantees. The oracle interface decouples inference from the representation of the symbolic component, enabling problem-specific reasoning within the same framework. Experiments on several neurosymbolic tasks show that DPNL and ApproxDPNL substantially extend the range of problem instances tractable by probabilistic neurosymbolic inference.

Allen's interval algebra is a qualitative calculus for temporal relations, but its thirteen base relations are crisp predicates over exact interval boundaries. This is inadequate for temporal information from language, perception, databases, or uncertain histories, where times, durations, and boundaries are uncertain and expressions such as "just before" or "roughly during" have graded meaning. We develop the probabilistic Allen algebra (PAA): a generative and complete extension in which relation probabilities are derived from distributions over interval boundaries rather than assigned as scores. Time points are Gaussian; intervals have Gaussian midpoints and truncated-Gaussian durations. Every relation is a boundary-ordering predicate in one common probability space: point-point relations reduce to error functions, and point-interval and interval-interval relations to multivariate Gaussian orthant probabilities induced by linear inequalities. Contact relations (meets, starts, finishes, equals) receive positive measure through a tolerance band, and under a single tolerance the thirteen relations form a true partition that recovers crisp Allen as the tolerance vanishes. The construction derives Allen's taxonomy rather than positing it: coarse predicates such as precedence, overlap, and containment are unions of leaves whose probabilities are leaf sums, and this hierarchy is preserved as intervals collapse to points and thirteen relations reduce to five and then three. Each relation further decomposes into correlation-aware temporal primitives in the spirit of CIDOC CRM. The algebra is scale-invariant and separates graded expressions such as "shortly before" from contact relations. All results are Monte-Carlo validated and shipped as an open, tested Python package.

Knowledge graphs describe reality in crisp assertions, while the systems now consuming them, foundation models and autonomous agents, reason natively in probabilities. We argue that this mismatch is why the integration of language models and knowledge graphs remains a data-feeding pipeline rather than a unified reasoning architecture. We envision Semantic Bayesian World Models (SBWMs): a Web that describes the world not as a database of facts but as a shared, evolving fabric of beliefs over knowledge graphs, where ontological axioms constrain priors, observations update beliefs by Bayesian conditioning, and actions intervene upon the world. We work through what an agent gains from such a model: a home-security agent deciding whether the figure at the gate is a courier or a burglar, an actuarial estimate aggregated by entailment rather than by string frequency, a planning task that language models reliably fail, and the estimation of quantities that no document has ever stated. We then set out what the community must build to make them possible: belief annotation over RDF~1.2, probabilistic entailment regimes, semantic calibration layers, and protocols by which agents that have never met can exchange, and disagree over, calibrated beliefs.

Standard probabilistic logic programming frameworks typically rely on grounding logic programs into discrete propositional representations. This operational requirement restricts exact inference to finite domains and discrete probability distributions. In this paper, we introduce Measure-Theoretic Probabilistic Definite Clause Logic (MT-PDCL), a generalized foundational framework that eliminates this finite-domain restriction. By explicitly defining stochastic variables over bounded index domains and equipping the interpretation space with standard Borel $σ$-algebras, MT-PDCL allows logical variables to operate natively over continuous measurable spaces. Building on Continuous Distribution Semantics, MT-PDCL models probabilistic rules as mutually independent causal events. However, rather than aggregating these derivations via finite boolean circuits, declarative entailment is formally defined through exact Lebesgue integration over the continuous measure space. We introduce a continuous immediate consequence operator that unifies the integration of continuous prior distributions with the evaluation of exact continuous observations. We demonstrate that this approach replaces the combinatorial bottleneck of discrete grounding with exact, algebraic, and structurally differentiable inference. While this transition trades discrete combinatorics for the geometric curse of dimensionality, it achieves the expressive power of continuous probabilistic models while preserving the pure declarative syntax of definite clause logic.

GK is a query-directed first-order prover that extends ordinary resolution-based proof search with explicit positive and negative claims, numerical confidence values, and prioritized default rules with exceptions. It works directly with non-ground clauses, including equality and function terms. Candidate proofs are found by bounded first-order proof search; exception conditions of defaults are checked by further bounded searches, recursively when exceptions themselves depend on defaults. This avoids requiring a finite global grounding, while allowing incomplete searches to be reported as such. This paper adds structure-preserving quantitative reporting to that framework. Retained proof histories are used in two calculations. The first reconstructs the uncertain ground premises used by each proof and computes the probability that at least one retained proof is available, without counting shared premises independently. The second resolves positive and negative support at intermediate atoms before that support is propagated through later rules; the same calculation evaluates uncertain exception conditions for individual rule applications. Reports separate positive support, negative support, conflict, and ignorance and identify detected incomplete calculations or fallbacks. The implementation performs bounded reconstruction and dependency traversal after proof search and still requires no global grounding. Analytic examples and independent simulators reproduce the reference calculations on their stated fragments. Comparisons with probabilistic logic, probabilistic ASP, default logic, and goal-directed ASP identify cases of agreement, semantic difference, unsupported translation, and incomplete computation.

In medical applications, raw data is frequently associated with significant privacy concerns, lending particular importance to the encoding of summary statistics from the literature. On the other hand, deep learning has become an invaluable tool for assessing symptoms based on visual or auditory sensor data. DeepProbLog allows for an extensible neuro-symbolic approach that accommodates connectionist components to analyse patient images within a transparent and rigorous probabilistic framework, namely probabilistic logic programming under the distribution semantics. Framed as a case study in stroke detection from multimodal data, this contribution explores the pathway from summary statistics available in the literature to a DeepProbLog-based diagnostic system. It suggests a workflow using established maximum entropy techniques to complete available probabilistic information and the probabilistic logic programming system ProbLog 2 to move from the entropy-maximising causal model to a discriminative neuro-symbolic model expressible within DeepProbLog. The relative performance of models derived from less complete data is analysed alongside the potential of the probabilistic inductive logic programming system ProbFOIL 2 for compressing large discriminative models, and the perspectives and implications of using DeepProbLog for diagnostic reasoning are discussed.

Probabilistic logic programming is a formalism of statistical relational artificial intelligence that supports causal queries, including interventions from outside the system. When the structure of a probabilistic logic program is learned from data, however, only probabilistic information is used, and a single probability distribution may be compatible with several causal orders. This leads to ambiguity in interventional reasoning, raising the question of when the causal order is uniquely determined by the distribution. Exploiting the relationship between acyclic probabilistic logic programs and Bayesian networks, we derive conditions under which the probabilistic information encoded in a program determines a unique causal order. We also incorporate constraints arising from relational structure by taking into account prescribed sets of causal symmetries induced by the underlying relational vocabulary. The result is a method for verifying when a learned probabilistic logic program supports well-defined intervention semantics.

Both expressions of uncertainty and inferences are ubiquitous in natural language, and valid inferences over natural-language expressions of uncertainty are necessary for not only everyday conversations but also for high-stakes domains such as medicine and law. While large language models are increasingly evaluated on logical reasoning tasks, disentangling principled, symbolic reasoning from clever surface-level pattern matching is fraught with difficulty. We introduce a benchmark for reasoning over probability operators--inference over sentences with gradable epistemic modals (e.g., probably, might, must) containing 14,320 procedurally-generated English prompts across fifteen inference templates, systematically varying question form, negation strategy, and surface content. Evaluating 29 models, we find that most show answer biases independent of the logical form, a systematic preference for Yes or No. We summarize this with a competence floor: the worse of a model's accuracy on Yes-correct and No-correct items. Only 9 of 29 models exceed random chance. We also test variations in question form, verb phrases/activity, and both the gender and origin of names used in the prompts, finding biases across every axis.

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.

Machine Learning models are widely used for automating classification tasks by extracting statistical patterns from data. However, their performance deteriorates if the data distribution changes, making them ill-suited to handle uncertain and evolving information. Moreover, they provide limited support for integrating prior knowledge. To address these limitations, we present CLARK (Closed-loop Learning for Adaptive Reasoning over Knowledge Graphs), a framework that integrates knowledge graphs, symbolic rule mining, and probabilistic reasoning under the Logic Programs with Markov Logic Networks (LP$^{\text{MLN}}$) formalism. Starting from CACTUS-derived KGs, CLARK translates graph structure into an LP$^{\text{MLN}}$ program and iteratively enriches it with candidate rules proposed by symbolic learners. These rules are calibrated through probabilistic weight learning, enabling reasoning under uncertainty and refinement of the underlying graph structure. We evaluate CLARK on two medical datasets, analysing both rule quality and downstream classification performance. Results demonstrate that CLARK leads to improved classification performance and more generalisable inference. Overall, CLARK provides a principled approach to constructing adaptive, interpretable, knowledge-driven models for classification.

Logic and optimization can, in combination, make valuable contributions to rule-based AI. Logic is the obvious medium for encoding a rule base and drawing inferences from it, while optimization provides a powerful technology for computing inferences. Their combination has taken on new relevance amid a growing concern for transparency in AI. which is important for reproducibility, explainability, trustworthiness, and fairness. Rule-based AI provides a natural solution to transparency that is becoming increasingly practical due to today's highly advanced optimization methods. This article surveys several areas of logic-optimization partnership, including probabilistic logic, Bayesian logic, belief logics and Dempster-Shafer theory, nonmonotonic (default) logic, many-valued logics, and inference of logical formulas from noisy data based on Boolean regression. It shows how to compute projections, the fundamental problem of both logic and optimization, using decision diagrams and logic-based Benders decomposition. It describes the use of postoptimality analysis to explain how conclusions are reached, further enhancing transparency, as well as the role of optimization in answer set programming modulo theories. The paper concludes by suggesting possible future research directions.

Neuro-symbolic AI based on $IFOL_B$ is a way to combine neural learning and symbolic reasoning to overcome limitations of purely neural systems (like lack of interpretability and logical structure) with formal logical machinery for self-reference. In this paper we expand the cognitive power of $IFOL_B$ by using the probability computation for the currently unknown sentences, based on Nilsson's probability structure for the $IFOL_B$. We introduce the global symmetry transformation that preserves the current knowledge database and logical deduction, and the local one used for real-time decisions about concrete (sub)problems that involve only a very strict subset of $IFOL_B$ predicates. The computation of probability density function $KI$ in both cases, based on the Shannon's maximum information entropy, is provided by neural networks of this probabilistic neuro-symbolic AGI.

Neurosymbolic systems such as DeepProbLog combine neural perception with probabilistic logic, but standard inference is associational. Counterfactual reasoning additionally requires a causal semantics for interventions and evidence. We introduce DeepSWIP, a single-world counterfactual semantics for DeepProbLog programs. Using neural materialization, we reduce fixed-context neural predicates to ordinary ProbLog choices, apply Single World Intervention Programs (SWIPs), and compute counterfactuals by weighted model counting (WMC) over a single transformed program. Under finite grounding and unique-supported-model assumptions, DeepSWIP is exact relative to the learned materialized FCM. The standard quotient-WMC form of ProbLog conditionals identifies active neural probabilities and explains intervention cleaning, calibration sensitivity, and rare-evidence instability. Experiments on MPI3D confirm the transformation against a DeepTwin construction against 12,000 queries, as predicted and a 2.14$\times$ inference speedup from avoiding the Twin's endogenous duplication. A SUMO HOV experiment shows that neural calibration degradation biases plug-in estimates, while a correctly scoped randomized-policy AIPW estimator removes most first-order bias for population mean and ATE estimands. Code is at https://github.com/saibib/deep_SWIP.

A Markov Logic Network (MLN) is a probabilistic relational model used in Statistical Relational Artificial Intelligence for defining a probability distribution on the set of possible worlds with domain $D$ for an arbitrary finite domain $D$. An MLN consists of soft constraints with associated weights which are nonnegative real numbers. In this study we consider a language speaking about a property $P(x)$ and a relation $R(x, y)$. We consider an MLN for which every Boolean combination of $P(x)$ and $R(x, y)$ is a soft constraint (with associated weight). Let $n$ denote the size (cardinality) of the domain. We show that, for every choice of weights, if the weights are scaled by $1/n$ then, for every first-order sentence $\varphi$, the probability that $\varphi$ holds tends to either 0 or 1 as $n \to \infty$; that is, a 0-1 law for first-order logic holds. Morover, the limit probability does {\em not} depend on the weights. If we instead use the standard semantics of MLNs, in the case of which the weights are {\em not} scaled, then the limit behaviour is more complicated and {\em depends} on the weights. With unscaled weights we get 7 qualitatively different cases which depend on the weights. In some cases we have a 0-1 law for first-order logic, in some cases not, but we may still have a convergence law. The influence of the weights on the asymptotic probability of a first-order sentence may be in the form of a sudden ``phase transition'' from one of the 7 cases to another. The presence of a convergence law has positive implications for inference on large domains.

ULLER (Unified Language for LEarning and Reasoning) offers a unified first-order logic (FOL) syntax, enabling its knowledge bases to be used directly across a wide range of neurosymbolic systems. The original specification endows this syntax with three pairwise independent semantics: classical, fuzzy, and probabilistic, each accompanied by dedicated semantic rules. We show that these seemingly disparate semantics are all instances of one categorical framework based on monads, the very construct that models side effects in functional programming. This enables the modular addition of new semantics and systematic translations between them. As example, we outline the addition of generalised quantification in Logic Tensor Networks (LTN) to arbitrary (also infinite) domains by extending the Giry monad to probability spaces. In particular, our approach allows a modular implementation of ULLER in Python and Haskell, of which we have published initial versions on GitHub.