Semi-Supervised Learning using Differentiable Reasoning

Emile van Krieken; Erman Acar; Frank van Harmelen

REVIEW 3 major objections 6 minor 33 references

Semi-Supervised Learning using Differentiable Reasoning

T0 review · 3 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Adding a differentiable logic-satisfaction loss on unlabeled images significantly improves a simple classifier's semantic image interpretation, the authors claim.

desk verdict A useful empirical study of MP/MT imbalance in a Real Logic loss, but the normalization's claimed mechanism is unverified because Equation 18, as written, does not detach its theta-dependent weights. read the letter →

arxiv 1908.04700 v1 pith:5CSGM457 submitted 2019-08-13 cs.AI cs.LGcs.LO

Emile van Krieken , Erman Acar , Frank van Harmelen This is my paper · ORCID

classification cs.AIcs.LGcs.LO

keywords semi-supervisedlearningdifferentiablereasoningproductreallogicsemanticimageinterpretationravenparadoxmodusponenstollensfirst-order

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper introduces Differentiable Reasoning (DR), a semi-supervised objective that adds a differentiable measure of how well a first-order logic knowledge base is satisfied on unlabeled data to the usual supervised loss. On the PASCAL-PART semantic image interpretation task, this extra loss raises type-classification precision from 0.440 (supervised only) to 0.455 without special handling, and to 0.517 when the contributions of Modus Ponens and Modus Tollens gradient updates are explicitly rebalanced. The authors show that the Modus Tollens updates dominate Modus Ponens by about two orders of magnitude, a symptom of the Raven Paradox, and that normalizing the two update types is what yields most of the improvement. A sympathetic reader would take the paper to establish that relational background knowledge, injected as a fuzzy-logic loss over unlabeled images, is a practical and effective source of training signal, and that the imbalance between positive and contrapositive reasoning is a central design issue for such losses.

What carries the argument

The machinery is the Differentiable Reasoning (DR) loss, a special case of Real Logic that the authors call Product Real Logic: each ground atom $P(o)$ has truth value $f^\theta_P(o) \in [0,1]$ from a parameterized model, conjunction is multiplication, negation is $1 - p$, and implication $\varphi\rightarrow\psi$ is the Reichenbach implication $1 - p(\varphi)(1 - p(\psi))$. The loss for a rule $\forall x\, \varphi$ is the negative sum of log-valuations over all instantiations of $x$ across all unlabeled images. This keeps the computation polynomial in the number of objects per image rather than exponential in the Herbrand base, at the cost of exactness: Appendix A shows equality with the true marginal likelihood $p(K|O_u,\theta)$ only under disjointness assumptions (Equations 22-23) that the actual rule set does not satisfy. The second piece of machinery is the normalized loss of Equation 18, which separates the Modus Ponens and Modus Tollens gradient contributions and reweights them with a hyperparameter $\mu$; this is what converts the dominant contrapositive signal into a tunable resource.

What would settle it

Take a tiny unlabeled domain and a knowledge base that violates the disjointness assumptions (e.g., two rules sharing the predicate $partOf$), enumerate all worlds to compute the exact $p(K|O_u,\theta)$, and compare it with the DR loss's value during training; any divergence shows the loss is an estimate, not the declared marginal, and quantifies the error the approximation introduces.

Watch

Extended reading notes

Core claim

The central claim is that a simple supervised classifier for semantic image interpretation can be significantly improved by adding a 'differentiable reasoning' loss term that evaluates the degree to which a first-order logic knowledge base is satisfied by the model's predictions on unlabeled images. The loss is the negative log of a product-real-logic valuation of each rule, computed by grounding the universally quantified formulas over objects in each image and treating connectives with the product t-norm and the Reichenbach implication. Under the paper's construction, optimizing the supervised loss plus this unsupervised term approximates maximizing the joint likelihood of the labeled worlds and the knowledge base given the unlabeled domains. The paper further claims that the gradient of the implication rule decomposes into a Modus Ponens part (pushing up the consequent when the antecedent is likely) and a Modus Tollens part (pushing down the antecedent when the consequent is unlikely), that these are strongly imbalanced in practice, and that renormalizing their relative contribution, controlled by a hyperparameter $\mu$, produces the largest empirical gains (precision $0.517$ at $\mu=0.25$ versus $0.440$ supervised-only).

Load-bearing premise

The whole training signal rests on treating the unlabeled loss as the probability that the knowledge base holds, but the paper's own appendix proves that equality only when no ground atom is shared between rules or subformulas, which the actual rules violate.

Editorial extensions

If this is right

Semi-supervised gains on sparse relational tasks can be obtained from generic background knowledge without hand-labeling the unlabeled set; the DR loss turns every rule into a training signal on every image.
The Raven Paradox is not a philosophical curiosity but a measurable engineering phenomenon: here Modus Tollens updates are roughly 100 times larger than Modus Ponens updates, so any neural-symbolic loss built on material implication inherits a strong contrapositive bias.
Renormalizing MP and MT contributions introduces a single hyperparameter $\mu$ that controls the inductive bias; the paper's results suggest intermediate values ($\mu \approx 0.25$) beat both pure MP and the unnormalized baseline.
Contrapositive reasoning is not merely noise: disabling MT entirely ($\mu=1$) destabilizes training and lowers accuracy, so the normalization should rebalance rather than eliminate either update type.
The approach scales to images with a few dozen objects because the Herbrand base is $O(|P| \cdot |D_i|^2)$, making full grounding feasible for semantic image interpretation and similar structured domains.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Because the paper's Appendix A shows exactness only under disjointness assumptions that the used rules violate (a predicate like $partOf$ appears in multiple formulas), the DR loss is best read as an approximation to $p(K|O_u,\theta)$; a natural testable extension is to measure, on tiny domains, how far the product-real-logic valuation drifts from the exact world-sum when those assumptions fail.
The MP/MT imbalance likely transfers to any fuzzy-logic loss built on the material implication, including semantic loss variants and logic tensor networks; reweighting schemes analogous to Equation 18 could be applied there without changing the underlying model.
One could test the Raven-Paradox explanation directly by constructing a synthetic dataset in which the class balance of antecedents is controlled; the paper's account predicts that the optimal $\mu$ tracks the prior rate of positive antecedents rather than being task-specific.
A practical extension the authors leave implicit: the same loss could regularize natural-language semantic parsing, where objects in a sentence play the role of objects in an image and relational rules (e.g., thematic role constraints) supply the knowledge base.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The new thing here is not the Differentiable Reasoning loss itself—the authors correctly call it a special case of Real Logic and a close relative of Semantic Loss—but the empirical analysis of how Modus Ponens and Modus Tollens gradient contributions behave, and the normalization proposed to control them. That analysis is clearly explained and genuinely useful. The headline result, type precision improving from 0.440 with the supervised baseline to 0.517 with the normalized loss at mu=0.25 over 20 random splits of PASCAL-Part, is credible and significant.

The paper is honest about its main theoretical limitation: Product Real Logic is exact only under disjointness assumptions that practical knowledge bases violate, and Appendix A says so plainly. I also appreciate that the Raven Paradox connection is not just naming: the dMP/dMT plots and the gradient derivation in Equations 15-17 make the imbalance concrete.

Now the soft spots. The stress-test note about Equation 18 is correct and matters. Equation 18 is written as a loss, but the weights dMP/sum dMP and dMT/sum dMT are functions of theta. If you differentiate that loss, you get extra theta-derivative terms from the weights themselves. The paper's stated claim is that normalization controls the relative contribution of MP and MT gradient updates, but to actually get that behavior you must stop the gradient through the coefficients or treat them as constants. The paper never says it does either. So the 'Normalized' results in Table 1 cannot currently be attributed to the advertised control of the Raven Paradox. This is not a fatal flaw—the fix is a one-line clarification and a code check—but it is load-bearing for the paper's second contribution.

A smaller issue: the experiments compare only against the supervised baseline and the unnormalized variant. There are no standard semi-supervised competitors, no pseudo-labeling, no entropy minimization. The authors promise AUC for partOf in Section 3 but Table 1 reports only type precision; the AUC never appears. That is a missing result.

Citation patterns are fine; the authors credit the works their method builds on. The paper is well situated.

Who would get value from this? Researchers working on neuro-symbolic learning, and anyone using real logic or semantic losses for semi-supervised structured prediction. It is a good case study of a failure mode that likely affects many such systems. I would send it to a venue where a neuro-symbolic audience will review it, with an explicit request that the authors clarify whether the normalization weights are detached, and run a stronger baseline comparison. It deserves a serious referee.

Referee Report

3 major / 6 minor

Summary. The paper proposes Differentiable Reasoning (DR), a semi-supervised learning objective that augments a supervised cross-entropy loss with a differentiable term L_DR (Eqs. 6-12) measuring satisfaction of a first-order logic knowledge base over unlabeled domains, using product real logic with the Reichenbach material implication. The framework is applied to semantic image interpretation on PASCAL-Part, with 7 labeled and 2128 unlabeled images over 20 random splits. The authors report that adding the unlabeled DR loss improves type-classification precision over a supervised baseline, and that the gradient contributions from Modus Ponens and Modus Tollens are highly imbalanced, which they attribute to the Raven Paradox. They propose a normalized loss (Eq. 18) with hyperparameter mu to reweight MP and MT contributions, obtaining further improvements (best at mu=0.25). An appendix proves that Product Real Logic equals the marginal likelihood p(K|D,theta) under a disjointness assumption on ground atom occurrences.

Significance. If the conclusions hold, the paper makes two contributions: a simple, general recipe for injecting relational background knowledge into neural training via an unlabeled-data loss, and an analysis of implication behavior showing that contrapositive (MT) updates can dominate training. Strengths include the explicit loss and gradient equations, the transparent experimental protocol with 20 splits and confidence intervals, and the honest appendix that states the exactness conditions for the probabilistic interpretation. The empirical improvement over the supervised baseline is plausible and the paper identifies a real bias in material-implication losses. However, two technical points currently limit the significance: the exact objective optimized in the 'Normalized' variant is not fully specified (the theta-dependence of the normalization weights), and the claimed probabilistic interpretation of L_DR is not valid for the actual knowledge base used. The comparison only against a supervised baseline also leaves the semi-supervised claim under-supported.

major comments (3)

[Section 2.5, Eq. (18)] Equation (18) is written as a loss, but the surrounding text and Section 4 treat it as a direct reweighting of MP and MT gradient contributions. Because the coefficients dMP_phi(o) and dMT_phi(o) (Eqs. 15-16) and the normalization sums depend on the current parameters theta, differentiating Eq. (18) with respect to theta generates additional terms proportional to the partial derivative of dMP_phi(o)/sum dMP (and symmetrically for MT) with respect to theta. The paper neither derives these extra terms nor states that the weights are detached (stop-gradient). Without this information, the 'Normalized' rows in Table 1 cannot be attributed to the advertised control of Modus Ponens versus Modus Tollens reasoning; they may be the result of an unanalyzed objective. Please report the exact gradient used, including any detach operations, or provide a full derivation of the objective minimized.
[Section 2.2 and Appendix A] The transition from the probabilistic objective (Eqs. 3-5) to the practical loss (Eqs. 6-12) is not derived in the main text, and Appendix A shows equality only under the disjointness assumptions in Eqs. (22)-(23). These assumptions are violated by the knowledge base used in Section 3: partOf appears in multiple rules (e.g., forall x not partOf(x,x), forall x,y partOf(x,y) -> not partOf(y,x), and inside the antecedent of the chair/cushion rule), and formulas also share other predicates. Hence L_DR is not the marginal likelihood p(K|Ou,theta) for the actual experiments. The paper should explicitly acknowledge this, provide a justification for the approximation (for instance by benchmarking against the exact semantic loss on a small problem), or temper the claim that the loss 'computes the correct probability' (end of Section 2.2). This is load-bearing because the empirical results are obtained with the approximate loss.
[Section 3, Table 1] The paper introduces DR as a semi-supervised learning technique, yet Table 1 compares only against a fully supervised baseline. An improvement over supervised training shows that the unlabeled data carry useful signal, but it does not show that logical reasoning is the cause; a simple pseudo-labeling, entropy-minimization, or consistency regularizer on unlabeled images could yield similar gains. Adding at least one generic semi-supervised baseline (or a DR variant without the knowledge-base rules) would materially strengthen the central claim that background knowledge, rather than unlabeled data per se, drives the improvement.

minor comments (6)

[Section 2.4, Eq. (15)] The denominator contains an unbalanced parenthesis; it should be 1 - p_hat(phi|o,theta) * (1 - p_hat(psi|o,theta)), without the extra closing parenthesis after the equality.
[Appendix A, Eq. (22)] As written, union of A_phi over phi in K equals the empty set is impossible when formulas contain ground atoms; the intended condition is that the sets A_phi are pairwise disjoint (A_phi intersect A_psi = empty set for distinct phi, psi).
[Section 3] The experimental setup should specify how the test set for Table 1 is defined separately from the 'held-out validation set of 200 images' and the test set <Ot,Wt> used in Section 4, including the number of images in each.
[Section 3, Table 1] The statement 'All results are significant' should state the exact paired test used and whether any multiple-comparison correction was applied, given that seven values of mu are compared.
[Section 4, Figure 6] The claim that 'the unnormalized method seems to be best at reasoning correctly' is based on visual inspection of a single training run; report the variance over the 20 splits or provide quantitative support.
[Section 2.4 and Section 4.1] The imbalance between dMP and dMT is a mathematical consequence of the chosen Reichenbach implication and the predicted probabilities, as shown in Eqs. (15)-(16); the empirical contribution is the magnitude of the effect during training, and the narrative should distinguish these two aspects.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the core DR loss is an independently evaluated semi-supervised objective, and the flagged MP/MT and Appendix A issues are correctness concerns rather than derivation-by-construction.

full rationale

I walked the claimed derivation chain and found no step where a prediction is forced by its inputs by construction. The primary claim is empirical: adding the differentiable reasoning loss L_DR (Eqs. 6-12) to the supervised cross-entropy loss improves type and partOf prediction on PASCAL-PART, measured on a held-out validation set. The loss is defined directly from the knowledge base and the model probabilities, not from the reported accuracies, so the improvement is not an identity. The Appendix A proof that Product Real Logic equals p(K|D,theta) is explicitly conditional on disjointness assumptions (Eqs. 22-23) that the paper does not claim hold in general; that is a soundness/approximation limitation, not circularity. The MP/MT imbalance is indeed derived mathematically in Eqs. 15-16 from the chosen Reichenbach implication, but the paper does not use that derivation as evidence of an empirical prediction; it is an analysis of the method, and the experimental section then evaluates the proposed normalization on external data. Eq. 18 reweights MP and MT contributions using theta-dependent coefficients, and a careful reader may question whether the actual gradient matches the advertised normalized MP/MT updates, but this is an unverified correctness point about the objective, not a case of fitted parameters being renamed as predictions. The paper also does not rely on load-bearing self-citations: the cited Real Logic and LTN work is external, and the background knowledge is taken from Donadello et al. rather than from the authors' own prior results. Overall, the central contribution is self-contained and benchmarked externally, so the circularity score is 0.

Assumptions & free parameters 1 free parameters · 3 assumptions · 0 invented entities

The central claim rests on a hand-written knowledge base, a chosen fuzzy semantics, and a tuned hyperparameter mu. The method itself does not introduce new entities such as particles or forces.

free parameters (1)

mu = 0.25
In Equation 18, mu controls the relative weight of Modus Ponens versus Modus Tollens gradient contributions; 0.25 gave the best validation precision in Table 1, chosen from a small grid.

assumptions (3)

domain assumption Background knowledge rules (e.g., chair(x) and partOf(y,x) implies cushion(y) or armRest(y)) are true in the target domain.
The gain from the unsupervised loss depends on these hand-written rules from [7] being correct; no learning or verification of the rules is attempted.
domain assumption Product Real Logic with Reichenbach implication is an acceptable fuzzy semantics for the knowledge base.
The choice of product norm and implication in Eqs. 8-12 determines the gradient balance; the paper adopts it from [1] without comparing alternatives.
ad hoc to paper Ground atom occurrences across formulas and subformulas are disjoint (Eq. 22-23).
Appendix A requires this to show L_DR equals the true marginal probability, but the experiments reuse predicates across rules, so this axiom is violated.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Semi-Supervised Learning using Differentiable Reasoning." pith.science (2026). https://pith.science/paper/5CSGM457

@misc{pith2026190804700,
  author       = {Pith},
  title        = {Pith review of: Semi-Supervised Learning using Differentiable Reasoning},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5CSGM457}},
  note         = {Machine review of arXiv:1908.04700}
}

read the original abstract

We introduce Differentiable Reasoning (DR), a novel semi-supervised learning technique which uses relational background knowledge to benefit from unlabeled data. We apply it to the Semantic Image Interpretation (SII) task and show that background knowledge provides significant improvement. We find that there is a strong but interesting imbalance between the contributions of updates from Modus Ponens (MP) and its logical equivalent Modus Tollens (MT) to the learning process, suggesting that our approach is very sensitive to a phenomenon called the Raven Paradox. We propose a solution to overcome this situation.

Figures

Figures reproduced from arXiv: 1908.04700 by the authors.

**Figure 3.** The Bayesian network associated with grounding of the formula [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

**Figure 2.** We can deduce that b is a cushion if we are confident about the truth value of chair(a) and partOf(b, a) using the formula ∀x, y chair(x) ∧ partOf(y, x) → cushion(y). A convenient property of the SII task is that we consider just binary relations between objects appearing on the same pictures. The Herbrand base then contains O(|P| · |Di | 2 ) ground atoms, which is feasible as there are often not more than a few do… view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: Plots of d MP ϕ (o) (Equation 15) and d MT ϕ (o) (Equation 16). Note that the y axis is using a log scale. We analyze a naive way of dealing with this phenomenon. We normalize the contribution to the total gradient of MP and MT reasoning by replacing the loss function …

**Figure 5.** Figure 5: The average magnitude of Modus Ponens and Modus Tollens gradients. Next, we will analyze how accurate our approach is at reasoning by comparing its ’decisions’ to what should have been the correct ’decision’. We sample 2000 pairs of bounding boxes from the PASCAL-PART …

**Figure 6.** Figure 6: The left plot shows crMP and crMT and the right plot cuMP and cuMT for the Unnormalized method (denoted as Unnorm) and the Normalized methods with µ = 0.25 and µ = 1. time, the unnormalized method seems to be best at reasoning correctly for both MP and MT. Another inte…

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

33 extracted references · 30 canonical work pages

[1]

An introduction to many-valued and fuzzy logic: semantics, algebras, and derivation systems

Merrie Bergmann. An introduction to many-valued and fuzzy logic: semantics, algebras, and derivation systems. Cambridge University Press, 2008

work page 2008
[2]

Semi-supervised learning

Olivier Chapelle, Bernhard Scholkopf, and Alexander Zien. Semi-supervised learning. 2006

work page 2006
[3]

Detect what you can: Detecting and representing objects using holistic models and body parts

Xianjie Chen, Roozbeh Mottaghi, Xiaobai Liu, Sanja Fidler, Raquel Urtasun, and Alan Yuille. Detect what you can: Detecting and representing objects using holistic models and body parts. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 1971–1978, 2014

work page 1971
[4]

SDD: A new canonical representation of propositional knowledge bases

Adnan Darwiche. SDD: A new canonical representation of propositional knowledge bases. IJCAI International Joint Conference on Artiﬁcial Intelligence, pages 819–826, 2011

work page 2011
[5]

Lifted rule injection for relation embeddings

Thomas Demeester, Tim Rocktäschel, and Sebastian Riedel. Lifted rule injection for relation embeddings. In Proceedings of the 2016 Conference on Empirical Methods in Natural Language Processing, pages 1389–1399. Association for Computational Linguistics, 2016

work page 2016
[6]

Regularizing relation representa- tions by ﬁrst-order implications

Thomas Demeester, Tim Rocktäschel, and Sebastian Riedel. Regularizing relation representa- tions by ﬁrst-order implications. In AKBC2016, the Workshop on Automated Base Construction, pages 1–6, 2016

work page 2016
[7]

d’Avila Garcez

Ivan Donadello, Luciano Seraﬁni, and Artur S. d’Avila Garcez. Logic tensor networks for semantic image interpretation. In IJCAI, pages 1596–1602. ijcai.org, 2017

work page 2017
[8]

Indicative conditionals

Dorothy Edgington. Indicative conditionals. In Edward N. Zalta, editor, The Stanford Ency- clopedia of Philosophy. Metaphysics Research Lab, Stanford University, winter 2014 edition, 2014

work page 2014

Show all 33 references

[9]

Fast r-cnn

Ross Girshick. Fast r-cnn. International Conference on Computer Vision, pages 1440–1448, 2015

2015
[10]

Studies in the logic of conﬁrmation (i.)

Carl G Hempel. Studies in the logic of conﬁrmation (i.). Mind, 54(213):1–26, 1945

1945
[11]

Harnessing deep neural networks with logic rules

Zhiting Hu, Xuezhe Ma, Zhengzhong Liu, Eduard Hovy, and Eric Xing. Harnessing deep neural networks with logic rules. In Proceedings of the 54th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers), pages 2410–2420. Association for Computation...

2016
[12]

Fuzzy Implications, volume 231

Balasubramaniam Jayaram and Michal Baczynski. Fuzzy Implications, volume 231. 2008

2008
[13]

Image retrieval using scene graphs

Justin Johnson, Ranjay Krishna, Michael Stark, Li-Jia Li, David Shamma, Michael Bernstein, and Li Fei-Fei. Image retrieval using scene graphs. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 3668–3678, 2015

2015
[14]

Semi-supervised classiﬁcation with graph convolutional networks

Thomas N Kipf and Max Welling. Semi-supervised classiﬁcation with graph convolutional networks. 2016

2016
[15]

Visual genome: Connecting 14 DIFFERENTIABLE REASONING language and vision using crowdsourced dense image annotations

Ranjay Krishna, Yuke Zhu, Oliver Groth, Justin Johnson, Kenji Hata, Joshua Kravitz, Stephanie Chen, Yannis Kalantidis, Li-Jia Li, David A Shamma, et al. Visual genome: Connecting 14 DIFFERENTIABLE REASONING language and vision using crowdsourced dense image annotations. Intern...

2017
[16]

Pseudo-label: The simple and efﬁcient semi-supervised learning method for deep neural networks

Dong-Hyun Lee. Pseudo-label: The simple and efﬁcient semi-supervised learning method for deep neural networks. 2013

2013
[17]

Deepproblog: Neural probabilistic logic programming

Robin Manhaeve, Sebastijan Duman ˇci´c, Angelika Kimmig, Thomas Demeester, and Luc De Raedt. Deepproblog: Neural probabilistic logic programming. arXiv preprint arXiv:1805.10872, 2018

2018 arXiv
[18]

Kevin Murphy, Yair Weiss, and Michael I. Jordan. Loopy Belief Propagation for Approximate Inference: An Empirical Study. pages 467–476, 2013

2013
[19]

Cubuk, and Ian J

Avital Oliver, Augustus Odena, Colin Raffel, Ekin D. Cubuk, and Ian J. Goodfellow. Realistic evaluation of semi-supervised learning algorithms. 2018

2018
[20]

Combining representation learning with logic for language processing

Tim Rocktäschel. Combining representation learning with logic for language processing. CoRR, abs/1712.09687, 2017

2017 arXiv
[21]

End-to-end differentiable proving

Tim Rocktäschel and Sebastian Riedel. End-to-end differentiable proving. In Advances in Neural Information Processing Systems, pages 3791–3803, 2017

2017
[22]

Injecting logical background knowl- edge into embeddings for relation extraction

Tim Rocktäschel, Sameer Singh, and Sebastian Riedel. Injecting logical background knowl- edge into embeddings for relation extraction. In Proceedings of the 2015 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technolog...

2015
[23]

On the hardness of approximate reasoning

Dan Roth. On the hardness of approximate reasoning. Artiﬁcial Intelligence, 82(1-2):273–302, 1996

1996
[24]

Berg, and Li Fei-Fei

Olga Russakovsky, Jia Deng, Hao Su, Jonathan Krause, Sanjeev Satheesh, Sean Ma, Zhiheng Huang, Andrej Karpathy, Aditya Khosla, Michael Bernstein, Alexander C. Berg, and Li Fei-Fei. ImageNet Large Scale Visual Recognition Challenge. International Journal of Computer Vision (IJC...

2015
[25]

Modeling Relational Data with Graph Convolutional Networks

Michael Schlichtkrull, Thomas N Kipf, Peter Bloem, Rianne van den Berg, Ivan Titov, and Max Welling. Modeling Relational Data with Graph Convolutional Networks. In Aldo Gangemi, Roberto Navigli, Maria-Esther Vidal, Pascal Hitzler, Raphaël Troncy, Laura Hollink, Anna Tordai, an...

2018
[26]

Logic tensor networks: Deep learning and logical reasoning from data and knowledge

Luciano Seraﬁni and Artur d’Avila Garcez. Logic tensor networks: Deep learning and logical reasoning from data and knowledge. 2016

2016
[27]

Reasoning with neural tensor networks for knowledge base completion

Richard Socher, Danqi Chen, Christopher D Manning, and Andrew Ng. Reasoning with neural tensor networks for knowledge base completion. In Advances in neural information processing systems, pages 926–934, 2013

2013
[28]

Logic and structure

Dirk Van Dalen. Logic and structure. Springer, 2004

2004
[29]

Peter B.M. Vranas. Hempel’s raven paradox: A lacuna in the standard Bayesian solution.British Journal for the Philosophy of Science, 55(3):545–560, 2004

2004
[30]

The effect of class distribution on classiﬁer learning: an empirical study

Gm Weiss and Foster Provost. The effect of class distribution on classiﬁer learning: an empirical study. Rutgers University, (September 2001), 2001

2001
[31]

A semantic 15 VAN KRIEKEN , ACAR AND VAN HARMELEN loss function for deep learning with symbolic knowledge

Jingyi Xu, Zilu Zhang, Tal Friedman, Yitao Liang, and Guy Van den Broeck. A semantic 15 VAN KRIEKEN , ACAR AND VAN HARMELEN loss function for deep learning with symbolic knowledge. In Jennifer Dy and Andreas Krause, editors, Proceedings of the 35th International Conference on ...

2018
[32]

Cohen, and Ruslan Salakhutdinov

Zhilin Yang, William W. Cohen, and Ruslan Salakhutdinov. Revisiting semi-supervised learning with graph embeddings. In Proceedings of the 33rd International Conference on International Conference on Machine Learning - Volume 48, ICML’16, pages 40–48. JMLR.org, 2016

2016
[33]

Semi-supervised learning using gaussian ﬁelds and harmonic functions

Xiaojin Zhu, Zoubin Ghahramani, and John D Lafferty. Semi-supervised learning using gaussian ﬁelds and harmonic functions. In Proceedings of the 20th International conference on Machine learning (ICML-03), pages 912–919, 2003. Appendices A Conditional Optimality of Product Rea...

2003

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

An introduction to many-valued and fuzzy logic: semantics, algebras, and derivation systems

Merrie Bergmann. An introduction to many-valued and fuzzy logic: semantics, algebras, and derivation systems. Cambridge University Press, 2008

work page 2008

[2] [2]

Semi-supervised learning

Olivier Chapelle, Bernhard Scholkopf, and Alexander Zien. Semi-supervised learning. 2006

work page 2006

[3] [3]

Detect what you can: Detecting and representing objects using holistic models and body parts

Xianjie Chen, Roozbeh Mottaghi, Xiaobai Liu, Sanja Fidler, Raquel Urtasun, and Alan Yuille. Detect what you can: Detecting and representing objects using holistic models and body parts. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 1971–1978, 2014

work page 1971

[4] [4]

SDD: A new canonical representation of propositional knowledge bases

Adnan Darwiche. SDD: A new canonical representation of propositional knowledge bases. IJCAI International Joint Conference on Artiﬁcial Intelligence, pages 819–826, 2011

work page 2011

[5] [5]

Lifted rule injection for relation embeddings

Thomas Demeester, Tim Rocktäschel, and Sebastian Riedel. Lifted rule injection for relation embeddings. In Proceedings of the 2016 Conference on Empirical Methods in Natural Language Processing, pages 1389–1399. Association for Computational Linguistics, 2016

work page 2016

[6] [6]

Regularizing relation representa- tions by ﬁrst-order implications

Thomas Demeester, Tim Rocktäschel, and Sebastian Riedel. Regularizing relation representa- tions by ﬁrst-order implications. In AKBC2016, the Workshop on Automated Base Construction, pages 1–6, 2016

work page 2016

[7] [7]

d’Avila Garcez

Ivan Donadello, Luciano Seraﬁni, and Artur S. d’Avila Garcez. Logic tensor networks for semantic image interpretation. In IJCAI, pages 1596–1602. ijcai.org, 2017

work page 2017

[8] [8]

Indicative conditionals

Dorothy Edgington. Indicative conditionals. In Edward N. Zalta, editor, The Stanford Ency- clopedia of Philosophy. Metaphysics Research Lab, Stanford University, winter 2014 edition, 2014

work page 2014