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REVIEW 3 major objections 5 minor 53 references

Logic could be learned from images

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read The paper claims that logical operations over numbers can be learned directly from images of those numbers, with no reasoning pattern supplied in advance, and that a divide-and-conquer network using extra label images lifts multiplication…

desk verdict The datasets and negative results on multiplication are the real contribution; the DCM's 84.5% success relies on human-supplied carry labels, so the 'no preset reasoning patterns' claim is overreach. read the letter →

arxiv 1908.01931 v2 pith:OYYS7YN2 submitted 2019-08-06 cs.CV

Qian Guo , Yuhua Qian , Xinyan Liang , Yanhong She , Deyu Li , Jiye Liang This is my paper · ORCID
classification cs.CV MSC 00-0199-00
keywords logicreasoningdataconceptLitasklogicalpatternnetworkdivideandconquermodelarithmeticfromimagesbitwiseoperationsneuralnetworks
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

This paper introduces a task it calls learning logic from images (LiLi): a network sees two images, each containing an integer rendered as pixels, and must produce an output image containing the result of an operation it was never told about. The authors report that standard networks reach about 100 percent accuracy on bitwise AND, OR, and XOR, and about 99 percent on addition and subtraction once the training set has 150,000 samples. Multiplication stays near chance for every standard model, so the paper proposes a divide-and-conquer model that splits the product into carry and non-carry parts using extra label images; that model reaches 84.5 percent test accuracy. The paper's position is that logical relations over numbers can be mined from pixel data alone, and that decomposition is a workable remedy when end-to-end learning fails.

What carries the argument

The load-bearing objects are the data concept logic, defined as a triple $(I,R,O)$ in which $R$ is an unknown relation from input sequences to outputs; the LiLi task, which instantiates $I$ as pairs of images and $O$ as a result image; the Logical Pattern Network (LPN), a neural network trained by mean-square error that is supposed to absorb $R$ from data; and the divide-and-conquer model (DCM), an MLP-based pipeline that splits multiplication into carry, non-carry, and synthetic subtasks. The DCM carries the argument for the hard case by showing that a hand-chosen decomposition with intermediate label images reduces the uncertainty of each predicted digit and lifts multiplication accuracy from below one percent to 84.5 percent.

What would settle it

Regrade the same test predictions by exact pixel comparison with ground-truth images, and also measure the OCR software's accuracy on clean ground-truth output images; if exact-pixel accuracy differs substantially from the OCR-based percentages, or if the OCR reader misreads clean ground-truth digits, then the reported results are artifacts of the recognition step rather than of logic learning.

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Extended reading notes

Core claim

The central claim is that a logic relation can be treated as a high-dimensional mapping $R:[-1,1]^{2K}\to\{0,1\}^K$ between pixel values and learned by a neural network called the Logical Pattern Network, with no reasoning pattern specified in advance. On six synthetic image datasets, the paper reports that this succeeds almost perfectly for bitwise operations, succeeds for addition and subtraction once training data reaches 150,000 samples, and fails for multiplication with every tested standard architecture. The paper's explanation is that each output digit of a product depends on all preceding digit positions, so the uncertainty per digit is far larger than for addition or subtraction. The divide-and-conquer model attacks this by training separate MLP branches to predict the carry part and the non-carry part of the product, then a synthetic branch to combine them, reducing per-digit uncertainty from roughly 900 possibilities to 90 and yielding 84.5 percent test accuracy as the paper measures it.

Load-bearing premise

Every reported accuracy depends on the OCR software's ability to read digits from the predicted images; the paper itself notes that at least some correctly predicted digits are not recognized by the OCR, so if OCR errors are non-negligible, both the failures of standard models and the 84.5 percent success of the DCM could be misstated.

Editorial extensions

If this is right

  • If the results hold, bitwise logic operations can be acquired from raw pixels with no symbolic supervision: all tested models reach about 100 percent accuracy even with only 10,000 training samples.
  • Addition and subtraction, which require tracking a carry or borrow between adjacent digit positions, become learnable by CNN-MLP, Autoencoder, and ResNet models once the training set grows to 150,000 samples, supporting the paper's data-scaling strategy for hard logic.
  • Multiplication marks a sharp boundary for end-to-end neural networks: every standard model stays below 0.35 percent accuracy even at 150,000 samples, so the LiLi task isolates a genuinely hard reasoning step.
  • The DCM result implies that decomposing a complex visual logic task into subtasks, with extra images as labels, turns an unsolved problem into one that is mostly solvable: 84.5 percent test accuracy.
  • The same decomposition idea can be carried to other complex logic tasks, such as computing decimal bit operations by first converting to binary, as the paper's conclusion suggests.

Reading between the lines

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

  • If the central claim is right, the LiLi task could serve as a cleaner probe of machine reasoning than question-answering style benchmarks, because the only way to answer is to actually compute the relation from the two input images.
  • The OCR-based accuracy measure probably makes the reported numbers conservative: the paper states that some correct predicted digits are not recognized by the OCR software, so exact pixel-level grading might raise the reported accuracies.
  • The DCM's carry/non-carry decomposition is a human-specified reasoning pattern, which sits in tension with the paper's 'no reasoning patterns beforehand' slogan; the pure data-driven claim is really established only for the base tasks, while the hard-task success comes from injecting domain knowledge.
  • A testable extension is whether a sufficiently scaled end-to-end model, or an architecture with explicit positional alignment, can eventually match the DCM without decomposition; the paper's data-scaling results for addition suggest the gap may shrink with data.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. This paper introduces a 'data concept logic' (DCL) and a 'learning logic from images' (LiLi) task in which a model receives two rendered images of numbers and must output the rendered image of the result of an unknown binary operation. Six datasets are constructed: bitwise AND/OR/XOR, addition, subtraction, and multiplication. The authors train CNN-LSTM, MLP, CNN-MLP, autoencoder, and ResNet variants; with 150,000 training samples these models reach near-100% accuracy on the bitwise tasks and roughly 99% on addition/subtraction, while all remain below 0.4% on multiplication. To address multiplication, the paper proposes a divide-and-conquer model (DCM) that is trained with additional human-defined carry and non-carry images and reports 84.5% accuracy. The paper concludes that logic patterns can be learned from images without preset reasoning patterns, while also stating that the DCM combines domain expert knowledge with data-driven models.

Significance. The LiLi benchmark is a useful diagnostic: it has a clear single-answer evaluation, a controlled difficulty gradient, and the experiments show that ordinary architectures can learn arithmetic from raw pixel pairs surprisingly well. The consistently high accuracies on bitwise and addition/subtraction tasks across several architectures are credible and constitute the paper's main empirical contribution. The complete failure of all tested networks on multiplication, and the substantial improvement from a hand-decomposed training target, are informative for the community even though the decomposition is provided by the authors. The paper does not provide code or data links, and no formal or machine-checked proofs are included, so reproducibility rests on the textual dataset description and hyperparameter tables.

major comments (3)
  1. [Section 3.2 / Section 5] The claim in Section 3.2, after Eq. (9), that 'the LPN merely needs to be provided some training data to automatically learn the logical patterns... without providing any reasoning patterns beforehand' is not supported for the hard task. In Section 5 and Fig. 13, the DCM is trained on a regenerated multiplication dataset in which the carry image C and non-carry image D are human-defined intermediate targets; this is exactly a preset decomposition of the multiplication reasoning pattern. The paper's own conclusion states that the DCM works by 'combing the domain expert knowledge with data-driven model.' The central claim should be revised to apply only to the easy and intermediate tasks, or the DCM should be presented as a hybrid method rather than evidence for unsupervised logic discovery.
  2. [Section 4.2] Accuracy is measured by Tesseract OCR on predicted images, and the authors note that 'Some are right but OCR can not recognize them.' Because no OCR accuracy on the ground-truth images is reported, the percentages in Tables 2-5 are uncalibrated with respect to actual image-level correctness. The 100% easy-task results are robust to this concern, but the DCM's 84.5% and the multiplication failure rates could shift if OCR errors are not symmetric; an OCR baseline on ground-truth outputs or a pixel/character-level metric should be provided.
  3. [Table 5 / Fig. 14] The relationship between the reported 84.5% DCM accuracy and the per-subtask accuracies (carry 86.25%, non-carry 98.38%, synthetic 84.46%) is unclear. The testing procedure passes predicted carry/non-carry images c' and d' into the synthetic subtask, so errors in the first two subtasks propagate; if the 84.5% is instead the synthetic subtask accuracy evaluated with ground-truth c and d, then the end-to-end accuracy on the original a,b inputs is not reported. The authors should state exactly which pipeline is evaluated and report end-to-end accuracy.
minor comments (5)
  1. [Section 2.2] Formulas 5-7 contain repeated typos, including 'squence' for 'sequence' and 'T he' for 'The.'
  2. [Section 4.2] The text contains 'CNN-NLP' where 'CNN-MLP' is clearly intended, and 'verity' should be 'verify.'
  3. [Section 5] Eq. (14) is not operationalized: no definition or proxy for 'difficulty' is given, so the inequality H > f(h1,...,hk) reads as a heuristic rather than a testable criterion.
  4. [Section 4.1] Table 1 does not include learning rates, optimizers, or epoch counts for all models, despite the text saying that the hyper-parameter settings are shown in Table 1; adding these details would improve reproducibility.
  5. [Section 1] The relationship between the LiLi task and the Fashion-Logic dataset of reference [23] should be stated explicitly, since both are described as data-driven visual logic tasks.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation; the DCM's human-specified decomposition weakens the headline claim but is not a circular prediction.

full rationale

The core LiLi experiments (bitwise, addition, subtraction) train standard networks on (x1,x2)->y image pairs and report genuine generalization on held-out images; these results are self-contained empirical findings, not derived from the claim being tested. The DCM section is the only place where a circularity concern could arise, but it does not reduce by construction: although the paper defines the multiplication target as E = C + D and trains the synthetic subtask on (c,d)->e, at test time the DCM receives only the original images a and b and must predict c', d' and then e; the reported 84.5% is an end-to-end generalization result, not an identity or a refit of the labels. The real problem is scope: Section 3.2 claims the LPN learns 'without providing any reasoning patterns beforehand,' yet Section 5 regenerates the multiplication set with carry/non-carry labels and the conclusion concedes the DCM works by 'combining the domain expert knowledge with data-driven model.' That makes the 'logic learned from images' claim unsupported for the difficult multiplication task, but it is a claim-support gap and a correctness risk rather than a circular step. Likewise, Section 4.2's note that 'Some are right but OCR can not recognize them' makes reported accuracies lower bounds, a measurement caveat rather than circularity. Reference [23] is a same-author citation, but it is only used as related work and is not load-bearing. No circular step is present; the paper is transparent about the DCM's extra label information.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central empirical claims rest on standard learning assumptions plus two paper-specific choices: the unstated rendering pipeline and the carry/non-carry decomposition. No new physical or mathematical entities are introduced. The carry/non-carry labels are the main additional information that makes the DCM work.

assumptions (4)
  • standard math Universal approximation theorem: a sufficiently large neural network can represent the logical relation R.
    Invoked in Section 2.1 to justify using a neural network as an LPN; standard but assumes sufficient capacity, data, and optimization success.
  • domain assumption The rendered digit images present numbers in a fixed visual layout that CNNs and Tesseract OCR can reliably extract.
    The dataset description in Section 3.1 gives image sizes and number ranges but not the font, rendering function, or preprocessing; this assumption is necessary for the entire evaluation pipeline.
  • ad hoc to paper Multiplication can be decomposed into carry and non-carry components C and D such that the product E equals C plus D.
    Introduced in Section 5 for the DCM. This human-designed decomposition is the key auxiliary supervision that makes the hard multiplication task tractable, and it contradicts the stated goal of learning without preset reasoning patterns.
  • domain assumption OCR errors are rare enough that reported accuracies approximately equal model accuracies.
    Section 4.2 uses Tesseract OCR to judge correctness; the paper admits OCR failures, so this assumption is only partially satisfied and no OCR baseline is provided.

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Pith. "Pith review of Logic could be learned from images." pith.science (2026). https://pith.science/paper/OYYS7YN2

@misc{pith2026190801931,
  author       = {Pith},
  title        = {Pith review of: Logic could be learned from images},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OYYS7YN2}},
  note         = {Machine review of arXiv:1908.01931}
}
read the original abstract

Logic reasoning is a significant ability of human intelligence and also an important task in artificial intelligence. The existing logic reasoning methods, quite often, need to design some reasoning patterns beforehand. This has led to an interesting question: can logic reasoning patterns be directly learned from given data? The problem is termed as a data concept logic. In this study, a learning logic task from images, called a LiLi task, first is proposed. This task is to learn and reason the logic relation from images, without presetting any reasoning patterns. As a preliminary exploration, we design six LiLi data sets (Bitwise And, Bitwise Or, Bitwise Xor, Addition, Subtraction and Multiplication), in which each image is embedded with a n-digit number. It is worth noting that a learning model beforehand does not know the meaning of the n-digit numbers embedded in images and the relation between the input images and the output image. In order to tackle the task, in this work we use many typical neural network models and produce fruitful results. However, these models have the poor performances on the difficult logic task. For furthermore addressing this task, a novel network framework called a divide and conquer model by adding some label information is designed, achieving a high testing accuracy.

Figures

Figures reproduced from arXiv: 1908.01931 by the authors.

Figure 1
Figure 1. The differences among these popular computer vision tasks (1) Object recognition (sometimes object classification) is to classify individual objects. (2) Object detection is to classify individual objects and localize each using a bounding box. (3) Semantic segmentation is to classify each pixel into a fixed set of categories without differentiating object instances. (4) Image captioning is to describe the content o… view at source ↗
Figure 2
Figure 2. The workflow of a DCL task. hidden in the LPN, and mining R from data can be regarded as the iterative optimization process of parameter W of LPN. At each iteration, the value of W changes in the direction that the loss L becomes smaller. When the loss is small enough, the iteration stops and R is obtained. The workflow of a DCL task is illustrated in [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. The comparative analysis between the DCL and the LOH. [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (13 more)
Figure 4
Figure 4. Figure 4: The samples of six LiLi data sets. shown in [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]
Figure 5
Figure 5. Figure 5: The workflow of a LiLi task. 3.3. Inference form of a LiLi task Based on the inference form of the DCL 2.2, a LiLi task can be written as the following inference form based on the IF THEN rule. Antecedent 1 : I f two input images are x1 1 and x2 1 then the output image…
Figure 6
Figure 6. Figure 6: The validation losses of Bitwise And, Bitwise Or, Bitwise Xor, Addition, Subtraction and Multiplication on [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]
Figure 7
Figure 7. Figure 7: The validation losses of Bitwise And, Bitwise Or, Bitwise Xor, Addition, Subtraction and Multiplication on [PITH_FULL_IMAGE:figures/full_fig_p014_7.png]
Figure 8
Figure 8. Figure 8: The architectures of CNN-MLP and CNN2-MLP. [PITH_FULL_IMAGE:figures/full_fig_p016_8.png]
Figure 9
Figure 9. Figure 9: The validation losses of CNN-MLP and CNN2-MLP on Bitwise And, Bitwise Or, Bitwise Xor, Addition, [PITH_FULL_IMAGE:figures/full_fig_p017_9.png]
Figure 10
Figure 10. Figure 10: The test visual effects of Addition on 10,000 training data set and 150,000 training data set. (a) 10,000 training data set (b) 150,000 training data set [PITH_FULL_IMAGE:figures/full_fig_p018_10.png]
Figure 11
Figure 11. Figure 11: The test visual effects of Subtraction on 10,000 training data set and 150,000 training data set. predicted image e’ at the output of the synthetic subtask. Specifically, the inputs are firstly passed through the carry subtask and non-carry subtask to get a carry pred…
Figure 15
Figure 15. Figure 15: In Fig. 15(a), both DCM and MLP get correct predicted images. In Fig. 15(b), the DCM gets the correct [PITH_FULL_IMAGE:figures/full_fig_p018_15.png]
Figure 12
Figure 12. Figure 12: The test visual effects of Multiplication on 10,000 training data set and 150,000 training data set [PITH_FULL_IMAGE:figures/full_fig_p019_12.png]
Figure 13
Figure 13. Figure 13: The procedure of multiplication. subtask, non-carry subtask and synthetic subtask, each subtask only learns one aspect of the task. This helps to reduce uncertainty of each predicted digit embedded in the image e’. In order to explain the reason for the effectiveness …
Figure 14
Figure 14. Figure 14: Training and testing procedure. (a) Both get correct predicted images. (b) The DCM gets correct predicted images, but MLP does not. (c) Both get wrong predicted images [PITH_FULL_IMAGE:figures/full_fig_p020_14.png]
Figure 15
Figure 15. Figure 15: The visual effects of multiplication task on 150,000 training set. to choose one of 0∼9. Hence, there are 90 possibilities (C 1 9C 1 10) for “d 2 3 ”. The DCM largely reduces the number of possible values from 900 to 90. Therefore, the DCM confirms more digits than th…

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Pith tools

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