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From Statistical Relational to Neurosymbolic Artificial Intelligence: a Survey

Giuseppe Marra, Sebastijan Dumančić, Robin Manhaeve, Luc De Raedt

arXiv:2108.11451v4cs.AI

TL;DR

Integrating learning and reasoning is a shared aim of NeSy and StarAI, which approach it through neural networks and probabilistic graphical models, respectively. The survey organizes their systems along seven shared dimensions and uses comparisons to identify analogies and opportunities for cross-fertilization.

  • Problem

    NeSy and StarAI both integrate learning and reasoning, yet they have developed with different paradigmatic emphases and relatively little interaction.

    Source: §1 · p. 2

  • Method

    The survey identifies seven shared dimensions and positions representative StarAI and NeSy systems along them to compare their approaches.

    Source: §1 · pp. 2–3

  • Results

    The survey highlights analogies between the fields and identifies opportunities to adapt techniques from one to the other.

    Source: §1 · p. 2

  • Takeaways & Limitations

    The shared dimensions provide concepts for understanding how learning and reasoning are integrated across StarAI and NeSy.

    Source: §1 · p. 2

    In sketch-based approaches, users must know the program structure approximately, and making the sketch less strict enlarges the search space.

    Source: §5.2 · p. 30

Abstract

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This survey explores the integration of learning and reasoning in two different fields of artificial intelligence: neurosymbolic and statistical relational artificial intelligence. Neurosymbolic artificial intelligence (NeSy) studies the integration of symbolic reasoning and neural networks, while statistical relational artificial intelligence (StarAI) focuses on integrating logic with probabilistic graphical models. This survey identifies seven shared dimensions between these two subfields of AI. These dimensions can be used to characterize different NeSy and StarAI systems. They are concerned with (1) the approach to logical inference, whether model or proof-based; (2) the syntax of the used logical theories; (3) the logical semantics of the systems and their extensions to facilitate learning; (4) the scope of learning, encompassing either parameter or structure learning; (5) the presence of symbolic and subsymbolic representations; (6) the degree to which systems capture the original logic, probabilistic, and neural paradigms; and (7) the classes of learning tasks the systems are applied to. By positioning various NeSy and StarAI systems along these dimensions and pointing out similarities and differences between them, this survey contributes fundamental concepts for understanding the integration of learning and reasoning.

1. Introduction

StarAI and NeSy pursue the shared goal of integrating learning and reasoning, but have developed with different emphases and relatively little interaction. The survey compares them through shared dimensions and positions representative systems to identify similarities and opportunities for cross-fertilization.

1. Introduction

  • StarAI emphasizes probabilistic logic, semantics, and tractable inference, while NeSy extends neural networks with symbolic knowledge and focuses on scalable approximate models.The survey notes that the two communities have interacted relatively little despite sharing the goal of integrating learning and reasoning.Source: §1 · p. 2
  • The survey identifies seven shared dimensions: inference approach, logic syntax, semantics, parameter or structure learning, representations, integration of paradigms, and learning tasks.They are intended to categorize both StarAI and NeSy approaches.Source: §1 · p. 2
  • It positions a wide range of StarAI and NeSy systems along these dimensions, highlighting analogies and opportunities to adapt techniques across fields.The survey also illustrates the dimensions with existing methods to provide an intuitive, concrete overview.Source: §1 · pp. 2–3
  • The survey takes a logical perspective on NeSy and focuses on representative, prototypical systems rather than aiming for completeness.It contrasts its focus on underlying principles with earlier surveys centered on reasoning, graph neural networks, composition patterns, or points where symbolic knowledge enters neural networks.Source: §1 · p. 3
  • Its seven subsequent sections introduce concepts from logic, probability, or machine learning and show how they appear in StarAI and NeSy systems.The dimensions cover inference, syntax, semantics, learning, representation, paradigm integration, and learning tasks.Source: §1 · pp. 2–3

2. Proof- vs Model-theoretic View of Logic

This section contrasts proof-based inference, which derives query proofs, with model-based reasoning, which finds truth assignments satisfying clauses, and traces how these views shape StarAI probabilistic models and NeSy architectures or losses. It also considers how proof-based and objective-based NeSy systems incorporate logic, including their tradeoff between control and interpretability and scalability.

2. Proof- vs Model-theoretic View of Logic

  • Proof-theoretic inference seeks a proof for a query, whereas model-theoretic reasoning seeks a truth assignment that satisfies the theory.The survey focuses on clausal logic; proof search in logic programs can use AND/OR trees and forward or backward chaining, while model-theoretic theories constrain propositions without imposing directed inference.Source: §2 · pp. 5–7

2.1. Implications for StarAI

  • StarAI extends clauses into probabilistic models through directed definite-clause programs or undirected weighted-clause constraints.Probabilistic logic programs annotate facts with probabilities and define possible-world semantics; MLNs and PSL use weighted clauses as soft constraints, with infinite weights enforcing hard constraints.Source: §2.1 · pp. 7, 9

2.2. Implications for NeSy

  • Proof-based NeSy systems compile query proofs into differentiable neural computation graphs, making inference graph evaluation and enabling parameter learning.Weighted facts or rules label leaves or edges, while activation functions label AND/OR nodes; examples include TensorLog, Neural Theorem Provers, DeepProbLog, and LRNNs.Source: §2.2 · pp. 10–11
  • Model-based NeSy systems turn logical constraints into neural-network loss terms, training networks to satisfy the theory while learning subsymbolic tasks.Approaches include fuzzy-logic regularizers in SBR and LTN, probability-based Semantic Loss, and DL2 [40], which can include numerical variables; unlike proof-based models, the logic is not used at inference.Source: §2.2 · pp. 12–14
  • Proof-based, directed NeSy models use logical clauses to define network architecture and constrain inference, giving designers control over where and how logic is used.Source: §2.2 · p. 14
  • Proof-based systems can modify logical knowledge at test time without retraining, supporting modularity and out-of-distribution generalization.Source: §2.2 · p. 14
  • Objective-based systems encode logic latently in network weights, reducing control and interpretability but often improving inference-time scalability.Source: §2.2 · p. 14
  • Balancing control and interpretability against scalability remains an open and important research question in NeSy.Source: §2.2 · p. 14

3. Logic - Syntax

This section distinguishes propositional, relational, and first-order syntax through the structure of literals and grounding, then traces their implications for StarAI and NeSy. First-order rules can template and share parameters in probabilistic models, while structured logical terms support neural-model unrolling and subsymbolic inputs.

3. Logic - Syntax

  • Propositional literals are unstructured truth-valued atoms, whereas first-order literals use predicates over terms; relational logic excludes functors.First-order terms can include constants, variables, and structured terms formed using functors.Source: §3 · p. 15
  • Grounding applies substitutions that replace variables with terms, producing ground expressions and potentially infinitely many ground clauses when functors are present.For finite domains, the passage says grounding yields a polynomially larger set of clauses.Source: §3 · p. 15

3.1. Implications for StarAI

  • First-order rules template graphical models, and shared parameters across their groundings compress representations for more efficient learning and better generalization.Grounding unrolls the template; factors corresponding to the same rule share its weight.Source: §3.1 · p. 16

3.2. Implications for NeSy

  • Grounding relational or first-order theories can unroll neural architectures or loss functions, supporting model capacity, modularization, generalization, and formal control.The survey relates this unrolling to recurrent, recursive, and graph neural networks.Source: §3.2 · p. 16
  • First-order NeSy systems can treat subsymbolic samples as domain elements and encode neural inputs by grounding predicates over them.When literals lack internal structure, as in Semantic Loss, this mapping must be handled outside the logical framework.Source: §3.2 · p. 17
  • First-order logic can represent real-valued functions with functors, allowing FOL-based systems to address regression beyond relational logic's conventional classification tasks.The passage illustrates this with a segmentation function that returns an object's bounding box in an image.Source: §3.2 · p. 17

4. Logic - Semantics

The survey traces model-theoretic semantics from Boolean models to fuzzy and probabilistic extensions, then compares how StarAI and NeSy use these semantics for inference and learning.

4.1. Model-theoretic semantics

  • Definite-clause theories have a unique least Herbrand model, general clause sets may have multiple models, and ASP can represent disjunction using stable models.The least Herbrand model contains all ground facts entailed by the theory and is smallest under set inclusion.Source: §4.1 · p. 17

4.2. Fuzzy semantics

  • Fuzzy semantics assigns atoms truth degrees in [0, 1] and maps logical connectives to real-valued t-norm functions, allowing formulas to be evaluated through differentiable circuits.The circuit is compiled from the sentence's expression tree using the corresponding t-norm operations.Source: §4.2 · pp. 18–19

4.3. Implications for StarAI

  • StarAI places probability distributions over possible worlds and uses knowledge compilation to turn weighted model counting into bottom-up evaluation of an arithmetic circuit.A d-DNNF supports this compilation by making conjunctions decomposable and disjunctions deterministic.Source: §4.3 · pp. 19, 21, 22
  • Probabilistic Soft Logic uses fuzzy-valued atoms and factors, enabling gradient-based approximation of MAP inference where Boolean counterparts are related to NP-hard maxSAT.Its fuzzy interpretation makes the objective continuous and differentiable.Source: §4.3 · pp. 22–23

4.4. Implications for NeSy

  • NeSy systems can replace scalar probabilities or truth values with neural-network outputs, as DeepProbLog does for probabilities of probabilistic facts.DeepProbLog retains the arithmetic-circuit structure while neural networks supply some fact-probability leaves.Source: §4.4 · pp. 23–24
  • SL preserves probabilistic logic semantics but compilation can yield formulas exponential in grounded-theory size, whereas SBR directly produces a linear-size differentiable loss by using fuzzy semantics.The contrast is between a probabilistic compilation route and a direct fuzzy-logic loss.Source: §4.4 · p. 26

5. Structure versus Parameter Learning

The survey distinguishes learning a logical theory from learning its weights, then compares how StarAI and NeSy systems approach those tasks. StarAI commonly separates clause discovery from weight learning, while NeSy includes neural guidance, sketching, and other ways to handle program structure.

5. Structure versus Parameter Learning

  • Structure learning discovers clauses and their weights, whereas parameter learning keeps the logical theory fixed and learns only its probabilities or weights.In discriminative settings, structure learning targets a relation; in generative settings, it explains interactions among relations.Source: §5 · pp. 26–27

5.1. Implications for StarAI

  • StarAI structure search faces combinatorial explosion, so language bias or clause-wise learning can constrain the search, although restrictive bias may exclude the target theory.Structure learning is NP-complete; parameter learning is easier in principle but depends strongly on the quality of supplied clauses.Source: §5.1 · p. 27, §5 · p. 27
  • StarAI structure learning searches valid theories, often identifies clauses before learning their weights, and can use a clause-at-a-time procedure such as ProbFoil.ProbFoil refines candidate clauses, scores their coverage of positive and negative examples, then learns probabilities for selected clauses.Source: §5.1 · pp. 27–29

5.2. Implications for NeSy

  • NeSy learning spans neurally guided search, structure learning via parameter learning, program sketching, and implicit structure learning.These approaches combine or shift the boundary between symbolic structure search and parameter optimization.Source: §5.2 · p. 29
  • Neural guidance ranks symbolic candidates to prioritize search, addressing uninformed exploration, though such methods can require large training sets.Guidance may score predicates or clauses; one described approach uses a neural heuristic to choose which candidate clause to expand, with reported computation-time improvements of several orders of magnitude.Source: §5.2 · pp. 29–30
  • Program sketching reduces the search space by asking users to specify most of a program and leave selected holes for symbolic or neural completion.The sketch imposes a strong bias; making it less restrictive enlarges the search space.Source: §5.2 · p. 30
  • Structure learning via parameter learning optimizes weights for clauses enumerated from user templates, avoiding combinatorial search but retaining many potentially irrelevant clauses.Large clause sets can make optimization difficult, and irrelevant clauses may cumulatively create spurious interactions.Source: §5.2 · pp. 30–31
  • NMLNs learn relational features and potentials over data fragments without necessarily relying on a provided or learned symbolic model structure.This avoids combinatorial search and uses gradient-based learning, but limits fragment size and loses the ability to inspect discovered structure.Source: §5.2 · p. 32

6. Symbolic vs subsymbolic representations

The section contrasts symbolic representations, which support exact structural matching, with subsymbolic representations, which support numerical similarity. NeSy systems combine them through distinct translation and reasoning arrangements.

6. Symbolic vs subsymbolic representations

  • Symbolic representations use entities, numbers, variables, and structured terms, while subsymbolic representations encode entities as fixed-size numerical vectors, matrices, or tensors.Unification finds substitutions that make terms syntactically equal; subsymbolic methods can compare representations numerically but do not directly determine entity identity.Source: §6 · pp. 32–33
  • Translating symbols into one-hot encodings or embeddings does not itself enable symbolic manipulation, and encoding relations or graph structures remains challenging.Translating vectors back to classes can use softmax, arg-max, or Gumbel-softmax, depending on the desired output.Source: §6 · p. 33

6.1. Implications for StarAI and NeSy

  • NeSy methods vary in whether symbolic and subsymbolic representations occur sequentially, alternately, or simultaneously during reasoning.The survey contrasts these arrangements with StarAI's symbolic input, intermediate, and output representations and neural systems' subsymbolic inputs and intermediates.Source: §6.1 · p. 33
  • Many NeSy methods translate symbolic inputs into subsymbols once, then reason with subsymbolic intermediate representations; Logic Tensor Networks make this grounding explicit.They define grounding as a mapping of symbolic entities to subsymbolic counterparts and ground clauses by combining these mappings with a t-norm.Source: §6.1 · pp. 33–34
  • Other NeSy systems alternate symbolic and subsymbolic processing, as in ABL's perception, consistency-checking, and abductive-reasoning components.ABL revises pseudo-labels to obtain logically consistent hypotheses and uses them to retrain the perception model.Source: §6.1 · p. 34
  • Neural Theorem Prover interweaves symbols and subsymbols through soft-unification, using embedding similarity when ordinary unification fails.Parameterized predicates can also support rule learning; soft-unification may create additional proofs and computational problems.Source: §6.1 · pp. 35–36

7. Logic vs Probability vs Neural

The survey classifies systems by how fully they preserve logic, probability, and neural computation. It describes contrasting StarAI and NeSy combinations and identifies a trade-off between scalability and preserving the original paradigms.

7. Logic vs Probability vs Neural

  • The survey marks full recovery of a paradigm with uppercase L, P, or N and partial recovery with lowercase letters.Preservation means the ability to replicate the original paradigm's model and inference algorithm.Source: §7 · p. 36

7.1. StarAI: Logic + Probability

  • Knowledge-based model construction is classified lP because logic generates a probabilistic graphical model that supplies semantics and inference.This can make the effects of logical inference rules on the resulting distribution difficult to predict.Source: §7.1 · p. 36
  • Probabilistic logic programs are classified Lp: logical operations remain clear, but exploiting probabilistic independencies needed for efficient inference is harder.Source: §7.1 · pp. 36–37

7.2. NeSy: Logic + Probability + Neural

  • LTNs and SBRs are classified lN because they use logic as a training regularizer but drop the logical loss at test time and do not integrate probability.Source: §7.2 · p. 37
  • Logic-focused differentiable methods are classified Ln: they build networks through symbolic inference and relax semantics for learning, but are often unlike deep-stacked neural architectures.Source: §7.2 · p. 37
  • An extended LRNN framework is classified LN and can treat logic rules as deep-architecture layers, implementing existing neural architectures.Source: §7.2 · p. 37
  • NeSy systems extending knowledge-based model construction are classified lPN, while DeepProbLog and NeurASP extend probabilistic logic programs as LpN.Source: §7.2 · p. 37
  • Systems in lN or Ln are usually more scalable but may relax logic and omit probability distributions; lPN or LpN preserve paradigms at the cost of more complex inference.The survey notes that these paradigm-preserving systems usually resort to exact probabilistic inference.Source: §7.2 · p. 37

8. Tasks

The survey reviews learning tasks addressed by NeSy, from prediction with supplied knowledge to tasks that induce symbolic knowledge. These tasks combine neural models and logical reasoning in different ways.

Distant Supervision

  • Distant supervision uses logic programs to map neural-network concepts to labels, training the neural subtasks indirectly through the label and background knowledge.The logic program is essential to inference; considering all label-compatible concept combinations creates a scaling challenge.Source: §8 · p. 38

MNIST Addition

  • In MNIST addition, the training target is the sum of two images’ digit labels, so digit classification receives no direct label supervision.Logic links the model’s digit outputs to the supplied sum, reasoning about digit pairs that could produce it.Source: §8 · pp. 38–39

Semi-supervised Classification

  • Semi-supervised classification combines direct supervision for some concepts with additional, often noisy logical knowledge that can improve predictions.When the knowledge connects concepts across patterns, the task is collective classification.Source: §8 · p. 39

Document classification in citation networks

  • Document classification in citation networks balances document-word features with neighboring papers’ labels to make collective predictions.NeSy systems can express label passing with a rule weighted differently across domains.Source: §8 · pp. 39–40

Knowledge Graph Completion

  • Knowledge graph completion predicts missing graph edges and links NeSy reasoning with knowledge graph embeddings and graph neural networks.The survey identifies NTPs, NMLN, DLM, DiffLog, and TensorLog as systems applied to this task.Source: §8 · p. 40

Generative Tasks

  • Generative tasks model input-data distributions to sample new patterns, combining deep generative models with symbolic features and logic reasoning.Examples include molecule generation in NMLNs and image generation in VAEL or.Source: §8 · p. 40

Knowledge Induction

  • Knowledge induction makes symbolic knowledge the learning target, as in program synthesis from positive and negative input-output examples.Some induced programs explain the behavior of a black-box neural model, connecting the task to explainability.Source: §8 · pp. 40–41

9. Open Challenges

The survey identifies open challenges in semantics, probabilistic and fuzzy reasoning, structure learning, inference scaling, data efficiency, and symbolic representation learning. It frames these as research directions for understanding and improving NeSy and StarAI systems.

Semantics

  • NeSy still needs clearer model semantics and better understanding of how its competing formalisms relate.The survey compares the current situation to early statistical relational learning, where competing formalisms were described as an “alphabet soup.”Source: §9 · p. 41

Probabilistic reasoning

  • Probabilistic integration of logic and neural methods is considered natural, but inference is costly and often requires approximation.The survey calls for comparing probabilistic approximate inference with approximations based on logic relaxations such as fuzzy logic.Source: §9 · p. 41

Fuzzy semantics

  • Fuzzy approaches vary in their t-norms and connective translations, and often leave unclear which Boolean-logic properties they preserve.The survey suggests studying natively fuzzy models and characterizing Boolean approximations relative to properties one wants to preserve.Source: §9 · p. 41

Structure learning

  • StarAI structure learning remains difficult because inference is complex and the search is combinatorial, with neural components adding further difficulty.Existing NeSy methods are limited and mostly domain-specific; systems supporting structure learning still require clause templates or a model sketch from users.Source: §9 · p. 42

Scaling inference

  • Scaling inference remains a major challenge for StarAI and for NeSy systems with explicit logical or probabilistic reasoning.The survey identifies lifted or approximate inference and reasoning from intermediate representations as research directions.Source: §9 · p. 42

Data efficiency

  • StarAI methods can learn efficiently from small datasets but do not scale to big datasets, whereas neural methods are data-hungry but handle big datasets.The survey presents understanding how the approaches might help each other overcome these complementary weaknesses as a research direction.Source: §9 · p. 42

Symbolic representation learning

  • Changing data representations helps deep learning make target tasks easier, while changing representations at the symbolic level remains an open NeSy challenge.The survey suggests neurally inspired methods could help make progress on symbolic representation learning.Source: §9 · p. 42

Appendix A. Knowledge graphs embeddings and Graph Neural Networks for neurosymbolic AI

The appendix examines knowledge graph embeddings and graph neural networks through the survey’s seven dimensions. It presents them as sharing features with NeSy systems and as lying at the intersection of statistical relational and geometric deep learning.

Appendix A

  • The appendix analyzes knowledge graph embeddings and graph neural networks using the survey’s seven dimensions and finds they share features with NeSy systems.It proposes that NeSy can also be found at the intersection of statistical relational and geometric deep learning approaches.Source: Appendix A · p. 57

Knowledge graphs

  • A knowledge graph is a multi-relational graph of entities and relations, with edges commonly represented as head-relation-tail triples.The appendix gives (homer, fatherOf, bart) as an example triple.Source: Appendix A · p. 57

Knowledge graph tasks

  • StarAI has been used for knowledge graph completion, entity-property prediction, and clustering entities by connectivity patterns.The appendix notes that explicit probabilistic dependencies among relationships make StarAI well suited to knowledge graph reasoning.Source: Appendix A · p. 57

Latent representations

  • To scale to larger knowledge graphs, StarAI probabilistic dependencies have been relaxed into latent-feature models that predict relationships through simpler interactions.Knowledge graph embeddings and graph neural networks are two ways to encode these latent representations.Source: Appendix A · p. 58

Appendix A.1. Knowledge Graph Embeddings

Knowledge graph embedding models score triples using entity and relation embeddings, with models differing in scoring functions and embedding spaces. Some approaches incorporate logical clauses or queries, connecting embeddings to NeSy systems.

Knowledge Graph Embeddings

  • Knowledge graph embedding models assume triples are conditionally independent given global latent embeddings for entities and relations.A scoring function over the head-entity, relation, and tail-entity embeddings predicts whether a triple exists.Source: Appendix A.1 · p. 58

Scoring functions

  • Embedding models mainly differ in their scoring functions and embedding spaces, including translation-based and semantic-matching approaches.Translation-based methods score how closely a relation-specific translation of the head matches the tail; semantic matching uses similarity-based scoring.Source: Appendix A.1 · p. 58
  • Standard multilayer perceptrons are often used to learn semantic-matching similarity measures, resembling neural interfaces in lPN and LpN models.The survey relates this to the neural predicates of DeepProbLog in Example 13.Source: Appendix A.1 · p. 58

Logical information

  • Knowledge graph embeddings learn entity and relation embeddings from ground relations, though some methods also incorporate first-order clauses or logical queries.The latter approaches bridge embeddings with NeSy systems such as SL [133] and SBR [33].Source: Appendix A.1 · p. 58

Seven dimensions

  • Knowledge graph embedding models are mostly model-based and undirected, rely heavily on subsymbolic representations, and learn task-specific embedding parameters.When logical rules are injected, a fuzzy-logic interpretation of the scoring function is often used.Source: Appendix A.1 · p. 58

Appendix A.2. Graph Neural Networks

Graph neural networks process relational data through neural message passing, but their role in logic-based reasoning remains limited. Their subsymbolic, proof-based computation also raises questions about expressivity and tractable learning.

Appendix A.2. Graph Neural Networks

  • At each layer, nodes send messages to outgoing neighbors and update their states by aggregating incoming messages; final representations classify nodes or graphs.Messages are computed by standard neural networks.Source: Appendix A.2 · pp. 58–59
  • GNNs applied to knowledge graphs generally classify known structures rather than infer whether an edge exists; relational reasoning often adds KGE-based graph reconstruction.A GNN encodes the graph, then a KGE-based factorization technique reconstructs it.Source: Appendix A.2 · p. 59
  • GNN inference relies exclusively on neural computation, with no clear direction for injecting external knowledge such as logical rules.Source: Appendix A.2 · p. 59
  • GNNs can be characterized as directed, proof-based systems with subsymbolic node representations and implicitly learned inference rules.Original GNNs update node states until a fixed point, resembling forward-chaining in logic programming.Source: Appendix A.2 · p. 59
  • GNNs have been shown to be as expressive as two-variable counting logic C2, a first-order fragment with counting quantifiers.Source: Appendix A.2 · p. 59
  • A shared challenge is increasing expressivity to higher-order fragments while keeping learning and inference tractable.GNNs learn logical structure implicitly without expert-provided knowledge or heavy combinatorial search, but their expressivity is limited compared with many logic-based NeSy models.Source: Appendix A.2 · p. 60

Appendix B. Fuzzy logic, fuzzyfication and soft-Satisfability

Fuzzy logic is often used in StarAI and NeSy to make logical inference differentiable and more scalable. However, optimizing a fuzzy relaxation and then thresholding its values can fail to preserve Boolean satisfaction, and some inferences can be counterintuitive.

Appendix B. Fuzzy logic, fuzzyfication and soft-Satisfability

  • Differentiable t-norms can make first-order logical inference more scalable than standard combinatorial optimization, and some fuzzy fragments yield convex inference problems.The survey highlights this use in undirected and regularization-based methods such as PSL and LTN.Source: Appendix B · p. 60
  • Maximizing a fuzzy relaxation of A ∨ B ∨ C can yield (0.34, 0.34, 0.34), which thresholding at 0.5 maps to all False, violating the Boolean disjunction.The example uses the Łukasiewicz t-norm and gradient-based optimization.Source: Appendix B · p. 60
  • Gradient-based fuzzy inference for universally quantified rules can be counterintuitive, particularly with the product t-norm.The cited example concerns rules such as ∀x: A(x) → B(x).Source: Appendix B · p. 61
  • It remains unclear which Boolean-logic properties fuzzy inference should preserve or whether a t-norm can guarantee them.Source: Appendix B · p. 61

Appendix B.1. Distribution semantics and fuzzy logic semantics

Probabilistic and fuzzy semantics assign different meanings to logical scores: one is grounded in possible worlds, while the other assigns truth degrees to formulas or proofs. The example shows that repeated proof structure can change a fuzzy score even when the probability remains unchanged.

Appendix B.1. Distribution semantics and fuzzy logic semantics

  • Under distribution semantics, an atom's probability sums the probabilities of worlds where it is true; fuzzy logic instead assigns truth degrees to formulas or proofs.Source: Appendix B.1 · p. 61
  • With p(b) = 0.3, both p(a1) and p(a2) equal 0.3, while product-t-norm fuzzy scores are t(a1) = 0.3 and t(a2) = 0.09.The rules are a1 <- b and a2 <- b,b.Source: Appendix B.1 · p. 61
  • Choosing a different t-norm, such as the minimum t-norm, can remove this particular discrepancy, but similar issues arise under other definitions.Source: Appendix B.1 · p. 61
  • The discrepancy reflects possible-world versus proof-or-derivation semantics, rather than probability versus fuzzy semantics alone.A similar behavior is reported for Stochastic Logic Programs under the name memoization.Source: Appendix B.1 · pp. 61–62
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