Differentiable Rule Induction from Raw Sequence Inputs
Authors: Kun Gao
Katsumi Inoue
Yongzhi Cao
Hanpin Wang
Yang Feng
Venue: ICLR 2025 Poster (2025)
Keywords: Neuro-Symbolic Methods
Interpretability
Inductive Logic Programming
Primary area: neurosymbolic & hybrid AI systems (physics-informed, logic & formal reasoning, etc.)

Abstract: Rule learning-based models are widely used in highly interpretable scenarios due to their transparent structures. Inductive logic programming (ILP), a form of machine learning, induces rules from facts while maintaining interpretability. Differentiable ILP models enhance this process by leveraging neural networks to improve robustness and scalability. However, most differentiable ILP methods rely on symbolic datasets, facing challenges when learning directly from raw data. Specifically, they struggle with explicit label leakage: The inability to map continuous inputs to symbolic variables without explicit supervision of input feature labels. In this work, we address this issue by integrating a self-supervised differentiable clustering model with a novel differentiable ILP model, enabling rule learning from raw data without explicit label leakage. The learned rules effectively describe raw data through its features. We demonstrate that our method intuitively and precisely learns generalized rules from time series and image data.

Published as a conference paper at ICLR 2025
DIFFERENTIABLE RULE INDUCTION FROM RAW SEQUENCE INPUTS
Kun Gao', Katsumi Inoue?, Yongzhi Cao', Hanpin Wang', Feng Yang Institute of High Performance Computing. Agency for Science, Technology and Research National Institute of Informatics
KKey Laboratory of High Confidence Software Technologies, School of Computer Science Peking University
gaok e sihpc... arstar edu. . sg, inoue@nii ac. jp,
[caoyz, whpxhy) ) @pku. edu. en, yangfe ihpc. a sstrrrrar.eee edu.. sg
ABSTRACT
Rule learning- based models are widely used in highly interpretable scenarios due to their transparent structures. Inductive logic programming (ILP), form of ma- chine learning, induces rules from facts while maintaining interpretability. Dif- ferentiable ILP models enhance this process by leveraging neural networks to im- prove robustness and scalability. However, most differentiable ILP methods rely symbolic datasets, facing challenges when learning directly from raw data. Specifically, they struggle with explicit label leakage: The inability to map con- tinuous inputs symbolic variables without explicit supervision of input feature labels. In this work, we address this issue by integrating a self-supervised dif- ferentiable clustering model with novel differentiable ILP model, enabling rule learning from raw data without explicit label leakage. The learned rules effec- tively describe raw data through its features. We demonstrate that our method intuitively and precisely learns generalized rules from time series and image data.
INTRODUCTION
The deep learning models have obtained impressive performances on tabular classification, time se- ries forecasting, image recognition, etc. While in highly trustworthy scenarios such as health care, finance, and policy-making process (Doshi Velez & Kim 2017), lacking explanations for decision- making prevents the applications of these complex deep learning models. However, the rule-learning models have interpretability intrinsically to explain the classification process. Inductive logic pro- gramming (ILP) is form of logic-based machine learning that aims to learn logic programs for gen- eralization and interpretability from training examples and background knowledge (Cropper et al.) 2022). Traditional ILP methods design deterministic algorithms to induce rules from symbolic data to more generalized formal symbolic first-order languages (Quinlan 1990 Blockeel & De Raedi, 1998]. However, these symbolic II.P methods face robustness and scalability problems when learn- ing from large-scale and ambiguous datasets (Evans et al., 2021 Hocquette et al. 2024). With the sake of robustness of neural networks, the neuro-symbolic ILP by combining neural networks and
ILP methods can learn from noisy data (Evans & Grefenstette 2018 Manhaeve et al. 2018 Gao
et al. 2022a) and can be applied to large-scale datasets (Yang et al.] 2017 Gao et al.] 2024 Phua & Inoue 2024). However, existing neuro-symbolit ILP methods are mainly learned from discrete symbolic data or fuzzy symbolic data that the likelihoods are generated from a pre-trained neural network module (Evans al. 2021, [Shindo et al., 2023). Learning logic programs from raw data is prevented because of the explicit label leakage problem, which is common in neuro-symbolic research (Topan et al. (22II): The leakage happens by introducing labels of ground objects for in- ducing rules (Evans& & Grefenstette 2018 (hindo et al. 2023). In fact, generating rules to describe objects in raw data without label information is necessary, especially when some objects are easily overlooked or lack labels yet are important for describing the data.
"Corresponding author.

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In this study, we introduce Neural Rule Learner (NeurRL), framework designed to learn logic programs directly from raw sequences, such as time series and flattened image data. Unlike prior approaches prone to explicit label leakage, where input feature labels are extracted using pre- pre-trained supervised neural networks and then processed with differentiable IIP methods to induce rules trained with input labels, generate symbolic labels. Instead, we leverage a pre-trained clustering model in an unsupervised manner to discretize data into distinct features. Subsequently, differentiable clustering module and a differentiable rule-learning module are jointly trained under supervision using raw input labels. This enables the discovery of rules that describe input classes based on feature distributions within the inputs. As result, the model achieves efficient training in a fully differentiable pipeline while avoiding the explicit label leakage issue. The contributions of this study include: (1) We formally define the IIP learning task from raw inputs based on the interpretation transition setting of IIP. (2) We design a fully differentiable framework for learning symbolic rules from raw sequences, in- cluding novel interpretable rule-learning module with multiple dense layers. (3) We validate the model's ss effectiveness and interpretability on various time series and image datasets.
method bypasses the need for supervised pre-trained networks to
RELATED WORK
Inductive logic programming (ILP), introduced by Muggleton & Feng (1990); Muggleton & De
(19944, learns logic programs from symbolic positive and negative examples with background knowledge, Inoue et al. (2014) proposed learning from interpretation transitions, and Phua & Inoue (2021) applied ILP to Boolean networks. Mannaave et al. (201 8); Evans & Grefenstette (2018) adapted neural networks for differentiable and robust ILP, while Gao et al. (2022b) introduced a neural beworr-bbased ILP model for learning from interpretation transitions. This model was later extended for scalable learning from knowledge graphs (Gao et al.] [022a] 2024). Similarly, Liu eet al. (2024) proposed deep neural network for inducing mathematical functions. In our work, present a novel neural network-based model for learning logic programs from raw numeric inputs.
In the raw input domain, Evans & Grefenstette (018) proposed öILP to learn rules from symbolic relational data, using a pre-trained neural network to map raw input data to symbolic labels. Unlike JILP, which enforces a strong language bias by predefining logic templates and limiting the num- ber of atoms, NeurRL uses only predicate types language bias. Similarly, [Evans et al.] (202T) used pre-trained networks to map sensory data to disjunctive sequences, followed by binary neural networks to learn rules. [Shindo et al. (2023) introduced alL.,, leveraging object recognition mod- els to convert images into symbolic atoms and employing top-k searches to pre-generate clauses, with neural networks optimizing clause weights. Our approach avoids pre-trained large-scale neural networks for mapping raw inputs to symbolic representations. Instead, we propose a fully differen- tiable framework to learn rules from raw sequences. Additionally, unlike the memory-intensive rule candidate generation required by OILP and alLP, NeurRL eliminates this step, enhancing scalability.
Adapting autoencoder and clustering methods in the neuro- -uuuo-yymmolic domain shows promise. San- sone & Manhaeve (2023) applics conventional clustering on input embeddings for deductive logic programming tasks. Misino et al. (2022) and Zhha al. (2022) autoencoders and embeddings to calculate probabilities for predefined symbols to complete deductive logic programming and pro- gram synthesis tasks. In our approach, we use an autoencoder to learn representations for sub-areas of raw inputs, followed by a differentiable clustering method to assign ground atoms to similar pat- terns. The differentiable rule-learning module then searches for rule embeddings with these atoms in a bottom-up up manner (Cropper& & Dumancic 2022). Similarly, DIFFNAPS (Walter et al.) 2024) also uses an autoencoder to build hidden features and explain raw inputs. Additionally, BotCL (Wang et al. 2023) uses attention-baser features to explain the ground truth class. However, the logical connections between the features used in DIFFNAPS and BotCL to describe the ground truth class unclear. In contrast, rule-based explainable models like NeurRL use feature conjunctions to describe the ground truth class.
Azzolin et al. (2023) use a post-hoc rule-based explainable model to globally explain raw inputs from local explanations. In contrast, our model directly learns rules from raw inputs, and NeurRL's perfor- mance is unaffected by other explainable models. Das et al. (1998) used clustering to split sequence data into subsequences and symbolize them for rule discovery. Our model combines clustering and rule-learning in fully differentiable framework to discover rules from sequence and image data,
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with the rule-learning module providing gradient information to prevent cluster collapse (Sansone 2023). where very different subsequences are assigned to the same clusters. He et al. (2018) viewed similar subsequences, called motifs, as potential rule bodies. Unlike their approach, we do not limit the number of body atoms in a rule. Wang et al. (019)) introduced SSSI, which uses shapelets as body atoms to learn rules and maximize information gain. Our model extends this by using raw data subsequences as rule body atoms and evaluating rule quality with precision and recall, a feature absent in SSSL.
PRELIMINARIES
3.1 LOGIC PROGRAMS AND INDUCTIVE LOGIC PROGRAMMING
A first-order language L = CD,F. (Loyd 1984) consists of predicates D, function symbols F. constants C, and variables V. A term is constant, variable, or expression f(i,,..., (.) with I as an n-ary function symbol. An atom is formula p(!,,..., (n), where is an n-ary predicate symbol. A ground atom (fact) has no variables. A literal is an atom or its negation: positive literals are atoms, and negative literals are their negations. A clause is a finite disjunction of literals, and a rule (definite clause) a clause with one positive literal, e.g., OA V Vaan V-a2 V... V.raaV,..V-.,. VV rule is written as: ON < 011, ..,,.... On, where or, is the head (head(r)), and {a1, (01,02,-- 02,,.... O.n) is the body (body(r)), with each atom in the body called a body atom. A logic program is a set of rules. In first-order logic, a substitution is a finite set (v1/t1, /2///////2.....///////// u2/t2,.... Un /t, where each v, is variable, t is a term distinct from v,, and 01, V2,,... Un are distinct (Lloyd 1984). A ground substitution has all as ground terms. The ground instances of all rules in P are denoted ground(P).
The Herbrand base Bp of a logic program P is the set of all ground atoms with predicate symbols from P. and an interpretation is a subset of Bp containing the true ground atoms (Lloyd 1984). Given T, the immediate consequence operator Tp: 2Br + 2Br for a definite logic program P is defined (T) = (head(r) E ground (P). body(r) S T) (Apt et al. 1988). A logic program P with mn rules sharing the same head atom aj is called a same-head logic program. A same-head logic program with n possible body atoms can be represented as a matrix Mp E [0, 1] Each element 0aj in Mp is defined as follows (Gao et al. 2022a)) If the k-th rule is ag + Ojz Aapp then akj. 4. where 4, E (0. 1) and -==1 [ If the k-th rule Og + Ctj, then 0zj 1. Otherwise, 0ıj = Each row of Mp represents a rule in P. and each column represents body atom. An interpretation vector V, E {0. 1}" corresponds to an interpretation I, where [i = if the i-th ground atom is true in and v [i Given logic program with rules and atoms, along with an interpretation vector vi, the function Dp : {0, 11 + {0, 1)m² (Gao et al. 2024) calculate the Boolean value for the head atom of each rule in P. It is defined as:
1 < p. 1 < j4 S n, 1 S k S m).
otherwise.
Dp(v)) ((M)1))
where the function is a threshold function: 0(x) = 1 if x 2 1, otherwise ((x) = 0. For a same-head logic program, the Boolean value of the head atom v(a,) is computed v(ax) = V (vv [i]. Additionally, Gao et al. (2024) replaces 0(x) with a differentiable threshold func- tion and use fuzzy disjunction [
II x [ii to caleulate the Boolean value of the head atom in a same-head head logic program with neural networks.
x
Inductive logic programming (ILP) aims to induce logic programs from training examples and back- ground knowledge (Muggleton et al. 2012). ILP learning settings include learning from entail- ments (Evans& & Grefenstette 2018). interpretations (De Raedt & Dehaspe 1997), proofs (Passerini et al., 2006). and interpretation transitions (Inoue et al.] [0144). In this paper, we focus on learning from interpretation transitions: Given set E C 2 x BBP of interpretation pairs (I, J), the goal is to learn logic program P such that (T(T)= = J for all JI E.
3.2 SEQUENCE DATA AND DIFFERENTIABLE CLUSTERING METHOD
A raw input consists of an instance (x, y). where x E Th XT2xxxxxxxx x TT represents realevalued obser- vations with d variables, T, indicates the length for i-th variable. and y {0, 10. 1.......22 1} is the class label, with u classes. A sequence input is a type of raw input where x E R is an ordered se- quence of real-valued observations (Wang al. 2019). A subsequence s; of length from sequence
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(Z1, zT) is contiguous sequence (Ia,.... 11 Das et al. 1998. All possible
subsequence with length include
S4-141-
Clustering can be used to discover new categories (Rokach & Maimon 2005). In the paper. we adapt the differentiable k-means method (Fard et al. 2020) to group raw data x E X. First, an autoencoder A generates embeddings h for x, where " represents the parameters. Then, we set K clusters to discretize all raw data x. The representation of the k-th cluster is r4 E RR, with p as the dimension, and R = (ri,...,, r&) the set of all cluster representations. For any vector y IP, the function cy R) returns the closest representation based on a fully differentiable distance function f. Then, the differentiable k-means problems are defined as follows:
K Σ (3,,
min
5(x + A R,Y XEX
where the parameter À regulates the trade-off between seeking good representations for x and the representations that are useful for clustering. The weight function G is differentiable minimum function proposed by Jang et al. (2017):
(b, a:
Σ =k=1 e (x))))))))))))))))
where a E [0, (.o). A larger value of a causes the approximate minimum function to behave more like a discrete minimum function. Conversely, smaller results in smoother training proces.
METHODS
4.1 PROBLEM STATEMENT AND FORMALIZATION
In this subsection, we define the learning problem from raw inputs using the interpretation transition setting of ILP and describe the language bias of rules for sequence data. We apply a neuro-symbolic
description method from Marconato et al. (202) to describe raw input: (i) assume the label y depends entirely on the state of K symbolic concepts B = (a1, a2...., ax which capture high- level aspects of x, (ii) the concepts B depend intricately on the sub-symbolic input x' and are best extracted using deep learning, (iii) the relationship between and can be specified by prior knowledge B, requiring reasoning during the forward computing of deep learning models.
In this paper, we treat each subset of raw input as a constant and each concept as ground atom. For example, in sequence data X, if concept increases from the point 0 to 10, the corresponding ground atom is increeseee a continuous sequence x into discrete symbolic concepts, so that the set of all symbolic concepts B across all raw inputs X holds B = (X).. For binary class raw inputs, we use target atom he to represents the target class, where h, being true means the instance belongs to the positive class. Then, head symbolization function L,, maps binary input x to a set of atoms: if the class of x is the target class f, then [x (x) = {h+]: otherwise, (x) = U. Hence, a logic program P describing binary class raw inputs consists of rules with the same head atom [ We define the Herbrand base Bp of a logic program as the set all symbolic concepts B in all raw inputs and the head atom /tp For input x, [x) represents set of symbolic concepts I Br, which can be considered an interpretation. The task of inducing a logic program P from raw inputs based on learning from interpretation transition is defined as follows:
10)). We define the body symbolization function La to convert
Definition 1 (Learning from Binary Raw Input) Given set of raw binary label inputs X, learn a same-head logic program P. where (T([(xx)) = Ta(x)) holds for all raw inputs x E X.
In the paper, aim learn rules describe the target class with the body consisting of multiple sequence features. Each feature corresponds to subsequence of the sequence data. Besides, each feature includes the pattern information and region information of the subsequence. Based on the
We set a to 1000 the whole experiments. We can transfer multiple class raw inputs the hinary class by setting the class of interest as the target class and the other classes the negative class.
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Encoder
Decocer
All aucoory
Dense
bayer
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Dense layar
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Reconstuctec subquences
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(a) Learning pipeline.
(b) Deep rule-learning module.
Figure I: The learning pipeline of NeurRL and the rule-learning module.
pattern information, we further infer the mean value and tendency information of the subsequence. Using the region predicates, we can apply NeurRL to the dataset, where the temporal relationships between different patterns play crucial role in distinguishing positive from negative examples. Specifically, we use the following rules to describe the sequence data with the target class:
hr - pattern: (X,,), regions, (Xi)), pattern
(X5,g) (4)
where the predicate pattern, indicates the i-th pattern in all finite patterns within all sequence data, the predicate region, indicates the k-th region in all regions in a sequence, and the variable X, can be substituted by a subsequence of the sequence data. For example, pattern, (x(0= : 5)) and regi Beggion(XX) (x[0 5]) indicate that the subsequence x(0 5) matches the pattern with index one and belongs to the region with index zero, respectively. A pair of atoms, pattern,(X) (X)rggggi onx(X) corresponds to feature within the sequence data. In this pair, the variables are identical, with one predicate representing a pattern and the other representing a region. We infer the following information from the rules in format (4): In a sequence input x, if all pairs of ground patterns and regions atoms substituted by subsequences in x are true, then the sequence input x belongs to the target class represented by the head atom hT-
4.2 DIFFEENTIABLE SYMBOLIZATION PROCESS
In this subsection, we design a differentiable body symbolization function L,, inspired by differen- tiable k-means (Fard al. 2020), transform numeric sequence data x into a fuzzy interpretation vector V, E [0, 1 This vector encodes fuzzy values of ground patterns and region atoms substi- tuted by input subsequences. A higher value [i] indicates the i-th atom in the interpretation is likely true. Using the head symbolization function J (x), we determine the target atom's 's Boolean value ehh,), where v(h,) = 1 if J
{h+} and vh,,) if 6. Building on (Gao al., 2024), we learn a logic program matrix Mp E [o, 1]mxx such that V (v v(h.) holds. The rules format (4) are then extracted from Mr. generalized from the most specific clause with all pattern and region predicates.
The architecture of NeurRL is shown Fig. la To learn a logic program P from sequence data, each input sequence x is divided into shorter subsequences s of length 1 with a unit step stride. An encoder maps subsequences s to an embedding space z, and a decoder reconstructs from The differentiable k-means algorithm described in Section 3.2 clusters embeddings z, grouping subsequences s with similar patterns into groups r. This yields fuzzy interpretation vectors v and Boolean target atom value v(ha) for each sequence. Finally, the differentiable rule-learning module uses V, as inputs and v(h,) as labels to learn high-level rules describing the target class.
Now, we describe the method to build the differentiable body symbolization function i. from se- quence x to interpretation vector v as follows: Let K be the maximum number of clusters based on the differentiable k-means algorithm, each subsequence with the length in x and the correspond- ing embedding has a cluster index e (1 I K), and each sequence input x can be transferred
We use fully connected layers the encoder and decnder structures.
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into vector of cluster indexes E {0, 1] [[xxx(x))++) Additionally, to incorporate temporal or spatial information into the predicates of the target rules, we divide the entire sequence data into R equal regions. The region of subsequence is determined by the location of its first point, s(0]. For each subsequence s, we calculate its cluster index vector C2 E [0, 1] using the weight function R) (1 S k S K and
Ga. as defined Eq. (2) where c_(A] = (h,
=i=11 C2 D higher value the i-th element C, indicates that the subsequence s is more likely to grouped into the i-th cluster. Hence, the possibilities of cluster indexes of subsequences in all regions C₂ E [0. 1]x [Xxlxx1
1). A
transfer the sequence imput x E R² to cluster index tensor with x R. where lp is the number of subsequence in one region of input sequence data [|x] / /R] To calculate the cluster index possibility of each region, we sum the likelihood of cluster index of all subsequence within one re- gion in cluster index tensor C2 and apply softmax function to build region cluster matrix Cp R RRxR as follows:
cii,j] = Σ c,li,k,jl, coli,
(5)
Since the region cluster matrix Cp contains clusters index for each region. Besides, each cluster corresponds to pattern. Hence, we flatten Cp into fuzzy interpretation vector V1, which serves as the input to the deep rule-learning module of NeurRL.
4.3 DIFFERENTIABLE RULE-LEARNING MODULE
In this section, we define the novel deep neural network-based rule-leariing module denoted Na to induce rules from fuzzy interpretation vectors. Based on the label of the sequence input x, we can determine the Boolean value of target atom v(ha). Then, using fuzzy interpretation vectors V as inputs and Boolean values of the head atom v(ha) as labels y, we build a novel neural network- based rule-learning module follows: Firstly, each input node receives the fuzzy value of a pattern or region atom stored in VJ- Secondly, one output node in the final layer reflects the fuzzy values of the head atom. Then, the neural network consists of dense layers and one disjunction layer. Lastly, let the number of nodes in the k-th dense layer be m, then the forward computation process formulated follows:
T
i=1 (9k 9k-1 o ..... oi (v:))) (1)
(6)
where the i-th dense layer is defined as:
g (xx-1)
ReLU (Mx;-1 - -d),
(7)
d
with d as the fixed bia The matrix M; is the softmax-activated of trainable weights M, E R Maa in i-th dense layer of the rule-learning module:
M.
M kl
(8)
E M,
With the differentiable body symbolization function Lt and NR. we now define target function that integrates the autoencoder module, clustering module, and rule-learning module as follows:
ne Exexxex fi (s, ((:r))) A DEE, fi (h,,
(8), r2) Gr.f. (b,, R)) + Aafa (Na (Lo (x):
(9)
where Nee and ? represent the trainable parameters in the encoder and rule-learning module, re- spectively. The loss function fi and fz are set to mean square error loss and cross-entropy loss correspondingly. The parameters X₁ and A2 regulate the trade-off between finding the concentrated representations of subsequences, the representations of clusters for obtaining precise patterns, and the representations of rules to generalize the date 5 Fig. Ia illustrates the loss functions defined in
"In our experiments, we set the fixed hias as 0.5. 'In our experiments, assigned equal weights finding representations, identifying clusters, and discov- ering rules by setting d d

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The supervised loss functions applied to the autoencoder (highlighted in orange boxes) and the sule-learning module (highlighted in blue boxes), respectively. The unsuper- vised loss function is applied to the differentiable k-means method (highlighted in gray box).
the target function (9.
We analyze the interpretability of the rulc-learning module Nn- In the logic program matrix Mp defined in Section 3.1 the sum of non-zero elements in each row, Σド= Myli, is normalized to one, matching the threshold in the function @ in Eq. ( The conjunction of the atoms corresponding to non-zero elements in each row of Mp can serve as the body of rule. Similarly, in the rule- learning module of NeurRI., the sum of the softmax-activated weights in each layer is also one. Due to the properties of the activation function ReI.U(== d)/(1 d) in each node, a node activates only when the sum of the weights approaches one; otherwise, it deactivates. This behavior mimics the threshold function 0 in Eq. (.. Similar to the logic program matrix Mp, the softmax- weights in each layer also have interpretability. When the fuzzy interpretation vector and Boolean value of the target atom fit the forward process in Eq. (6), the atoms corresponding to non-zero elements in each row of the i-th dense noftmax-activatedd weight matrix M, form a conjunction. From a neural network perspective, the j-th node in the i-th dense layer, denoted as ni,, represents conjunction of atoms from the previous (i - 1)-th dense layer (input layer). The likelihood of these atoms appearing in the conjunction is determined by the softmax-aativatee weights M. connecting to node nj. In the final k-th dense layer, the disjunction layer computes the probability of the target atom h4. The higher likelihood conjunctions, represented by the nodes in the final k-th layer, form the body of the rule headed by h2s To interpret the rules headed by h4, compute the product of all softmax-activated weights M, as the program tensor: Mp = I M., where Mr E [0, 1jmxx. The program tensor has the same interpretability as the program matrix, with high-value atoms in each row forming the rule body and the target atom as the rule head. The number of nodes in the last dense layer determines the number of learned rules in one learning epoch. Fig. [bb shows a neural network with two dense layers and one disjunction layer in blue. The weights in orange represent significant boftmax-activatedd values, with input nodes as atoms and hidden nodes as conjunctions. Multiplying the softmax weights identifies the atoms forming the body of rule headed by the target atom.
We use the following method to extract rules from program tensor Mp: We set multiple thresholds E [0, 1]. When the value of the element in of Mp is larger than threshold then we regard the atom corresponding to these elements as the body atom in a rule with the format (4). We compute the precision and recall of the rules based on the discretized fuzzy interpretation vectors generated from the test dataset. The discretized fuzzy interpretation vector is derived as the flattened version of cp(i,]] = 1 [cpp[i.j] = max; Cp [k, .i). where max; [k: j] represents the maximum value in the j-th column of Cy. Then, we keep the nigh-precision rules as the output. The precision and recall of a rule are defined as: precision
Thody head/Mbot and recall = Tbody A head/ head/ Thhead where Moody head denotes the number of discretized fuzzy interpretation vectors v, that satisfy the rule body and have the target class as the label. Similarly, Thody represents the number of discretized fuzzy interpretation vectors that satisfy the rule body, while Thhend refers to the number of instances with the target class. When obtaining rules in the format 4 we can highlight the subsequences satisfying the pattern and region predicates above on the raw inputs for more intuitive interpretability.
To train the model, we first pre-train an autoencoder to obtaín subsequence embeddings, then ini- tialize the cluster embeddings using k-means clustering (Lloyd] 1982) based on these embeddings. Finally, we jointly train the autoencoder, clustering model, and rule-ne model using the target function (9) to optimize the embeddings, clusters, and rules simultaneously.
5 EXPERIMENTAL RESULTS
5.1 LEARNING FROM SYNTHETIC DATA
In this subsection, we evaluate the model on synthetic time series data based on triangular pulse signals and trigonometric signals. Each signal contains two key patterns, increasing and decreasing, with each pattern having a length of five units. To test NeurRL's learning capability on a smaller dataset, we set the number of inputs in both the positive and negative classes to two for both the training and test datasets. In each class, the difference between two inputs at each time point is a random number drawn from a normal distribution with mean of zero and a variance of 0.1. The
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A (atlem, ) regiun,(X)
patrem, region,(X) (#
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(a) The signals based on triangular pulse.
(b) The signals based on trigonometry function.
Figure The synthetic data and the learned rules.
4, (a) region, (X) patrem (Y) (V) (p 0.83, 0.89)
A, pattemm/(X) (Xgion,()) 0.99. 0.90)
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(a) rule from ECG dataset.
(b) rule from ItalyPow.Dem dataset.
Figure 3: Selected rules from two UCR datasets.
positive test inputs are plotted blue, and the negative test inputs in orange in Fig. 2 the length of each region and the subsequence length to five units, and the number of clusters is set to three in this experiment. NeurRL is tasked with learning rules to describe the positive class, represented by the head atom hp. If the ground body atoms, substituted by subsequences of an input, hold true, the head atom hp is also true, indicating that the target class of the input is positive.
We set both
The rules with 1.0 precision (p) and 1.0 recall (r) are shown in Fig. The rule in Fig. states that when pattern, (cluster index 0) appears in region, (from time points to 9), the time series label positive. Similarly, the rule in Fig, [2bjindicates that when patterny appears in regi onlo (from time points 10 to 14), the label is positive. We highlight subsequences that satisfy the rule body in red in Fig. 2 inferring that the patterng indicates decreasing. These red patterns perfectly distinguish positive from negative inputs.
5.2 LEARNING FROM UCR DATASETS
In this subsection, we experimentally demonstrate the effectiivnness of NeurRL on 13 randomly lected datasets from UCR (Dau et al. 2019). as used by Wang et al. (001)). To evaluate NeurRL's performance, we consider the classification accuracy from the rules extracted by the deep rule- learning module (denoted as NeurRL(R)) and the classification accuracy from the module itself (denoted as NeurRL(N)). The number of clusters in this experiment is set five. The subsequence length and the number of regions vary for each task. We set the number of regions to approxi- mately 10 for time series data. Additionally, the subsequence length set to range from two to five, depending the specific subtask.
The baseline models include SSSL (Wang et al., 2019). Xu (Xu Funaya 2015), and BoW (Wang et al. 2013). SSSL uses regularized least squares, shapelet regularization, spectral analysis, and eseudo-labeling to auto-learn discriminative shapelets from time series data. Xu's 's's method constructs graph to derive underlying structures of time series data in sumi-supervised way. BoW generates a bag-offwordd representation for time series and uses SVM for classification. Statistical details, such the number of classes (C.), inputs (I.), series length, and comparison results shown in
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Table 1: Classification accuracy on 13 binary UCR datasets with different models.
Dataset
C.
I.
Length
Xu
BoW
SSSL
NeurRL(R)
NeurRL(N)
Coffee
2
56
286
0.588
0.620
0.792
0.964
1.000
ECG
200
96
0.819
0.955
0.793
0.820
0.880
Gun point
2
200
150
0.729
0.925
0.824
0.760
0.873
ItalyPo..Dem
1096
24
0.772
0.813
0.941
0.926
0.923
Lighting2
121
637
0.698
0.721
0.813
0.689
0.748
CBF
930
128
0.921
0,873
1.000
0,909
0,930
Face four
112
350
0.833
0.744
0.851
0.914
0.964
Lighting7
143
319
0.511
0.677
0.796
0.737
0.878
OSU leaf
442
427
0.642
0.685
0.835
0.844
0.849
Trace
200
275
0.788
1.00
1.00
0.833
0.905
WordsSyn
25
905
270
0.639
0.795
0.875
0.932
0.946
OliverOil
4
60
570
0.639
0.766
0.776
0.768
0.866
StarLightCurves
9236
2014
0.755
0.851
0.872
0.869
0.907
Mean accuracy
0.718
0.801
0.859
0.842
0.891
Table Comparisons with don-differentiable k-means clustering algorithm.
Non-differentiable k-means running time (s)
Differentiable k-means
Dataset
accuracy
accuracy running time (s)
Coffee
0.893
313
0.964
42
ECG
0.810
224
0.820
65
Gun point
0.807
102
0.740
35
ItalyPow. DalyPow.Dem
0.845
114
0.926
63
Lighting2
0.672
1166
0.689
120
Tab. [ with the best results in bold and second-bes underlined. The NeurRL(N) achieves the most best results, with seven, and NeurRL(R) achieves five second-best results.
The learned rules from the ECG and ItalyPow. Dem. datasets in the UCR archive are shown in Fig. 3 In Fig. 3a red highlights subsequences with the shape patterng in the region regionj, while green highlights subsequences with the shape pat pattern; in the region regiong. The rule suggests that when data decreases between time points 15 25 and then increases between time points 30 to 40, the input likely belongs to the positive class. The precision and recall for this rule are 0.83 and 0.89, respectively. In Fig. 3b red highlights subsequences with the shape patterng in region regiong. The rule indicates that lower value around time points 18 to 19 suggests the input belongs to the positive class, with precision and recall 0.99 and 0.90, respectively.
To demonstrate the benefits of the fully differentiable learning pipeline from raw sequence inputs to symbolic rules, we compare the accuracy and running times (in seconds) between NeurRI and its deep rule-learning module using the non-differentiable k-means clustering algorithm (Lloyd 1982). We use the same hyperparame- ter to split time series into subsequences for two methods. Results in Tab. show that the differentiable pipeline significantly reduces running time without sacrificing rule accuracy in most cases.
+
positive class
negative class
1,r
Figure 4: Leamnd rules from MNIST datasets.
5.3 LEARNING FROM IMAGES
In this subsection, we ask the model to learn rules to describe and discriminate two classes of images from MNIST datasets. We divide the MNIST dataset into five independent datasets, where each dataset contains one positive class and one negative class. For two- dimensional image data, first flatten the image data one-dimensional sequence data. Then, the

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Published as a conference paper at ICLR 2025
0.4
0.
02
NeurBLik NeurtLinI
02
NeurtiLN NeumLINN
00
0.0
Number Clusters
Langin of Regions
Langtn Subsequence
Number Clsstes
Length of Regions
Length of Supsequencc
(a) On ECG dataset.
(b) On ItalyPow.Dem dataset.
Figure 5: Results of ablation study. Hyperparameter values vs. accuracy.
sequence data can be the input for the NeurRI. model to learn the rules. The lengths of subsequence and region are both three. Besides, the number of clusters set five. After generating the highlighted patterns based on the rules, we recover the sequence data to the image for interpreting these learned rules.
We present the rule for learning the digit one from the digit zero in Fig. and the rules for learning other positive class from the negative class are shown in Fig. in Appendix We present the rule with the precision larger than 0.9 and the highlight features (or areas) defined in the rules. Each highlight feature corresponds to pair of region and pattern atoms in learned rule. We can interpret rules like importance attention (Zhang et al.) 2019), where the colored areas include highly discriminative information for describing positive inputs compared with negative inputs. For example, in Fig. 4 if the highlighted areas are in black at the same time, then the image class is one. Otherwise, the image class is zero. Compared with attention, we calculate the precision and recall to evaluate these highlight features quantitatively.
5.4 ABLATION STUDY
We conducted ablation studies using default hyperparameters, except for the one being explored. In this study, the number of clusters and regions influences the number of atoms in the learned rules, while the length of subsequences depends the length of potential patterns. Hence, these three hyperparameters collectively describe the sensitivity of NeurRL. We present the accuracy of both NeurRL(N) and NeurRL(R) the time series tasks ECG and IalyPow.Dem. from the UCR archive.
From Fig. S we observe that NeurRL's ' sensitivity varies across tasks, and the subsequence length is sensitive hyperparameter when the sequence length is small, as seen in the ItalyPow.Dem dataset. Properly chosen hyperparameters can achieve high consistency in accuracy between rules and neural networks. Notably, increasing the number of clusters or length of regions does not reduce accuracy linearly. This suggests that the rule-learning module effectively adjusts clusters within the model. In addition, conduct another ablation study in Tab. of Appendix A to show that pre-training the autoencoder and clustering prevents cluster collapse (Sansone 2023) and improves accuracy without increasing significant training time.
CONCLUSION
Inductive logic programming (ILP) is rule-based machine learning method that supporting data interpretability. Differentiable ILP offers advantages in scalability and robustness. However, label leakage remains a challenge when learning rules from raw data, as neuro-symbolic models require intermediate feature labels as input. In this paper, we propose a novel fully differentiable ILP model, Neural Rule Learner (NeurRL), which learns symbolic rules from raw sequences using differen- tiable k-means clustering module and a deep neural network-based rule-learning module. The dif- ferentiable k-means clustering algorithm groups subcomponents of inputs based on the similarity of their embeddings, and the learned clusters are used as input for the rule-learning module to induce rules that describe the ground truth class of input based its features. Compared to other rule-based models, NeurRL achieves comparable classification accuracy while offering interpretability through quantitative metrics. Future goals include variables representing entire inputs to explain handwritten digits without label leakage (Evans et al., 2021). Another focus is learning from incomplete data (e.g., healthcare applications) address real- world challenges.
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ACKNOWLEDGMENTS
We express our gratitude to the anonymous Reviewers for their valuable feedback, which has con- tributed significantly to enhancing the clarity and presentation of our paper. We also thank Xingyao Wang, Yingzhi Xia, Yang Liu, Jasmine Ong, Justina Ma, Jonathan Tan, Yong Liu, and Rick Goh for their supporting.
This work has been supported by Al Singapore under Grant AISG2TC2022066, Singapore. This work has been supported by the NII international internship program, JSPS KAKENHI Grant Num- ber JP21H04905 and IST CREST Grant Number JPMICR22D3, Japan. This work has also been supported by the National Key R&D Program of China under Grant 2021 102IYFFI YFF1201 102 and the Na- tional Natural Science Foundation of China under Grants 61972005 and 62172016.
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A ABLATION STUDY ON PRE-TRAINING
In this section, we conduct experiments with and without pre-training the autoencoder and clustering model investigate whether pre-training improves accuracy. The results, including accuracy and running time (in seconds), are presented in Tab. 3
Table 3: Ablation study: With vs. without pre- tre-trainin autoencoder and clustering model. The first accuracy is achieved using NeurRI(NN), while the second accuracy is obtained with NeurRI(R).
With pre-training
Without pre-training
Dataset
Accuracy
Running time (8)
Accuracy
Running time (8)
Coffee
1.00, 0.96
42
0.83, 0.81
30
ECG
0.88, 0.82
65
0.87, 0.64
ItalyPow.Dem
0.92, 0.93
63
0.75, 0.80
61
Gun Point
0.87, 0.76
35
0.86, 0.43
Lighting2
0.75, 0.69
120
0.64, 0.60
63
The experiment results indicate the pre-training autoencoder and clustering model can improve the accuracy of both NeurRL(N) and NeurRI(() without increasing significant training time.
B THE LINK OF THE MODEL
The model and data can found here: http:://gthhbb.cc////oett//////e
C CLASSIFYING AND EXPLAINING MNIST DATA
Each column in Fig. corresponds learning task. There are six rules presented Fig. to describe the positive class from the negative class.
444
6
9
Positive Clees
3
5
7
8
Negative Class
pat.ss
p= (19) 0:43
p=
0190 0.76
Figure 6: Rules from MNIST datasets.
14


Reviews:
[{"review_id": "8QyC1xW0sl", "reviewer": "Reviewer_6MqZ", "summary": "This work introduces Neural Rule Learner (NeurRL), a system designed to learn interpretable rules from sequential data, such as from time-series sensor data (like spectrograms) and serialized higher-dimensional data (such as images).\n\nThe system uses symbolization functions to map raw data into a predetermined framework of pattern and region predicates, allowing discrete raw data sequences to be represented as logical atoms that can be used in rule learning. In the supplied reference implementation, patterns are learned automatically through a differentiable k-means clustering process, while regions are predefined subdivisions of the sequence. The combination of learned patterns and predefined regions allows the system to express discovered features as logical rules using this symbolic vocabulary.\n\nThe framework uses two types of symbolization functions: body symbolization functions that map discrete sequences to symbolic concepts, and head symbolization functions that map inputs to a set of atoms. The system avoids the \"label leakage\" problem by not requiring pre-trained neural networks to map raw inputs to symbolic labels. The entire pipeline is differentiable and can be trained end-to-end, learning directly from raw data without needing intermediate symbolic representations.\n\nThe implementation uses a deep rule-learning module with multiple dense layers, and rules are evaluated using precision and recall metrics. The reference implementation includes an autoencoder component for learning concentrated representations of subsequences.\nExperimental validation was conducted on both time series data (UCR datasets) and image data (MNIST).  Learning on UCR data-subsets  highlights low-data or data-efficient learning methods. The reference implementation achieves comparable or better classification accuracy compared to baseline methods, and provides interpretable rules. The generated rules are human-readable and capture patterns in the data.", "questions": "Question:  Please clarify the generalizability of your approach.  Do you expect the rule learning method to be sensitive to data errors that arise naturally in real-world collection, such as uncalibrated sensors or aspect/focus variation in images, or frame rate in video?   Consider the following comment, if that helps clarify the objective of the request.\n\nI am not familiar with the UCR datasets, and I don't think you provided a link, nor did the paper you cite have a currently valid link; I found data and descriptions here: https://www.cs.ucr.edu/~eamonn/time_series_data/.  Some datasets seem to have a small degree of possible coordinate sensitivity, such as the spectrogram data, while other would vary significantly with scale, placement of the signal within a collection window, and image orientation.  (This is also true of MNIST).  Please review the table below, and help me understand the sensitivity of your method to data collection errors.  This gets at the issue of generalizability / fragility of the method.  In the table, the first column has the dataset name, the second column records whether or not your method was the top score in your reported results, the third records a (possibly incorrect) sense of how sensitive the data collection is to getting all the coordinates right, and the fourth is an explanation of the assessment. \n\n| Dataset | NeurRL | Coordinate Sensitivity | Rationale for sensitivity assessment |\n|---------|-------------|----------------------|-------------|\n| Coffee | Top | Low | Based on spectrographic measurements which are inherently coordinate-independent. The frequency patterns would be consistent regardless of measurement setup, as long as calibration is maintained. |\n| ECG | -- | Medium | While heart rhythms have characteristic shapes, the absolute voltage measurements depend on electrode placement. However, the relative patterns are fairly robust across different setups. |\n| Gun Point | -- | High | Heavily dependent on camera angle, distance calibration, and the defined origin point for measuring hand position. The distinction between gun/no-gun relies on precise spatial coordinates. |\n| ItalyPow.Dem | Top | Low | Power demand measurements are absolute values independent of coordinate systems. The readings represent actual power consumption regardless of measurement setup. |\n| Lightning2 | -- | Low (?) | While electromagnetic measurements depend on sensor placement, the characteristic patterns of lightning types are relatively robust across different setups. |\n| CBF | -- | Low | As a synthetic dataset designed to test pattern recognition, the shapes (Cylinder-Bell-Funnel) are meaningful in their relative values rather than absolute coordinates. |\n| Face Four | Top | High | Face profile traces are highly dependent on camera angle, distance, and orientation. Changes in perspective would significantly alter the time series. |\n| Lightning7 | Top | Low (?) | Similar to Lightning2, with moderate dependence on sensor placement but relatively robust patterns. |\n| OSU Leaf | Top | High | Leaf contour measurements depend heavily on orientation, scale, and starting point of the trace. Different coordinate systems would produce very different time series. |\n| Trace | -- | Low | As synthetic nuclear instrument data, the patterns represent relative changes that are meaningful independent of absolute coordinate systems. |\n| WordsSyn | Top | High | Pen trajectories are highly dependent on writing orientation, scale, and starting position. The coordinate system directly affects the recorded patterns. |\n| OliveOil | Top | Low | Spectroscopic measurements are independent of coordinate systems. The chemical signatures would be consistent across different spectrometers (after calibration). |\n| StarLightCurves | Top | Low | Brightness measurements are relative values that remain meaningful regardless of the telescope's exact positioning (assuming proper astronomical calibration). |\n\nQuestion 2:  Can you explain \"leakage\" more clearly?  The coordinate dependence an example above creates an implicit bias in the data, for instance limited field-of-view and imaging system orientation in MNIST and handwriting digitization; is this an example of \"label leakage\"?", "limitations": "", "rating": 8, "confidence": 3, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "Originality: This provides a new and instructive implementation to address the raw-to-symbolic data problem, while avoiding the data leakage issue.\n\nQuality and Clarity: The paper explains and illustrates the method in a clear and compelling way.\n\nSignificance:  This appears to be an important contribution to the problem of addressing the problem of learning symbolic rules about raw sequential data.", "weaknesses": "The approach may require significant data invariance (orientation of images, sampling regularity).  Many important datasets have these properties, although image data often does not.  See questions for further clarification.\n\nAs a demonstration of the advantage of this approach as an interpretable and explainable method, it would be helpful to have the interpretation of rules on each dataset from UCR discussed in more detail.  How interpretable are the rules in terms of the data patterns?  Are there particular datasets or data types for which rules emerge that provide insight or match intuition?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "dcbG9bYDvi", "reviewer": "Reviewer_K7Gv", "summary": "The paper proposes a new differentiable Inductive Logic Programming framework where symbolic rules are learned via gradients from sequential data. To overcome the bottleneck of previous models that require pre-trained neural networks to perceive raw input, the authors propose Neural Rule Learner (NeurRL), which combines VAE and differentiable k-means to the differentiable rule learning approach. The resulting framework can learn classification rules that specify regions and corresponding patterns to be classified as positive. It is demonstrated in the experiments using simple datasets, including synthetic, UCR, and MNIST datasets.", "questions": "Differentiable ILP frameworks ($\\partial$ ILP and $\\alpha$ ILP) typically suffer from their intense memory consumption due to the tensor encoding. Does NeurRL share the same problem? \n\nFor other questions, please refer to the weakness section.", "limitations": "", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "This paper tackles an important topic in the field of differentiable ILP, specifically the need for pre-trained models to ground raw input into symbols for effective rule learning. The approach of combining differentiable k-means and VAE with gradient-based rule learners is noteworthy, even though the concept is relatively straightforward.\n\nThe manuscript is well-written, with a high-quality presentation overall. It clearly outlines the research question and effectively conveys its core idea.\n\nThis work is a valuable contribution to the neuro-symbolic research community, paving the way for developing more robust systems capable of learning with fewer priors on raw input.", "weaknesses": "My primary concern is that it remains unclear whether the main claim is adequately supported by the methods and experiments sections. The paper asserts:\n\n> (Line 42-45) Learning logic programs from raw data is hindered by the label leakage problem common in neuro-symbolic research (Topan et al., 2021): This leakage occurs when labels of ground objects are introduced for inducing rules (Evans & Grefenstette, 2018; Shindo et al., 2023).\n\nand\n\n> (Line 82-85) In our work, we do not require a pre-trained large-scale neural network to map raw input data to symbolic labels or atoms. Instead, we design an end-to-end learning framework to derive rules directly from raw numeric data, such as sequence and image data.\n\nIf this claim holds, the proposed system should be trainable end-to-end, grounding symbols in meaningful ways such that rule learners can generate rules based on them. However, the paper does not make it clear how this grounding is accomplished.\n\nDoes NeurRL perform grounding on the obtained clusters? Can the rule learners manage rules involving multiple clusters by understanding the meaning of each cluster? If not, I am skeptical that the proposed framework fully addresses the bottleneck identified in the introduction, namely the reliance on perception models in prior rule learning frameworks like $\\partial$ ILP and $\\alpha$ ILP.\n\nWithout grounding, the method would be confined to the rule format (Eq. 5), as the system would lack the means to learn from non-grounded symbols, limiting its applicability to other domains.\n\nWhile I appreciate the method's ability to learn from small datasets, it is unclear how it would scale to larger, more complex datasets. Given that neural models typically require large datasets, future development would benefit from the capacity to handle such data at scale to better integrate neural networks. It would be beneficial to provide runtime comparisons on different dataset sizes, or to discuss potential challenges and solutions for scaling the approach to larger datasets. How does the model's performance change with increasing dataset complexity and size? If scalability to large-scale data is an issue, this limitation should be discussed somewhere in the paper.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "LluxKFyonn", "reviewer": "Reviewer_2bKk", "summary": "The paper proposes a supervised neurosymbolic model for time series data, structured as a cascade of three main building blocks. First, an autoencoder extracts an embedding representation from raw data. Next, a differentiable clustering module maps the embedding to a symbolic representation, followed by a fuzzy neural rule learner that maps the symbolic representation to the target. The overall training objective includes three loss functions, one for each building block: a reconstruction error for the autoencoder, a k-means objective for the differentiable clustering, and a supervised cross-entropy loss for the rule learner.\nTraining occurs in two stages, beginning with the pre-training of the autoencoder, followed by the fine-tuning of the entire model. Additionally, a post-hoc heuristic is introduced to extract rules in propositional logic form from the weights learned by the rule learner. Experiments are conducted on a series of small datasets from UCR, comparing the model to existing neural and kernel-based classification baselines for time series data, with results demonstrating superior predictive accuracy. Lastly, a qualitative experiment on MNIST data is provided.", "questions": "In addition to discuss the above mentioned weaknesses, please find below some additional questions: \n\n1. The proposed solution introduces several hyperparameters, including parameters for data pre-processing (such as subsequence length and the number of temporal/spatial slots), fixed biases for the rule learner’s neurons, temperature parameters for the rule learner’s weights, and weights for the loss functions. How are these hyperparameters selected, and how sensitive is the model’s performance to their values? A sensitivity analysis could clarify this, and it should be feasible given the small size of the datasets. \n2. The post-hoc strategy achieves lower predictive accuracy compared to the full neural solution (see Table 1), with drops of up to 15 percentage points in some cases. This underscores the heuristic nature of the proposed strategy. Could you provide an explanation for this phenomenon? Additionally, what can be said about the faithfulness of the extracted rules in relation to the learned model’s weights? \n3. Why is neuro-symbolic learning necessary for the chosen MNIST task, and what is the rationale for this choice? The task could be solved by relying on superficial image cues (such as identifying regions that are either black or white), as illustrated by the qualitative demonstration in Figure 4, without needing to understand the underlying semantics of the digits. Therefore, a purely neural approach might suffice for this task. It would be interesting to assess the model’s capability to recognize digit classes within arithmetic tasks or, more broadly, in tasks defined by programs over integer sequences (e.g., sequences of even or odd numbers, counting). \n4. Are the codes learnable in the differentiable clustering layer and how do you ensure that meaningful clusters can be learnt, without experiencing any form of collapse? Please refer for instance to the work in [1].\n\n**References**\n\n[1] Sansone. The Triad Of Failure Modes and a Possible Way Out. In NeurIPS SSL Workshop 2023", "limitations": "", "rating": 5, "confidence": 3, "soundness": 2, "presentation": 1, "contribution": 2, "strengths": "1. The paper is overall clear **Clarity**. However the text should be polished to rewrite/rearrange words in some sentences and correct the several typos present throughout. Please refer the MINOR paragraph below for a non-exhaustive list.\n2. The considered problem is relevant and timely **Relevance**.\n3. Code is available, but no further check has been performed **Code availability**.\n\nMINOR \\\nL.109 -> $\\neg\\alpha_2$ \\\nL.129 -> a substitution or an interpretation \\\nL.151 -> is ordered real-valued observations with the dimension one \\\nL.159 -> discrete -> discretize \\\nL.161 -> closest \\\nL.195-197 -> could you rephrase the sentence? \\\nL.273 -> the its \\\nL.325-327 -> could you rephrase the sentence? \\\nL.381 -> synthetic", "weaknesses": "1. The paper lacks adequate context within the existing neuro-symbolic learning literature, making its contribution appear incremental in terms of novelty **Quality**. For example, neuro-symbolic (NeSy) autoencoders have been previously explored (see [1,2]), and differentiable clustering has also been investigated (refer to [3,4]). Additionally, the rule learner shows a strong resemblance to $\\delta$ILP. At a minimum, a discussion of related work should be included to clarify the paper's novelty and contribution. An experimental comparison with existing methods could also help demonstrate the advantages of the proposed approach. \n2. The paper lacks soundness due to several incorrect and overly ambitious claims that are not adequately supported **Soundness**. First, the paper states that this work is the first to enable joint training of the neural network and rule learner without pre-training. However, this claim is inaccurate, as the proposed training still requires a two-stage process. Additionally, contrary to what is mentioned, the work in [5] already achieves joint learning of a neural network and a rule learner. An experimental comparison on the datasets introduced in [5] would be valuable. Furthermore, the paper suggests that the proposed solution can uncover underlying rules without identifying conditions for failure. However, depending on the level of supervision and the complexity of the symbolic task, the neural learner may learn incorrect mappings from raw data to symbols. The work in [3,4] presents a basic sequential setting to analyze this issue, demonstrating that while representation learning (through differentiable clustering) is necessary to address the problem, it is not sufficient. Similar observations are subsequently reported in [6,7], where the authors label this issue as \"reasoning shortcuts.\" \n3. There is very limited discussion about the details of the experimental methodology, raising serious concerns regarding reproducibility **Reproducibility**\n4. Experiments are conducted on small datasets with relatively simple symbolic tasks **Quality**.  I encourage the authors to consider more traditional and more complex experimental settings for neuro-symbolic learning, including the arithmetic tasks on handwritten digits or the datasets in [5].\n\n**References**\n\n[1] Misino, Marra, Sansone. VAEL: Bridging Variational Autoencoders and Probabilistic Logic Programming. In NeurIPS 2022 \\\n[2] Zhan, Sun, Kennedy, Yue, Chauduri. Unsupervised Learning of Neurosymbolic Encoders. In TMLR 2022 \\\n[3] Sansone, Manhaeve. Learning Symbolic Representations Through Joint GEnerative and DIscriminative Training. In ICLR NeSy-GeMs Workshop 2023 \\\n[4] Sansone, Manhaeve. Learning Symbolic Representations Through Joint GEnerative and DIscriminative Training. IJCAI KBCG Workshop 2023 \\\n[5] Evans et al. Making Sense of Sensory Input. In Artificial Intelligence 2021 \\\n[6] Marconato et al. Neuro-Symbolic Continual Learning: Knowledge, Reasoning Shortcuts and Concept Rehearsal. In ICML 2023 \\\n[7] Marconato, Teso, Vergari, Passerini. Not All Neuro-Symbolic Concepts Are Created Equal: Analysis and Mitigation of Reasoning Shortcuts. In NeurIPS 2023", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "jXWMObhaqo", "reviewer": "Reviewer_iHmD", "summary": "This paper proposes a novel differentiable Inductive Logic Programming model called NeurRL, aimed at solving the problem of learning symbolic rules from raw data. Unlike previous methods that rely on pre-trained neural networks to extract labels for symbolic learning, NeurRL combines a differentiable clustering module and a deep neural network learning module, enabling end-to-end rule learning from raw data such as time series and images without leaking label information.", "questions": "1. Is there a way to reduce the number of hyperparameters, such as determining the number of categories K adaptively during training? If this is not feasible, please at least provide guidance on how to select these hyperparameters and show how different choices impact the experimental results.\n2. Please compare your work with more related studies (see Weakness 4)", "limitations": "", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The paper formally establishes the ILP learning task from raw data, utilizing the interpretation transition framework of ILP to guide the learning process.\n2. Experimental results demonstrate that NeurRL shows a notable improvement over previous methods in terms of effectiveness.\n3. The paper's overall presentation is clear, making the concepts and methodologies easy to follow.", "weaknesses": "1. The model has a large number of hyperparameters (especially the number of categories K, subsequence length l, and the number of regions P), which may negatively affect reproducibility and generalizability.\n2. The experimental details are insufficient, lacking information on how hyperparameters were chosen and whether tuning was done separately for different experiments. Moreover, it is unclear if the results reported are the best performance or averaged.\n3. The learned predicates lack high-level abstraction, instead offering post-hoc explanations for neighboring pixels. This limits the generalizability and interpretability of the induced rules.\n4. Although previous works did not formally define the task of extracting rules from raw data, some have achieved similar goals with comparable architectures (e.g., autoencoder + discriminator)[1, 2], or going further in technical sophistication[3] . This paper lacks a discussion of and comparison with such works.\n5. The model does not consider translation invariance in the data, meaning some patterns should only need to be present rather than bound to a specific region. This limitation may hinder the model's performance in tasks where the position of the target is irrelevant.\n\n[1] Azzolin, S., Longa, A., Barbiero, P., Liò, P., & Passerini, A. (2022). Global explainability of gnns via logic combination of learned concepts. arXiv preprint arXiv:2210.07147.\n\n[2] Walter, N. P., Fischer, J., & Vreeken, J. (2024, March). Finding interpretable class-specific patterns through efficient neural search. In Proceedings of the AAAI Conference on Artificial Intelligence (Vol. 38, No. 8, pp. 9062-9070).\n\n[3] Wang, B., Li, L., Nakashima, Y., & Nagahara, H. (2023). Learning bottleneck concepts in image classification. In Proceedings of the ieee/cvf conference on computer vision and pattern recognition (pp. 10962-10971).", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)