[1,2]Xiaoxuan Shen

[1,2]Jianwen Sun

[1,2,3]Zongkai Yang

1]National Engineering Research Center of Educational Big Data, Central China
Normal University, Wuhan, 430079, China 2]Faculty of Artificial Intelligence
in Education, Central China Normal University, Wuhan, 430079, China 3]National
Engineering Research Center for E-learning, Central China Normal University,
Wuhan, 430079, China

# Automated discovery of symbolic laws governing skill acquisition from
naturally occurring data

Sannyuya Liu  Qing Li<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>[  [  [

###### Abstract

Skill acquisition is a key area of research in cognitive psychology as it
encompasses multiple psychological processes. The laws discovered under
experimental paradigms are controversial and lack generalizability. This paper
aims to unearth the laws of skill learning from large-scale training log data.
A two-stage algorithm was developed to tackle the issues of unobservable
cognitive states and algorithmic explosion in searching. Initially a deep
learning model is employed to determine the learner’s cognitive state and
assess the feature importance. Subsequently, symbolic regression algorithms
are utilized to parse the neural network model into algebraic equations.
Experimental results show the algorithm can accurately restore preset laws
within a noise range in continuous feedback settings. When applied to Lumosity
training data, the method outperforms traditional and recent models in fitness
terms. The study reveals two new forms of skill acquisition laws and reaffirms
some previous findings.

##  1 Introduction

Skill acquisition theory explains how individuals develop skills through
practice and experience, emphasizing the processes of acquiring, refining, and
automating skills [1]. A key concept in this theory is deliberate practice,
which involves focused efforts to improve specific areas of performance [2].
This theory has had a significant impact on fields like sports, music,
education, and programming, offering valuable insights into effective skill
acquisition and improvement across different domains [3].

Skill acquisition is a complex psychological process involving practice [4, 5,
6], memory [7, 8, 9], and skill transfer [10, 11]. Researchers use symbolic
representation methods, like algebraic equations, to describe experimental
phenomena and identify patterns. Establish cognitive theoretical models based
on observed experimental patterns. Instance theory [12, 13] explains the power
law [14] of practice by suggesting that skill improvement results from
accumulating and refining relevant instances during practice repetitions.
Similarly, there are also theories such as multi-phase theory [15, 16] and
parallel distributed processing models [17, 18], along with cognitive
architectures like SOAR [19] and ACT-R [20, 21]. Symbolic patterns are
essential for developing cognitive theories, but their complexity and
controversy highlight the need for further study in understanding human
behavior.

Traditional psychological research often derives symbolic rules from
controlled experiments, leading to varying conclusions due to different
perspectives. For instance, the law of practice, initially believed to follow
the power law by many researchers [6, 14], but some studies suggesting there
are better symbol representations, like exponential or APEX functions [5], in
terms of data fitting [4]. Besides, biases and limitations in experimental
designs can hinder result replication and generalization [22]. Jenkins’
tetrahedral model [23] emphasizes considering subjects, encoding activities,
events, and retrieval conditions in memory experiments, indicating that
findings from controlled experiments may apply only to specific conditions.
Further research is required to evaluate the robustness and generalizability
of findings across diverse contexts.

Psychologists have turned to analyzing human behavior patterns using naturally
occurring data sets (NODS) to overcome challenges in traditional psychological
research. This approach, known as the ”big data” paradigm, complements lab
experiments by extracting psychological patterns from large real-world
datasets [22]. While lacking experimental control, NODS allow for statistical
separation of factors and uncover novel patterns that are not commonly
investigated in previous research. The use of empirical data from realistic
settings enhances the applicability of conclusions, and the vast scale of data
leads to more robust findings [22]. The emergence of online training systems
has provided learners with opportunities to improve skills through practice
sessions, generating extensive training log data with valuable behavioral
patterns related to skill acquisition. It presents opportunities to develop
theories and models using the ”big data” paradigm.

Symbolic regression, a machine learning technique, aims to find mathematical
expressions that best fit a dataset by combining operations, functions, and
variables [24]. It is valuable for uncovering data patterns and inspiring
further research [25]. As a search-based algorithm, symbolic regression
typically provides optimal results within the search space, making it suitable
for identifying skill acquisition patterns and applications in behavioral
sciences. While it has been widely used in physics [26, 27, 28, 29], chemistry
[30, 31], and materials science [32, 33]. However, the complexity of training
log data, including scale, dimensionality, and latent states, poses challenges
for applying symbolic regression algorithms directly. Traditional algorithms
like genetic algorithms struggle to find optimal symbolic models due to the
exponential growth in solution space with variables and operators [24]. The
numerous uncontrolled variables and factors affecting skill acquisition in
NODS hinder direct optimization of symbolic models. Moreover, the skill
mastery of learner is unobservable, present obstacles for current data
analysis algorithms to uncover symbolic rules related to skill acquisition
from training log data.

Deep learning models, or deep neural networks, have become popular in machine
learning for their ability to effectively analyze data in high-dimensional
space using the back-propagation algorithm [34]. These models excel at
capturing complex data patterns, enabling accurate estimation of learner skill
mastery. However, a drawback is their black-box nature, making it hard for
humans to interpret the decision-making process behind the fitted data
patterns.

Figure 1: Overall model architecture diagram. A two-stage model is proposed
for the automated discovery of symbolic laws that govern skill acquisition.
Specifically, (a) provides a toy example of training log data. The example
illustrates the relevant information of nine practice sessions for a specific
learner, where each record containing five main elements: learner, practice,
skill, score, and time. (b) depicts the constructed deep regressor. It
consists of three main modules: feature encoding, mastery inference, and score
prediction. The model takes the output of the previous n practice sessions and
predicts the score of the next practice session. The proposed model follows
the autoregressive paradigm to establish the optimization objective of the
model. (c) describes the process of extracting symbolic governing laws from
the trained deep regressor. First, we propose the symbolic distillation
method, which represents the black-box neural network model as the closest
symbolic representation. Secondly, for a trained deep regressor, it has
already embedded hidden patterns from the data into the model. The symbolic
distillation method is utilized to interpret the various modules of the deep
regressor and fuse them together to obtain symbolic governing laws.

This paper proposes a method to automatically discover hidden patterns in
skill acquisition from large training log data. A two-stage model is
presented, combining deep learning and symbolic regression algorithms to
reveal the fundamental principles of skill acquisition. This approach
guarantees accurate fitting and interpretability by extracting key patterns
from complex training logs and presenting them in algebraic equation,
enhancing understanding. Figure 1 illustrates the overall framework of the
proposed method.

The first stage of the proposed two-stage model focuses on developing a deep
learning regressor to accurately fit large-scale online training log data,
shown in Fig. 1b. This regressor aims to estimate crucial latent variables,
such as behavior feature encoding and user skill proficiency, and to fit key
mapping functions precisely. Drawing inspiration from the successful
transformer model [35] for sequence modeling, we devised a transformer-like
model. Initially, it evaluates the importance of behaviors in each skill
training by utilizing static behavior features and generates encoded
sequential behavior features. Subsequently, a non-linear neural network model
computes the user’s skill mastery state based on the encoded features.
Finally, the skill proficiency level is linked to the training score using the
three-parameter item response theory (IRT) model [36, 37, 38, 39], a well-
established cognitive diagnostic model. The model is trained using an
autoregressive approach that predicts future training outcomes based on
historical training sequences, enabling a comprehensive exploration of users’
behavior patterns during the training process.

In the second phase of the study, symbolic regression was used to extract core
patterns from a trained deep learning regressor into algebraic equations,
resulting in symbolic rules (Fig. 1c). Symbolic distillation was developed to
extract these patterns from the neural network by creating a smaller pattern
extraction dataset from a large training dataset. By combining this with a
gradient-based neural network feature importance estimation algorithm, crucial
features were selected, reducing the solution space for symbolic regression.
The symbolic model was optimized by minimizing prediction error between the
symbolic model and the trained neural network using the pattern extraction
dataset. Symbolic distillation techniques were then applied to behavior
importance estimation and mastery estimation networks post deep learning model
fitting. Finally, symbolized rules were consolidated to derive skill
acquisition rules.

Thus far, we have outlined the rationale and general process of the proposed
method. A detailed description of the model will be presented in sections 4.1
and 4.2 of the Method section. The effectiveness and feasibility of the method
are confirmed through experiments on simulated data in the Results section.
Following this, the method is applied to the Lumosity dataset for large-scale
skill training to reveal hidden patterns and regularities in the data.

##  2 Results

###  2.1 Model validation on simulated data

Validating patterns from data in skill acquisition is challenging, due to
their contentious nature. So the model validation is addressed through
simulated experiments. The validation aims to demonstrate the algorithm’s
ability to restore data patterns, enhancing reliability in real-world
applications. Algebraic equations from skill acquisition theories are used as
preset governing laws, shown in Table 1. Then, simulated learners, exercises
and training logs are generated with skill mastery and practice scores
computed by preset governing laws. The study tests if the algorithm can
restore these laws using the simulated data only. Extended Data Figure 2
provides details on simulated data generation, evaluation process, and sample
representation. Further details on the process of generating simulated data,
including data structure and parameter settings, will be elaborated in the
Method section.

Table 1: The predetermined practice rules - the relationship function between practice times and skill mastery. Here, $N_{1}$ and $N_{2}$ refer to the number of times the learner practiced skill 1 and skill 2, respectively. $\alpha$ and $\beta$ are the exercise rate parameters, $\mu$ is the mutualism factor, $\gamma$ represents the forgetting rate. In the simulation, we only establish mutualism for skill 1, meaning that practicing skill 2 aided in mastering skill 1. Therefore, the number of practice sessions for the skill 1 considering mutualism is defined as $((1-\mu)\cdot N_{1}+\mu\cdot N_{2})$. SETTINGS | SKILLS | GOVERNING LAWS
---|---|---
SETTING 1 | SKILL 1: Linear | $\beta\cdot\alpha\cdot((1-\mu)\cdot N_{1}+\mu\cdot N_{2})-\gamma\cdot(N_{1}+N_{2})$
Linear+ Exponential | SKILL 2: Exponential | $\beta\cdot\exp(\alpha\cdot N_{2})-\gamma\cdot(N_{1}+N_{2})$
SETTING 2 | SKILL 1: Exponential | $\beta\cdot\exp(\alpha\cdot((1-\mu)\cdot N_{1}+\mu\cdot N_{2}))-\gamma\cdot(N_{1}+N_{2})$
Exponential + Power | SKILL 2: Power | $\beta\cdot{N_{2}}^{\alpha}-\gamma\cdot(N_{1}+N_{2})$
SETTING 3 | SKILL 1: Power | $\beta\cdot{((1-\mu)\cdot N_{1}+\mu\cdot N_{2})}^{\alpha}-\gamma\cdot(N_{1}+N_{2})$
Power + Linear | SKILL 2: Linear | $\beta\cdot\alpha\cdot N_{2}-\gamma\cdot(N_{1}+N_{2})$
|  |
Figure 2: Results of the simulated data experiment. The results of the model
experiments on simulated data. Specifically, it presents more detailed
analyses of (a) the fitting degree of the deep learning fitter under different
parameter settings, (b) the numerical analysis of regularization terms under
different parameter settings, and (c) the analysis of the degree of accuracy
in reproducing governing laws. Feature, Structure, and Value are introduced as
three dimensions to evaluate the degree of formula restoration. Feature
measures whether the variables in the restored formula match those in the
assumed formula. Structure indicates whether the form and structure of the
restored formula align with the assumed formula. Value represents whether the
parameters in the formula match those in the assumed formula. In general, if
the Feature, Structure, and Value are correctly restored, the formula can be
considered completely restored and accurate. ”Yes” indicates that the
algorithm has successfully restored the predetermined pattern, whereas ”No”
indicates its failure to do so.

First, we investigated the fitting of the deep regressor to the simulated
data. We created four deep learning regressors with different parameters:
H100+R0, H100+R1, H1000+R0, and H1000+R1. Here, H represents the number of
hidden neurons per layer in the model [43, 44], with H set to 100 and 1000 to
simulate small and large fitting abilities. R indicates the use of
regularization for attention value, with R=1 or 0 representing with or without
regularization in the deep regressor’s optimization objective. Regularization
can simplify the restoration equation, improve model readability, but may
reduce fitting ability. We trained the four regressors for 5000 epochs on
simulated datasets at five error levels ($\delta\in\\{0,0.001,0.01,0.1,1\\}$).
The average fitting absolute error and regularization term’s value are plotted
in Fig. 2a and Fig. 2b, respectively.

Fig. 2a shows that models with more hidden neurons (H1000+R0 and H1000+R1)
perform slightly better than those with fewer neurons (H100+R0 and H100+R1),
consistent with prior neural network studies [45]. The model with
regularization (R1) has slightly lower fitting accuracy than the model without
regularization (R0), but the difference is not significant. In Fig. 2b, the
regularization term for the R1 model is much smaller (3 to 7 orders of
magnitude) than that of the R0 model. Meanwhile, the H1000+R1 model has a
slightly lower regularization term than the H100+R1 model. Overall, the R1
model shows minimal deviation in fitting error compared to the non-regularized
R0 model, while vastly reducing the regularization term value. The H1000 model
outperforms the H100 model in fitting performance and regularization term
value. Therefore, the H1000+R1 model is chosen as the deep learning regressor
for symbolic distillation and analysis of governing laws in subsequent
experiments.

Secondly, we analyze the extent to which the model reproduces the
predetermined governing laws. Based on the trained H1000+R1 model, the process
of symbolization distillation and extraction of governing laws is performed,
and the resulting governing laws are compared with the predetermined model to
assess the degree of model restoration.

In deep learning regressor, we used five basic features from two aspects:
exercise characteristics (E) and scheduling information (S). The E feature
involves one-hot encoding of exercise correlations with SKILL 1, SKILL 2, and
a general factor [46, 47], treated as a special skill [42]. The S feature
includes Count (number of skill practices) and Interval (time between practice
sessions). In the H1000+R1 model, the E feature weight is set to 1 due to its
extremely low regularization term value (1e-7 to 1e-4), resulting in encoded
features Sum_G, Sum_S1, and Sum_S2, representing historical practice times and
frequency for corresponding skills.

Model evaluation considers Feature, Structure, and Value dimensions. Feature
dimension focuses on identifying the importance of Sum G, Sum S1, and Sum S2.
Structure dimension assesses the correctness of the algebraic equation from
symbolic regression, allowing some error in parameter values. Value dimension
checks if the restored equation falls within an acceptable range of the true
value, with an error threshold of less than 1%. These dimensions are
interconnected in the evaluation process.

Simulation experiments were conducted to test the restoration of governing
laws on simulated datasets across three settings and four noise levels.
Results are shown in Fig. 2c, with some feature importance results in
Supplementary Figure 1. The proposed method effectively restored governing
laws for preset skills within a noise range ($\delta<0.1$). At $\delta=0.1$,
the model reproduced equations for SKILL 2 and Linear setting equation for
SKILL 1 but faced challenges with Power and Exponential settings due to errors
causing ambiguity in function determination. This challenge lessened with
smaller errors. At $\delta=1$, the model identified the core features but
struggled with equation structure due to many noise values exceeding the
rating range [0, 1]. Results under alternative error settings in Supplementary
Figure 2 showed similar results. The results indicate that the proposed method
is capable of accurately reconstructing the predefined equations within a
certain range of error in three different settings, demonstrating its
potential applicability in real-world scenarios.

###  2.2 Model application on large-scale real-world cognitive training data

In this section, we attempted to apply the method proposed in this paper to
large-scale real-world datasets to discover patterns from real data and
provide evidence for skill acquisition theory. For the application, we
selected Lumosity as the validation platform [48]. A more detailed description
of the dataset, data filtering, preprocessing methods, etc., will be provided
in the Method section.

Model fitting analysis was conducted to assess the performance of a deep
learning regressor on actual data and perform model selection. Six models were
evaluated: H100+R0, H100+R1, H500+R0, H500+R1, H1000+R0, and H1000+R1. The
naming convention follows that of the simulation experiment, with H indicating
the number of hidden neurons per layer and R indicating the inclusion of a
regularization term in the optimization objective. Each model underwent 50
training epochs, with the mean fitting absolute error and regularization term
values plotted over epochs (Extended Data Figure 1a and 1b). Results showed
convergence of all models with error values ranging from 0.086 to 0.088. The
R1 model optimized the regularization term to a value near 1e-5, approximating
1 for attention value. The H1000+R1 model, exhibiting the best fitting
performance, was selected for further experiments.

Figure 3: The skill acquisition patterns discovered by the proposed method.
The symbolic governing laws discovered by the model on the Lumosity dataset.
The symbol regression results for six related skills and their corresponding
complexities (CX.$<$15) are presented, including algebraic equations and their
mean absolute error (MAE). The best-fitting equation with the same number of
features is marked in red.

To further assess the deep learning regressor’s performance, we randomly
sampled data and examined the model’s absolute fitting error across different
skills, shown in Extended Data Figure 1c. The error distributions for
Attention, Flexibility, Reasoning, and Language skills were similar. However,
Memory had the lowest average error, while Math had the highest, approximately
50% more than other skills. This indicates that Math skill training may
involve more complexity or unmodeled key variables. Additionally, the
proportion of training log data for each skill relative to the total data is
depicted in Extended Data Figure 1d, showing relatively low proportions for
Math, Reasoning, and Language.

Feature importance analysis was conducted for the H1000+R1 deep learning
regressor to identify key factors influencing skill mastery. Age and education
level were found to be important for all skills, indicating their impact on
skill acquisition. Game play can improve reaction speed and attention, while
education level reflects initial abilities, particularly for Math and
Language. Practice times, represented by ”#” symbols, were crucial for skill
acquisition. Additionally, certain subskills, such as SR_RE (All the features
used are explained in Extended Data Figure 2) for Reasoning, were found to
influence Language mastery, suggesting mutualism. These findings align with
previous research [49, 50].

Analysis of symbolic governing laws is shown in Figure 3, presenting optimal
equations for six skills at various levels of complexity. Symbol regression
algorithms perform better with more complex equations, but fitting error for
skills does not decrease significantly beyond a certain complexity threshold.
We used the number of features in equations as a metric and focused on
identifying the best fitting equations for each skill with different feature
quantities (highlighted in red in Fig. 3). Skills showed good fitting
performance with one feature, indicating a core pattern. Increasing complexity
did not improve fitting significantly. Therefore, we focused on the best
fitting equation with one feature (highlighted in yellow in the tables of Fig.
3).

###  2.3 Exploring auto-discovered patterns in real-world datasets

To analyze the identified patterns, we compared them with classical Power and
Exponential laws using data from Lumosity on coding features and skill mastery
across six skills. Parameters in these equations were optimized, and goodness
of fit was evaluated using the $R^{2}$ metric [51]. Results are summarized in
Table 2, with visual representations for pattern assessment. A sample of five
learners had their skill mastery estimation curves compared with outcomes from
the deep regressor, shown in Extended Data Figure 4.

Table 2: The comparisons on the goodness of fit between the discovered laws and the classical Power law and Exponential law for six skills using the Lumosity dataset. $R^{2}$ is introduced as a measure of goodness of fit, which represents the proportion of the target variable’s variance explained by the model and ranges from 0 to 1. A higher value indicates a better fit. The model with the highest goodness of fit within the same skill is bolded, while the model ranked second is underlined. Skill | Law Type | Equation | $R^{2}\uparrow$
---|---|---|---
Attention | Power | $0.340-0.821\times\\#AT^{-0.114}$ | 0.850
Exponential | $-0.049-0.194\times exp(-0.005\times\\#AT)$ | 0.809
Discovered | $0.544-{\\#AT}^{-0.081}$ | 0.856
Flexibility | Power | $-0.376+0.231\times\\#FL^{0.139}$ | 0.841
Exponential | $0.133-0.219\times exp(-0.011\times\\#FL)$ | 0.850
Discovered | $0.133-0.219\times exp(-0.011\times\\#FL)$ | 0.850
Language | Power | $0.026+0.033\times\\#LA^{0.354}$ | 0.527
Exponential | $-0.103+0.214\times exp(0.003\times\\#LA)$ | 0.477
Discovered | $0.051\times\\#LA^{0.295}$ | 0.525
Math | Power | $-0.139+0.113\times\\#MA^{0.121}$ | 0.202
Exponential | $-0.054+0.086\times exp(0.002\times\\#MA)$ | 0.115
Discovered | $0.955-\\#SA\\_AT^{-0.024}$ | 0.278
Memory | Power | $-0.020+0.020\times\\#ME^{0.427}$ | 0.829
Exponential | $0.269-0.279\times exp(-0.006\times\\#ME)$ | 0.885
Discovered | $0.268-0.637^{0.015\times\\#ME+2.60}$ | 0.891
Reasoning | Power | $-0.133+0.376\times\\#RE^{0.159}$ | 0.605
Exponential | $-0.041+0.513\times exp(-0.003\times\\#RE)$ | 0.265
Discovered | $0.126\times log(\\#SA\\_AT)$ | 0.698
|  |  |

The results in Table 2 show some interesting phenomenon. Firstly, the Power
law and Exponential law had varying fitting performances on different skills.
The Power law fit well for Attention, Language, Math, and Reasoning skills,
while the Exponential law was better for Flexibility and Memory skills. This
discrepancy is consistent with previous research findings [4, 5]. Secondly,
our method generally outperformed the traditional laws, especially for
Attention, Flexibility, Language, and Reasoning skills. Symbolic regression
equations provided better fitting results in most cases. Minor parameter
variations caused slight differences, highlighting the effectiveness of our
approach without prior knowledge of traditional skill acquisition laws.
Lastly, our method revealed new patterns for Math and Memory skills, deviating
from conventional laws and showing improved fitting performance. This
demonstrates the potential of our method to uncover novel patterns and offer
valuable insights for researchers, emphasizing a data-driven approach.

Upon examining the equations discovered, various practice patterns emerged.
Logarithmic law was identified in some equations, a type not previously
mentioned in research. Additionally, two inverse forms of power law were
observed for Attention ($0.544-{\\#AT}^{-0.0804}$) and Language
($0.0506\times{\\#LA}^{0.295}$), with parameters that exhibit opposite
effects. The absence of general factor theory in this study suggests it may
not significantly impact skill acquisition. Skill forgetting was not clearly
observed too. The main factors influencing Language and Math were found to be
the frequency of attention selection practice rather than other practice
frequency. This highlights the importance of attention in game-based skill
improvement. However, this phenomenon may imply improving Language and Math
skills through gamified practice may pose challenges. The experimental results
suggest that our algorithm has the capability to discover the pattern forms
previously observed by humans without any prior knowledge, and at the same
time, to identify unknown pattern forms with better fitting. This fully
demonstrates the application potential and value of the proposed method, which
can effectively serve researchers in providing insights as data analysis
tools.

Table 3: The results of the goodness of fit ($R^{2}$) and the Bayesian
information criterion (BIC) of all comparative models on the Lumosity dataset
are reported. All reported results are the average of 20 independent
experiments. The overall results of the dataset and the results for each skill
are separately reported. A t-test for significance testing was conducted
between the results of the best model and the second-best model, and the
corresponding p-values are reported. During the implementation of the t-test,
two-sided and heteroscedasticity settings are deployed, and the bootstrap
technique is employed to enhance the reliability of our findings. In each
independent experiment, we randomly sampled 50% of the dataset and
subsequently repeated this iterative procedure five times. Results with
significant differences (p-value$<$0.01) are indicated with an asterisk (*).
The results of the best-performing model are highlighted in bold, while the
second-best model is marked with an underline. $\uparrow$ represents the
larger the better, while $\downarrow$ represents the smaller the better. All
BIC values are expressed in units of $10^{7}$.

$R^{2}\uparrow$ | Overall | Attention | Flexibility | Language | Math | Memory | Reasoning
---|---|---|---|---|---|---|---
AFM | 0.255 | 0.254 | 0.362 | 0.305 | 0.170 | 0.162 | 0.298
LGM | 0.370 | 0.380 | 0.523 | 0.519 | 0.297 | 0.225 | 0.426
BLM | 0.439 | 0.459 | 0.606 | 0.573 | 0.448 | 0.267 | 0.462
TLM | 0.486 | 0.503 | 0.624 | 0.613 | 0.477 | 0.301 | 0.593
TLMF | 0.471 | 0.475 | 0.614 | 0.624 | 0.483 | 0.287 | 0.592
IM | 0.503 | 0.516 | 0.633 | 0.629 | 0.517 | 0.313 | 0.599
ADM | 0.627* | 0.646* | 0.645* | 0.641* | 0.423 | 0.643* | 0.628*
p-value | 8.93E-52 | 2.12E-45 | 2.76E-05 | 5.59E-08 | 1.86E-39 | 2.14E-50 | 8.43E-20
$BIC\downarrow$ | Overall | Attention | Flexibility | Language | Math | Memory | Reasoning
AFM | -3.814 | -3.951 | -3.983 | -4.017 | -3.486 | -3.601 | -4.112
LGM | -4.020 | -4.179 | -4.340 | -4.469 | -3.691 | -3.697 | -4.361
BLM | -4.217 | -4.400 | -4.624 | -4.691 | -4.058 | -3.812 | -4.505
TLM | -4.264 | -4.440 | -4.698 | -4.821 | -4.059 | -3.815 | -4.760
TLMF | -4.174 | -4.321 | -4.541 | -4.684 | -3.986 | -3.736 | -4.719
IM | -4.242 | -4.413 | -4.706 | -4.725 | -4.080 | -3.771 | -4.741
ADM | -4.779* | -4.979* | -4.820* | -4.912* | -4.135* | -4.771* | -4.882*
p-value | 5.59E-63 | 2.83E-46 | 7.59E-24 | 1.13E-15 | 7.97E-15 | 2.44E-69 | 3.08E-25
|  |  |  |  |  |  |

Our study model focuses on analyzing learners’ performance throughout the
learning process, a key research area. To evaluate our model’s performance, we
compare it with various recent learning models using large-scale data. The
comparative models include baseline learning model (BLM), two-timescale
learning model (TLM), two-timescale learning model with forgetting (TLMF), and
interactive model (IM) from [52]. In addition, we consider the classical
additive factor model (AFM) [53] and the latest learning growth model (LGM)
[54] for learner growth modeling. Our model, the Auto-discovered model (ADM),
combines discovered laws with the 3-parameter IRT model for predicting
learning performance. Unlike the comparison models that have unique parameters
for each learner, our model is a general learning model where all learners
share the same parameter set for learning patterns. We use two evaluation
metrics: $R^{2}$ index for fitting accuracy and Bayesian Information Criterion
(BIC) for model selection based on data likelihood and parameter count [55,
56]. Parameter estimation was performed on the Lumosity dataset, and the
optimal model results are summarized in Table 3.

Our model outperforms most comparative models in terms of goodness of fit. It
is comparable to the second-best model in Flexibility, Language, and Reasoning
skills, but excels in Attention, Memory, and overall data performance. While
slightly underperforming in Math skill compared to a specific model, our model
shows a significant advantage in BIC comparison with six other models due to
its smaller parameter size and higher fit level. These results indicate that
the auto-discovered symbolic patterns have certain advantages in terms of
simplicity and data fitting. This may suggest that in the realm of pattern
discovery, AI algorithms may possess certain advantages over humans and have
the potential to become important tools in related research.

##  3 Discussion

This paper presents a new method for researchers to identify hidden patterns
in data, offering evidence and insights for related studies. A deep regressor
is integrated into the model development process to streamline the discovery
of symbolic models. The deep regressor, incorporating deep learning
architecture (Transformer-like structure) and the IRT model, may include
researchers’ prior assumptions. Empirical findings show that using deep
learning regressors allows for accurate computation of latent states and
decreases the search space of symbolic regression. However, the inclusion of
priors does constrain the model’s generalizability to some extent.

This study used a heuristic model selection approach based on researchers’
understanding to establish model selection in symbolic regression. While no
universal method exists, exploring universal and rational model selection
methods is crucial. In addition, the methodology is limited to continuous
feedback scenarios and does not cover discrete feedback. Our future
exploration will focus on this, noting that discrete feedback may introduce
more noise and make it more difficult.

Finally, this paper examines skill acquisition in psychology research using
big data and artificial intelligence. By analyzing core patterns through
large-scale NODS, we provide new evidence and methods. However, like other
data-driven studies, our research may be influenced by data noise and biases,
potentially affecting the generalizability of findings. We recognize the
challenges in uncovering learning patterns and acknowledge that our study is
based on observations from the Lumosity dataset. Further research is needed to
validate generalizability and underlying mechanisms.

##  4 Method

###  4.1 Deep learning regression for training log data

####  4.1.1 Problem formulation

Training log data ($\mathcal{D}$) includes valuable information such as the
user set $\mathcal{U}=\\{u_{1},u_{2},…,u_{m}\\}$, the exercise set
$\mathcal{E}=\\{e_{1},e_{2},…,e_{n}\\}$, and the skill set
$\mathcal{S}=\\{s_{1},s_{2},…,s_{k}\\}$. A user’s training history, denoted as
$\mathcal{P}_{u_{i}}=\\{p_{u_{i}}^{1},p_{u_{i}}^{2},…,p_{u_{i}}^{l}\\}$, is a
sequence of exercises, where $u_{i}$ refers to the user, $p_{u_{i}}^{l}$ is
the $l$-th exercise of $u_{i}$ and $p_{u_{i}}^{l}\in\mathcal{E}$. The sequence
$\mathcal{P}_{u_{i}}$ is obtained by sorting the training history of user
$u_{i}$ by time. The corresponding grade for $\mathcal{P}_{u_{i}}$ is usually
normalized and can be denoted as
$G_{u_{i}}=\\{g_{u_{i}}^{1},g_{u_{i}}^{2},…,g_{u_{i}}^{l}\\}$, where
$g_{u_{i}}^{l}$ is the grade for the $l$-th exercise of user $u_{i}$ and
$g_{(}u_{i})^{l}\in[0,1]$. The index matrix indicating the relationship
between the exercises and skills is usually called the Q-matrix where
$Q\in\mathbb{R}^{n\times k}$. To simplify our notation system, we denote
$Q[p_{u_{i}}^{l}]$ as the skill index for exercise $p_{u_{i}}^{l}$, with the
skill index being represented by one-hot encoding on skills, where
$Q[p_{u_{i}}^{l}]\in\mathbb{R}^{k}$.

Exercise scores can generally reflect users’ level of skill mastery, and
exploring the evolution of these scores can reveal changes in knowledge
acquisition. By correlating these scores with practice behavior, the skill
acquisition patterns of learners can be determined. Observing the skill
acquisition patterns of numerous learners may reveal underlying governing laws
that involve psychological activities like practice, forgetting, and skill
transfer. An autoregressive self-supervised paradigm is introduced to capture
the core patterns of data. Its main objective is to construct a function f
which predicts the score of any training behavior $p_{u_{i}}^{l}$ using its
previous sequence of practice sessions, namely,
$f({p_{u_{i}}^{1},p_{u_{i}}^{2},…,p_{u_{i}}^{l-1}})\approx g_{u_{i}}^{l}$. If
$f$ can predict equivalent scores accurately, it is believed that there exist
core patterns or rules about skill acquisition within $f$. To construct $f$, a
deep learning model have devised which is inspired by the Transformer
architecture. The entire score prediction process is divided into three
primary stages: feature encoding, skill mastery inference, and prediction.

####  4.1.2 Feature encoding

The core task of the Feature Encoding module is to extract the features of the
historical exercise sequence. Our method involves characterizing the static
features of each training behavior, building a model to predict the
contribution level of each practice session to corresponding skill learning,
weighting the static feature sequence, and obtaining the feature encoding of
the practice behavior sequence.

Our study aims to develop the user’s practice behavior from three dimensions,
namely, user attributes, exercise characteristics, and scheduling information.
User attributes typically consist of personal details such as age, occupation,
and education, which can offer valuable information about motivation, initial
skill level, and learning speed that could impact skill learning. We will
denote $u_{i}$’s user attributes as $\textbf{{U}}[u_{i}]$. Exercise
characteristics aim at conveying relevant aspects of the exercise, notably the
relationship between the exercise and the skill, and the kind of stimulus
incorporated in the exercise. We will use the notation
$\textbf{{E}}[p_{u_{i}}^{l-1}]$ to denote the exercise characteristics of
exercise $p_{u_{i}}^{l-1}$. Scheduling information comprises details about the
user during the exercise process, such as the number of exercises intended to
learn a skill and the time between successive exercises. The scheduling
information pertains to the data about the user’s actions and behavior during
the ongoing exercise, which is subsequently used for prediction. We will refer
to the scheduling information of user $u_{i}$ during exercise $p_{u_{i}}^{l}$
by using the notation $\textbf{{S}}[p_{u_{i}}^{l}]$. Typically, features can
take on two main forms, namely, continuous and discrete. For discrete
features, we employ one-hot encoding for encoding purposes, while for
continuous variables, we directly use their continuous values as feature
values. In the end, the static features of a user’s practice behavior can be
documented as
$\textbf{{SF}}[p_{u_{i}}^{l-1}]=\textbf{{U}}[u_{i}]\oplus\textbf{{E}}[p_{u_{i}}^{l-1}]\oplus\textbf{{S}}[p_{u_{i}}^{l}]$,
where $\oplus$ means the concatenation operator.

Since different training behaviors may have distinct impacts on skill
learning, we employ neural networks to determine the significance of specific
behaviors for skill acquisition. First, to normalize the static features and
expedite model training while enhancing stability, we perform batch
normalization, given that the distinct dimensions of the static features have
dissimilar value ranges. Second, we employ a classical multilayer perceptron
(MLP) to model normalized features, where the perception machine varies
depending on the skill but shares the parameters for the same perception
machine. Consequently, for a user’s practice behavior $p_{u_{i}}^{l-1}$
related to skill $s_{k}$, we can quantify its importance as

$\alpha_{p_{u_{i}}^{l-1}}^{s_{k}}=\mathrm{MLP}_{att}\left(\mathrm{BN}\left(\textbf{{SF}}[p_{u_{i}}^{l-1}]\right),\bm{\Theta}_{att}^{s_{k}}\right)$
(1)

where MLP and BN are abbreviations for multilayer perceptron and batch
normalization, respectively. We will provide detailed information on the
methodology in the ”Method” section. $\bm{\Theta}_{att}^{s_{k}}$ refers to the
set of MLP parameters.

The user attributes, which make up one of the three static features, refer to
time-invariant variables and do not necessitate temporal feature fusion. Since
scheduling information inherently encompasses temporal information during
encoding, the temporal encoding of user exercise features becomes the primary
focus of sequence encoding. The encoded exercise feature of $u_{i}$’s exercise
sequence ${p_{u_{i}}^{1},p_{u_{i}}^{2},…,p_{u_{i}}^{l-1}}$ on skill $s_{k}$
can be represented by the following equation:

$\textit{{EE}}^{s_{k}}[\\{p_{u_{i}}^{1},p_{u_{i}}^{2},…,p_{u_{i}}^{l-1}\\}]=\sum_{j=1}^{l-1}\alpha_{p_{u_{i}}^{j}}^{s_{k}}\cdot\textit{{E}}[p_{u_{i}}^{j}]$
(2)

The encoded behavioral sequence feature for skill $s_{k}$ can be expressed as:
$\bm{\omega}^{s_{k}}[{p_{u_{i}}^{1},p_{u_{i}}^{2},…,p_{u_{i}}^{l-1}}]=\textit{{U}}[u_{i}]\oplus\textit{{EE}}^{s_{k}}[{p_{u_{i}}^{1},p_{u_{i}}^{2},…,p_{u_{i}}^{l-1}}]\oplus\textit{{S}}[p_{u_{i}}^{l}]$.

####  4.1.3 Mastery inference

Mastery inference focuses on constructing a mapping function (mastery inferer)
that maps encoded features of a behavioral sequence to the level of skill
mastery, which is the central component of the model. As the form of the
function is ambiguous, we propose utilizing a neural network model to fit the
mapping function and leverage its formidable non-linear fitting ability to
precisely estimate the skill mastery variable. Additionally, score prediction
necessitates employing another mapping function that translates the skill
mastery state into the final score. To this end, we are going to incorporate
the classic cognitive diagnostic model (item response theory [38, 39]) as the
mapping function.

The mastery inferer for each skill is independent. Therefore, we set up a
group of neural network models with the same structure but different
parameters to estimate the level of mastery for each skill. When user $u_{i}$
practices $p_{u_{i}}^{l}$, the level of mastery for skill $s_{k}$ can be
denoted as $\phi_{p_{u_{i}}^{l}}^{s_{k}}$, and it can be expressed as:

$\phi_{p_{u_{i}}^{l}}^{s_{k}}=\psi_{u_{i}}^{s_{k}}+\mathrm{MLP}_{msty}(\mathrm{BN}(\bm{\omega}^{s_{k}}[{p_{u_{i}}^{1},p_{u_{i}}^{2},…,p_{u_{i}}^{l-1}}]),\bm{\Theta}_{msty}^{s_{k}})$
(3)

where MLP and BN are short for multilayer perceptron and batch normalization,
respectively. We represent the level of mastery of each skill of user $u_{i}$
by $\bm{\Phi}_{p_{u_{i}}^{l}}$, where
$\bm{\Phi}_{p_{u_{i}}^{l}}=[\phi_{p_{u_{i}}^{l}}^{s_{1}},\phi_{p_{u_{i}}^{l}}^{s_{2}},…,\phi_{p_{u_{i}}^{l}}^{s_{k}}]$
and $s_{1},s_{2},…,s_{k}\in\mathcal{S}$. $\psi_{u_{i}}^{s_{k}}$ denotes the
initial proficiency level of skill $s_{k}$ possessed by user $u_{i}$ before
training.

####  4.1.4 Score prediction

Generally, a user’s score on an exercise is believed to be correlated with
their level of skill mastery, which has been extensively studied by scholars
who have developed numerous cognitive diagnostic models. In this paper, we
propose using the item response theory (IRT) model [38, 39] as the scoring
prediction function. Specifically, we introduce a three-parameter IRT model
that models user mastery, exercise difficulty, discrimination, and guessing,
enabling us to better estimate exercise scores using user mastery information.
Its specific form can be expressed as:

$\hat{g}_{u_{i}}^{l}=\gamma_{p_{u_{i}}^{l}}+\frac{1-\gamma_{p_{u_{i}}^{l}}}{1+exp\left(-d\cdot\eta_{p_{u_{i}}^{l}}\cdot\left(\bm{\Phi}_{p_{u_{i}}^{l}}-\beta_{p_{u_{i}}^{l}}\cdot
Q[p_{u_{i}}^{l}]\right)\right)}$ (4)

Here, $\eta_{p_{u_{i}}^{l}}$, $\beta_{p_{u_{i}}^{l}}$ and
$\gamma_{p_{u_{i}}^{l}}$ respectively denote the discrimination, difficulty,
and guessing of the exercise $p_{u_{i}}^{l}$. $Q[p_{u_{i}}^{l}]$ represents
the index of the exercise in the skill set, while $\hat{g}_{u_{i}}^{l}$
represents the predicted score for practice $p_{u_{i}}^{l}$.

The neural network model used in this article is implemented with the open-
source deep learning library Pytorch[57].

####  4.1.5 Model learning

In the preceding section, we denoted $\hat{g}_{u_{i}}^{l}$ as the forecasted
score for exercise $p_{u_{i}}^{l}$, obtained through the feature encoding,
transformation and forecasting process from the learning sequence
$\\{p_{u_{i}}^{1},p_{u_{i}}^{2},…,p_{u_{i}}^{l-1}\\}$. Firstly, since scoring
prediction is a typical regression task, we introduce mean absolute error
(MAE) loss function as our optimization objective. MAE is known to have better
resistance against outliers as compared to the more commonly used mean squared
error (MSE) loss. Our main objective is to extract knowledge that is
universally applicable and generalizable from the data. To accomplish this
objective, we endeavor to decrease the effect of score values that are unusual
or stand out on the model. Secondly, we intend to adopt a self-regressive
learning paradigm. Therefore, the established loss function needs to be
computed and accumulated on all the practice session data of every learner in
the dataset.

Considering the significance of user training behavior, shown in Eq. (1), we
seek to incorporate a constraint that ensures smoothness. The smoothness
constraint serves to restrict the importance values to a predefined range. If
the model is capable of constraining all the important values to a constant
value without compromising precision, it can diminish the overall complexity
of the model and alleviate the difficulty of interpreting it.

The learning objective function of the entire model can be represented as
follows.

$\mathcal{L}=\underbrace{\frac{1}{\left|\mathcal{D}\right|}\sum_{u_{i}\in\mathcal{U}}\sum_{p_{u_{i}}^{l}\in\mathcal{P}_{u_{i}}}\left|\hat{g}_{u_{i}}^{l}-g_{u_{i}}^{l}\right|}_{\text{MAE
loss}}+\lambda\cdot\underbrace{\frac{1}{\left|\mathcal{D}\right|}\sum_{u_{i}\in\mathcal{U}}\sum_{p_{u_{i}}^{l}\in\mathcal{P}_{u_{i}}}\frac{1}{\left|\mathcal{S}\right|}\sum_{s_{k}\in\mathcal{S}}\left(\alpha_{p_{u_{i}}^{l}}^{s_{k}}-1\right)^{2}}_{\text{regularization
term}}$ (5)

The objective function above contains two main components. The first component
is the fidelity term, which describes the discrepancy between the predicted
data and the actual data, represented by the MAE loss. The second component is
the regularization term, namely, the smoothness constraint on the importance
of training behavior, which is confined to a constant value near 1. $\lambda$
is a balancing factor used to balance the importance of the fidelity term and
the constraint term. $\left|\bullet\right|$ means the cardinality of a set. To
be more specific, $\left|\mathcal{D}\right|$ represents the number of training
records in the dataset, where
$\left|\mathcal{D}\right|=\left|\mathcal{U}\right|\times\left|\mathcal{P}_{u_{i}}\right|$.
Since the entire model is differentiable, it is capable of conducting
efficient learning of parameters through gradient-based iterative optimization
algorithms, for instance, stochastic gradient descent (SGD), and adaptive
moment estimation (Adam) [58].

###  4.2 Symbolic law extraction from deep learning regressor

####  4.2.1 Symbolic distillation

Formally, let T be the neural network model to be analyzed, and let S be the
symbolic regression model. The core idea of symbolic distillation is to ensure
that for any input X, $T(X)\approx S(X)$. Based on this, we can define the
optimization objective for the symbolic regression model as follows

$\pi=\mathop{\arg\min}\limits_{\pi}\mathbb{E}_{x\sim
X}\left|T(x,\bm{\Theta})-S(x,\pi)\right|$ (6)

where $\pi$ represents the algebraic equation learned by the symbolic
regression model, x represents the sampled input feature, $\bm{\Theta}$
represents all the parameters in the neural network model, which could be
considered constants after training.

Specifically, in this article, we construct $X$ using the input information
samples from the training data. In order to reduce the computational cost of
the symbolic regression model, we sample the input samples from the entire
dataset to obtain $X$, where $X\in\mathbb{R}^{s\times k}$. Here, $s$
represents the sampling size of the samples, which is a hyperparameter set
manually, while $K$ represents the feature dimension of the neural network
input information, i.e. the dimension of $\textbf{{SF}}[p_{u_{i}}^{l-1}]$ or
$\bm{\omega}^{s_{k}}[{p_{u_{i}}^{1},p_{u_{i}}^{2},…,p_{u_{i}}^{l-1}}]$, as
shown in Eqs. (1) and (3).

In our study, features have a high dimensionality. Symbolic regression is a
typical combinatorial optimization problem. When the number of variables
increases, the solution space grows exponentially, diminishing the efficiency
and stability of the symbolic regression model. As an exploratory study, our
objective is to identify core patterns and essential factors. Consequentially,
we use a trained neural network model and an interpretable analysis method,
based on gradient feature importance estimation, to screen features, select
the most important features, and use them to execute the symbolic regression
to identify key patterns. Specifically, the trained neural network model and
the gradient feature importance estimation method are incorporated to
calculate the importance of the $K$ features, among which the k most crucial
ones are chosen to form the input feature $C(X)$. Here,
$C(X)\in\mathbb{R}^{s\times k}$, and $k\ll K$. We will describe the detailed
gradient-based feature importance estimation method calculation in the
subsequent Method section. In addition, the neural network modules in the deep
learning regressor (attention-based encoders and mastery inferers for each
skill, shown in Fig. 1b) consist of two operators, namely BN and MLP. To avoid
feature scaling impact on importance estimation, we restrict our feature
importance estimation to the output features of the BN operators.

Thus, the objective of symbolic regression is formulated here as follows:

$\pi=\mathop{\arg\min}\limits_{\pi}\mathbb{E}_{x\sim
X}\left|T(x,\bm{\Theta})-S(C(x),\pi)\right|$ (7)

In order to avoid the over-influence of errors on the model, mean absolute
error is introduced as the objective function for the symbolic regression
model. Genetic algorithms are the primary approach employed to solve the
symbolic regression model, symbolic regression libraries such as Eureqa, PySR,
and geppy offer additional means of solving the model. The symbol regression
algorithm used in this article is implemented with the open-source symbol
regression library PySR.

####  4.2.2 Governing laws extraction

In this section, we are going to depict how we convert the prediction process
of trained deep regressor into a symbolic law using the symbolic distillation
method. We express the neural network models in each module of the deep
regressor as a symbolic law and ultimately combine them to obtain the symbolic
representation of the entire deep regressor. As the attention-based encoder
and mastery inferer for each skill are independent, it is feasible to perform
symbolic distillation on these components in parallel. Furthermore, since the
attention-based encoder and mastery inferer are coupled and exhibit sequential
order, the expansion process of the analysis follows a similar order. First,
for the attention-based encoder, a symbolic representation of
$\alpha_{p_{u_{i}}^{l}}^{s_{k}}$ is obtained by symbolically analyzing
equation (1), merged with feature encoding (equation (2)), producing a
symbolic representation of the encoded features, i.e.,
$\bm{\omega}^{s_{k}}[{p_{u_{i}}^{1},p_{u_{i}}^{2},…,p_{u_{i}}^{l-1}}]$. If the
model includes a smoothness constraint for attention value,
$\alpha_{p_{u_{i}}^{l}}^{s_{k}}$ can be approximated to 1 when the smoothness
component’s value is less than an extremely small threshold (e.g., 1e-4) to
reduce the complexity of the analysis equation. Next, the mastery inferer
undergoes a symbolic distillation, followed by the integration of symbolic
representation with the encoded features, reorganized to render the final
symbolic governing laws.

###  4.3 Data generation method and model parameter setting in the simulated
data experiment

To closely simulate real exercise patterns, three commonly accepted laws of
skill acquisition were considered: Power law ($\beta\cdot{N}^{\alpha}$ ),
Exponential law ($\beta\cdot\exp(\alpha\cdot N)$),and Linear law
($\beta\cdot\alpha\cdot N$). The concept of mutualism, indicating the positive
impact of exercising on multiple skills, was also integrated, where mastering
skill A could enhance proficiency in skill B. Additionally, a forgetting
factor and proficiency degradation model based on exercise interval were
introduced. Three sets of simulated skill mastering laws were established,
detailed in Table 1.

To generate comprehensive simulated training log data, learners and items were
first simulated with randomly initialized parameters. Subsequently, each
learner was assigned a random exercise sequence. By considering the exercise
process and pre- determined learning patterns, skill mastery for each exercise
was calculated for the learners. The IRT model, along with item parameters,
was utilized to compute scores for each exercise. Gaussian noise
($\varepsilon\sim N\left(0,\delta\right)$) was introduced during the
calculation of practice scores to assess the algorithm’s robustness to noise
and create data that mirrors real-world observations [41, 42].

The simulated training log data generation can be divided into three main
stages: learners and exercises generation, learning path generation, and
cognitive state and learning feedback generation, shown in Extended Data
Figure 2. In the learner simulation, a total of 50 independent virtual
learners are simulated. In the practice simulation, 20 simulated test
questions are generated for each of skill 1 and skill 2, each test question
containing three coefficients: difficulty, guessing, and discrimination, all
of which are random variables uniformly distributed between 0 and 1. Learning
path generation involves generating the practice sequence for each simulated
learner. To simplify the simulation environment setup process, the
corresponding learning sequence is generated by randomly selecting questions,
with each simulated learner’s learning sequence length being 20. Combining the
learner’s learning path with the predetermined skill acquisition function (in
Table 1), the learner’s skill mastery can be calculated at the beginning of
each practice. Finally, combining the parameters of the test questions, the
learner’s test score is calculated using the three-parameter IRT model (Eq. 4)
and Gaussian random noise with a mean of 0 and a variance of $\delta$ is
added.

Under the three settings, the parameters of the Linear law are $\alpha$=1,
$\beta$=0.3, $\gamma$=0.1, the parameters of the Exponential law are
$\alpha$=0.1, $\beta$=1, $\gamma$=0.1, and the parameters of the Power law are
$\alpha$=0.8, $\beta$=1, $\gamma$=0.2. In the three settings considered, the
mutualism factor $\mu$ is set to 0.5. In the simulation experiment, four
different levels of noise are selected as test standards, namely
$\delta\in\left\\{0,0.001,0.01,0.1\right\\}$. In the deep learning regressor,
the MLP models in feature encoding and mastery inference are both three-
layered (two hidden layers and one output layer), with 1000 neurons in each
hidden layer and Leaky ReLU as the activation function. The model optimizer is
set to Adam, and the learning rate is 1e-3. In the symbolic regression model,
the number of iterations is set to 2000, the population size is 100, and the
maximum function length is 15. The preset operators include ”+”, ”-”,
”$\times$”, ”pow”, ”exp”, ”log”, and ”sin”.

###  4.4 Data preprocessing method and model parameter setting in the real-
world data experiment

Lumosity (www.lumosity.com) is an online platform that offers brain training
games and exercises designed to improve cognitive functions such as memory,
attention, flexibility, speed of processing, and problem-solving. The platform
was created by Lumos Labs and is based on the concept of neuroplasticity [48],
which is the brain’s ability to change and adapt throughout a person’s
lifetime. Lumosity has been used by millions of people worldwide to enhance
their cognitive abilities and improve their overall mental fitness. In the
platform, users improve their proficiency in corresponding skills by playing
various games, which can be considered as cognitive training or practice as
mentioned earlier. By analyzing the extensive practice sessions of numerous
users, we aim to uncover the behavioral patterns of skill acquisition hidden
behind the data.

The dataset used in this study was shared by [42]. We performed data screening
and preprocessing based on the raw data of log data containing 50 million
exercises provided by them. Finally, we selected 6143 users as research sample
and extracted their first 2000 practice logs as the dataset. Secondly, based
on the data labeling foundation of [42], we identified 37 features as
behavioral features, including all U, E, and S type features, and the detailed
feature encoding, description, type, and abbreviation can be found in Extended
Data Figure 3. In the dataset, there are a total of 85 games, and each game is
labeled with its corresponding skills and subskills. Skills includes
Attention, Flexibility, Language, Math, Memory, and Reasoning, which are the
dimensions we considered when estimating users’ skill mastery. Subskill
represents more fine-grained skill types, which are included in skill, and the
relationship between them can be found in Extended Data Figure 3. In the
experiment, Subskill only serves as an input feature of the model and is not
the prediction target.

In this paper, we utilized raw data provided by [42]. To ensure the quality of
our data, we filtered out games with fewer contact times and related skills.
We then selected the first 2000 practice records of users with more than 2000
practice times as our training data. To avoid any issues with abnormal scores,
we normalized the score data of each game for the selected training data.
Specifically, we took the top 10% of scores as the full score and performed
maximum-minimum normalization on the data of each game. All scores were then
uniformly scaled to a range of [0.01,0.8] to prevent the Inf outlier problem
caused by the cognitive diagnostic model.

The deep learning regressor employs three-layered MLP models for feature
encoding and mastery inference, each consisting of two hidden layers and one
output layer. The activation function used is Leaky ReLU, and each hidden
layer comprises 1000 neurons. The model optimizer is set to Adam, with a
learning rate of 1e-4. The symbolic regression model is configured with a
population size of 100, a maximum function length of 15, and a fixed set of
operators, including ”+”, ”-”, ”*”, ”pow”, ”exp”, ”log”, and ”sin”. The model
is trained for 2000 iterations to achieve optimal performance. To increase the
interpretability of governing laws, operators other than ”+”, ”-”, and ”*” are
not allowed to be nested. In the symbolic regression model, a total of 2000
training log samples are selected randomly, and the most important 8 features
are screened through the feature importance analysis method. To increase the
interpretability of governing laws, operators other than ”+”, ”-”, and
”$\times$” are not allowed to be nested.

Data availability All data generated or analyzed during this study are
available. It can be found in https://github.com/ccnu-mathits/ADM. And it is
also available via Zenodo at https://doi.org/10.5281/zenodo.10938670 (ref.
[59]). Source data are provided with this paper.

Code availability The source code of this study is freely available on Github
at https://github.com/ccnu-mathits/ADM. And it is also available via Zenodo at
https://doi.org/10.5281/zenodo.10938670 (ref. [59]).

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Extended Data Fig. 1: Partial results on the Lumosity dataset. More
specifically, (a) the change curve of mean fitting absolute error during the
iteration process of the deep learning regressor; (b) the change curve of the
value of the regularization term during the iteration processes; (c) the
prediction error distribution of 1000 randomly selected records from the
trained H1000+R1 model. The box plot displays the interquartile range (IQR)
with the median line, while the whiskers extend to the minimum and maximum
values or a multiple of the IQR from the quartiles. Outliers are depicted as
individual points beyond the whiskers; (d) the proportion of the number of
practice times for each skill to the total number of practice times; (e) the
distribution of feature importance for each skill in H1000+R1. Extended Data
Fig. 2: Schematic representation of the simulation experiment. (a) Flowchart
depicting the process of generating simulated data and the data format. (b)
Schematic diagram illustrating the evaluation procedure of the simulation
experiment. (c) Sample representation of the simulated data. Extended Data
Fig. 3: The feature set utilized in the real-world dataset (Lumosity)
experiment, which comprises three components: user (U), exercise (E), and
scheduling (S). The features consist of two categories: discrete and
continuous. Discrete features are encoded through one-hot encoding, while
continuous features are encoded using real values. Game features are two-
dimensional, consisting of skill and subskill, both of which are discrete
features and characterize the relationship between the game and cognitive
skills. Subskill is a subdivision feature of skill. Extended Data Fig. 4:
Analysis of the difference between the mastery level of skills calculated by
the deep learning regressor and the discovered governing laws. Here, we
present the changes in skill proficiency at each time point during 2000
practice sessions for five learners. The blue line represents the mastery
curve computed by the deep regressor, while the red line represents the curve
computed by the discovered governing law. The governing law is the same for
all learners, but due to differences in their choices and order of practice,
there are variations in the independent variable of the governing law for each
learner.


# On compact operators between lattice normed spaces

Youssef Azouzi and Mohamed Amine Ben Amor
Research Laboratory of Algebra, Topology, Arithmetic, and Order
Department of Mathematics
Faculty of Mathematical, Physical and Natural Sciences of Tunis
Tunis-El Manar University, 2092-El Manar, Tunisia

###### Abstract

In this paper we continue the study of compact-like operators in lattice
normed spaces started recently by Aydin, Emelyanov, Erkurşun Özcand and
Marabeh. We show among others, that every $p$-compact operator between lattice
normed spaces is $p$-bounded. The paper contains answers of almost all
questions asked by these authors.

##  1 Introduction

In [3], the authors introduced a new notion of compact operators in Lattice-
normed spaces and studied some of their properties. These operators act on
spaces equipped with vector valued norms taking their values in some vector
lattices. Recall that an operator from a normed space $X$ to a normed space
$Y$ is said to be compact if the image of every norm bounded sequence
$\left(x_{n}\right)$ in $X$ has a norm convergent subsequence. This notion has
been generalized in the setting of lattice normed spaces giving rise to two
new notions: sequentially $p$-compactness and $p$-compactness ($p$ referred to
the vector valued norm). Notice that these notions coincide in the classical
case of Banach spaces. In general setting with vector lattice valued norms
boundedness and convergence are considered with respect to these ‘norms’. Also
as notions of relatively uniform convergence and almost order boundedness have
been generalized, new properties for the operator are considered like
semicompactness. Recall that if $(E,p,V)$ and $(F,q,W)$ are Lattice-normed
spaces and $T$ is a linear operator from $E$ to $F$, then $T$ is said to be
$p$-compact (respectively, $rp$-compact) if for every $p$-bounded net
$\left(x_{\alpha}\right)$ in $E$, there is a subnet
$\left(T{x}_{\varphi\left(\beta\right)}\right)$ that $p$-converges
(respectively, $rp$-converges) to some $y\in F$. The operator is said to be
sequentially $p$-compact if nets and subnets are replaced by sequences and
subequences above. In this paper we prove some new results in this direction.
Namely we show that every $p$-compact operator is $p$-bounded. As a
consequence we get that every $rp$-compact is $p$-bounded. Also we give an
example of sequentially $p$-compact operator which fails to be $p$-bounded. As
a consequence we deduce that a sequentially $p$-compact need not be
$p$-compact. In fact these two notions are totally independent. Example of
$p$-compact operators that fail to be sequentially $p$-compact is given. As
mentioned above the study of $p$-compact operators between lattice normed
spaces was started in [3]. That paper contains several new results but also
some open questions. Almost all these questions will be answered in our paper.

##  2 Preliminaries

The goal of this section is to introduce some basic definitions and facts. For
general informations on vector lattices, Banach spaces and lattice-normed
spaces, the reader is referred to the classical monographs [1] and [6].

Consider a vector space $E$ and a real Archimedean vector lattice $V$. A map
$p:E\rightarrow V$ is called a vector norm if it satisfies the following
axioms:

  1. 1)

$p(x)\geq 0;$ $p(x)=0\Leftrightarrow x=0$; $(x\in E)$.

  2. 2)

$p(x_{1}+x_{2})\leq p(x_{1})+p(x_{2});\,\,(x_{1},x_{2}\in E)$.

  3. 3)

$p(\lambda x)=|\lambda|p(x);\,\,(\lambda\in\mathbb{R},\,x\in E)$.

A triple $(E,p,V)$ is a lattice-normed space if $p(.)$ is a $V$-valued vector
norm in the vector space $E$. When the space $E$ is itself a vector lattice
the triple $\left(E,p,V\right)$ is called a lattice-normed vector lattice. A
set $M\subset E$ is called $p$-bounded  if $p\left(M\right)\subset[-e,e]$ for
some $e\in V_{+}.$ A subset $M$ of a lattice-normed vector lattice
$\left(E,p,V\right)$ is called $p$-almost order bounded if, for any $w\in
V_{+}$, there is $x_{w}\in E_{+}$ such that
$p((|x|-x_{w})^{+})=p(|x|-x_{w}\wedge|x|)\leq w$ for any $x\in M$.

Let $(x_{\alpha})_{\alpha\in\Delta}$ be a net in a lattice-normed space
$(E,p,V)$. We say that $(x_{\alpha})_{\alpha\in\Delta}$ is $p$-convergent to
an element $x\in E$ and write $x_{\alpha}\overset{p}{\longrightarrow}x,$ if
there exists a decreasing net $(e_{\gamma})_{\gamma\in\Gamma}$ in $V$ such
that $\inf_{\gamma\in\Gamma}(e_{\gamma})=0$ and for every $\gamma\in\Gamma$
there is an index $\alpha(\gamma)\in\Delta$ such that $p(v-v_{\alpha})\leq
e_{\gamma}$ for all $\alpha\geq\alpha(\gamma)$. Notice that if $V$ is Dedekind
complete, the dominating net $\left(e_{\gamma}\right)$ may be chosen over the
same index set as the original net. We say that $\left(x_{\alpha}\right)$ is
$p$-unbounded convergent to $x$ (or for short, $up$-convergent to $x$) if
$|x_{\alpha}-x|\wedge u\overset{p}{\longrightarrow}0$ for all $u\in V_{+}$. It
is said to be relatively uniformly $p$-convergent to $x\in X$ (written as,
$x_{\alpha}\overset{rp}{\longrightarrow}x)$ if there is $e\in E_{+}$ such that
for any $\varepsilon>0$, there is $\alpha_{\varepsilon}$ satisfying
$p(x_{\alpha}-x)\leq\varepsilon e$ for all $\alpha\geq\alpha_{\varepsilon}$.

When $E=V$ and $p$ is the absolute value in $E,$ the $p$-convergence is the
order convergence, the $up$-convergence is the unbounded order convergence,
and the $rp$-convergence is the relatively uniformly convergence. We refer to
[5] and [4] for the basic facts about nets in topological spaces and vector
lattices respectively. We will use [6, 8] as unique source for unexplained
terminology in Lattice-Normed Spaces. Since the most part of this paper is
devoted to answer several open questions in [3], the reader must have that
paper handy, from which we recall some definitions.

###### Definition 1

Let $X,\ Y$ be two lattice-normed spaces and $T\in L(X,Y)$. Then

  1. 1.

$T$ is called $p$-compact if, for any $p$-bounded net
$\left(x_{\alpha}\right)$ in $X$, there is a subnet $x_{\alpha_{\beta}}$ such
that $Tx_{\alpha_{\beta}}\overset{p}{\longrightarrow}y$ in $Y$ for some $y\in
Y$.

  2. 2.

$T$ is called sequentially $p$-compact if, for any $p$-bounded sequence
$x_{n}$ in $X$, there is a subsequence $\left(x_{n_{k}}\right)$ such that
$Tx_{n_{k}}\overset{p}{\longrightarrow}y$ in $Y$ for some $y\in Y$.

  3. 3.

$T$ is called $p$-semicompact if, for any $p$-bounded set $A$ in $X$, the set
$T(A)$ is $p$-almost order bounded in $Y$.

##  3 $p$-compact operators are $p$-bounded

It is well known that compact operators between Banach spaces are bounded.
This result remains valid for general situation of $p$-compact operators as it
will be shown in our first result, which answers positively Question 2 in [3].

###### Theorem 2

Every $p$-compact operator between two Lattice-normed spaces is $p$-bounded.

Proof. Assume, by contradiction, that there exists a $p$-compact operator
$T:(E,p,V)\longrightarrow(F,q,W)$ which is not $p$-bounded. Then there exists
a $p$-bounded subset $A$ of $E$ such that $T\left(A\right)$ is not
$q$-bounded. So, for every $u\in W^{+}$ there exists some $x_{u}\in A$
satisfying $q(T(x_{u}))\not\leq u$. Since the net $(x_{u})_{u\in W^{+}}$ is
$p$-bounded there is a subnet
$\left(y_{v}=x_{\varphi\left(v\right)}\right)_{v\in\Gamma}$ and an element
$f\in F$ such that $\left(Ty_{v}\right)$ $\overset{q}{\longrightarrow}f$. It
follows that the net $\left(Ty_{v}\right)$ has a $q$-bounded tail, which means
that for some $v_{0}$ in $\Gamma$ and some $w\in W^{+}$ we have,

$q\left(Tx_{\varphi\left(v\right)}\right)\leq w,\qquad\text{for }v\geq v_{0}.$
(1)

Pick $v_{1}$ in $\Gamma$ such that $\varphi(v)\geq w$ for all $v\geq v_{1}$.
It follows that for $v\geq v_{0}\vee v_{1},$ we have
$q(Tx_{\varphi\left(v\right)})\not\leq\varphi(v)$ and so

$q(Tx_{\varphi(v)})\not\leq w,$

which is a contradiction with 1. and the proof comes to its end.

The following lemma, which connects unbounded order convergence with pointwise
convergence, is a known fact, although a quick proof is included for the sake
of completeness.

###### Lemma 3

Let $E=\mathbb{R}^{X}$ be the Riesz space of all real-valued functions defined
on a nonempty set $X.$ The following statements are equivalent:

  1. (i)

The net $\left(f_{\alpha}\right)_{\alpha\in A}$ is $uo$-convergent in $E.$

  2. (ii)

for every $x\in X,$ the net $\left(f_{\alpha}\left(x\right)\right)_{\alpha\in
A}$ is convergent in $\mathbb{R}.$

Proof. (i) $\Longrightarrow$ (ii) Assume that
$f_{\alpha}\overset{uo}{\longrightarrow}f$ in the Dedekind complete Riesz
space $E.$ Then there is a net $\left(g_{\alpha}\right)_{\alpha\in A}$ which
decreases to $0$ and for some $\alpha_{0}$ we have

$\left|f_{\alpha}-f\right|\wedge 1\leq g_{\alpha}\text{ for all
}\alpha\geq\alpha_{0}.$ (2)

Since $\left(g_{\alpha}\left(x\right)\right)$ decreases to $0$ for every $x\in
X,$ it follows easily from 2 that $f_{\alpha}\left(x\right)-f\left(x\right)$
converges to $0,$ as desired.

(ii) $\Longrightarrow$ (i) Assume now that $f_{\alpha}$ converges simply to
some $f\in E$ and let $h\in E^{+}.$ Define a net $\left(g_{\alpha}\right)$ by
putting

$g_{\alpha}\left(x\right)=\sup\limits_{\beta\geq\alpha}\left(\left|f_{\beta}-f\right|\wedge
h\right)\left(x\right),\text{ }x\in X.$

it is clear that $g_{\alpha}$ decreases to $0$ and
$\left|f_{\alpha}-f\right|\wedge h\leq g_{\alpha}.$ This shows that
$f_{\alpha}\overset{uo}{\longrightarrow}f$ and we are done.

Consider the Riesz space $F$ of all bounded real valued functions defined on
the real line with countable support and denote by $E$ the direct sum
$\mathbb{R}\mathbf{1}\oplus F,$ where $\mathbf{1}$ denotes the constant
function taking the value $1.$ This example will be of great interest for us.
The following lemma establishes some of its properties. Recall that a vector
sublattice $Y$ of a vector lattice $X$ is said to be regular if every subset
in $Y$ having a supremum in $Y$ has also a supremum in $X$ and these suprema
coincide. For more information about this notion and nice characterizations of
it via unbounded order convergence the reader is referred to [4].

###### Lemma 4

The space $E$ introduced above has the following properties.

  1. (i)

$E$ is a regular vector sublattice of $\mathbb{R}^{\mathbb{R}}.$

  2. (ii)

$E$ is Dedekind $\sigma$-complete but not Dedekind complete.

Proof. (i) It is clear that $E$ is a vector sublattice of
$\mathbb{R}^{\mathbb{R}}.$ To show that it is regular assume that
$\left(g_{\alpha}\right)_{\alpha\in A}$ is a net in $E$ satisfying
$g_{\alpha}\downarrow 0$ in $E.$ Let $g=\inf\limits_{\alpha}g_{\alpha}$ in
$\mathbb{R}^{\mathbb{R}}$ and $x\in\mathbb{R}.$ Then
$h=g\left(x\right)\mathbf{1}_{\left\\{x\right\\}}\in E$ and $0\leq h\leq
g_{\alpha}$ for all $\alpha.$ This implies that $h=0$ and then
$g\left(x\right)=0.$ Hence $g=0$ and the regularity is proved.

(ii) Let $\left(g_{n}\right)$ be an order bounded sequence in $E$ and write
$g_{n}=\lambda_{n}+f_{n},$ with $\lambda_{n}\in\mathbb{R}$ and $f_{n}\in F.$
Let $\Omega$ be the union of the supports of $f_{n},$ then $\Omega$ is
countable. Let $g$ be the supremum of $\left(g_{n}\right)$ in
$\mathbb{R}^{\mathbb{R}},$ that is,

$g\left(a\right)=\sup g_{n}\left(a\right),\text{ for all }a\in\mathbb{R}.$

It will be sufficient to show that $g\in E.$ To this end observe that
$g\left(x\right)=\alpha:=\sup\alpha_{n}$ for all
$x\in\mathbb{R}\backslash\Omega.$ Now put
$f=\left(g-\alpha\right)\mathbf{1}_{\Omega}.$ Then $f\in F$ and $g=\alpha+f\in
E$ as required. Next we show that $E$ is not Dedekind complete. Consider the
net $\left(g_{x}\right)_{x\in\left[0,1\right]}$ in $E$ defined by
$g_{x}=x\mathbf{1}_{\left\\{x\right\\}}.$ It is a bounded net in $E$ and its
supremum in $\mathbb{R}^{\mathbb{R}}$ does not belong to $E.$ As $E$ is
regular in $\mathbb{R}^{\mathbb{R}}$ this net can not have a supremum in $E.$

###### Remark 5

Consider the following operator:

$T:L_{1}\left[0,1\right]\longrightarrow c_{0};\text{ }\qquad f\longmapsto
Tf=\left(\int_{0}^{1}f\left(t\right)\sin ntdt\right)_{n\geq 1}.$

It is mentioned in [1], that $T$ is not order bounded; it is perhaps more
convenient to consider the same operator defined on
$L_{1}\left[0,2\pi\right].$ In this case if we define $u_{n}$ by
$u\left(t\right)=\sin nt$ for $t\in\left[0,2\pi\right],$ then
$\left|u_{n}\right|\leq 1,$ however $\left(Tu_{n}\right)=\left(e_{n}\right)$
is not bounded in $c_{0},$ where $\left(e_{n}\right)$ denotes the standard
basis of $c_{0}.$ This statement implies also that $T$ is not sequentially
order compact. Because $\left(e_{n}\right)$ has no order bounded subsequence,
it follows that $\left(Tu_{n}\right)$ can not admit an order convergent
subsequence. So the statement made in [3] that $T$ is $p$-bounded is not
correct.

The above example is presented in [3] to show that sequentially $p$-compact
operators need not be $p$-bounded. Although the operator given in that example
fails to be sequentially $p$-compact, the assertion that sequentially
$p$-compact operators need not be $p$-bounded is true. This will be shown in
our next example.

###### Example 6

Consider the Riesz spaces $E$ and $F$ defined just before Lemma 4 and let $T$
be the projection defined on $E$ with range $F$ and kernel
$\mathbb{R}\mathbf{1}.$ We claim that $T$ is sequentially order compact, but
not order bounded. Let $\left(f_{n}\right)$ be an order bounded sequence in
$E.$ Then $\left|f_{n}\right|\leq\lambda$ for some real $\lambda>0$ and for
all $n.$ Write $f_{n}=g_{n}+\lambda_{n}$ with $\lambda_{n}$ real and $g_{n}\in
F$ and observe that $\left|g_{n}\right|\leq 2\lambda$ for all $n.$ We have
also $\left|g_{n}\right|\leq 2\lambda\mathbf{1}_{A}\in F$ where $A$ is the
union of the supports of $g_{n},$ $n=1,2,...$ A standard diagonal process
yields a subsequence $\left(g_{k_{n}}\right)$ of $\left(g_{n}\right)$ which
converges pointwise on $A$ and then on $\mathbb{R}$ since all functions
$g_{n_{k}}$vanish on $\mathbb{R}\backslash A.$ Hence $\left(g_{k_{n}}\right)$
is uo-convergent in $\mathbb{R}^{\mathbb{R}}.$ As $\left(g_{k_{n}}\right)$ is
order bounded this implies that $\left(g_{k_{n}}\right)$ is order convergent
in $\mathbb{R}^{\mathbb{R}}.$ Observe moreover that $\sup\limits_{p\geq
n}g_{k_{n}}$ belongs to $F,$ which shows that $\left(g_{k_{n}}\right)$ is
order convergent in $F.$ The fact that $T$ is not order bounded is more
obvious: it is clear that the image of the net
$\left(\mathbf{1}_{\left\\{x\right\\}}\right)_{x\in\left[0,1\right]}$ by $T$
is not order bounded in $F.$

As an immediate consequence of Theorem 2 and Example 6 we deduce that
sequentially $p$-compactness does not imply $p$-compactness. At this stage one
might expect that the converse is true. Does $p$-compactness imply
sequentially $p$-compactness? This is an open question left in [3].
Unfortunately the answer is again negative.

###### Example 7

Let $X$ be the set of all strictly increasing maps from $\mathbb{N}$ to
$\mathbb{N}$ and $E=\mathbb{R}^{X}$ be the space of all real-valued functions
defined on $X,$ equipped with the product topology.

  1. 1.

First we will prove that the identity map, $\mathfrak{I},$ is a $p$-compact
operator on the lattice-normed space $(E,|\ |,E)$. To this aim, pick a
$p$-bounded net $\left(f_{\alpha}\right)_{\alpha\in A}$ in $E$, that is,
$|f_{\alpha}|\leq f$ for some $f\in E^{+}$ and for every $\alpha\in A$. It
follows that

$f_{\alpha}\in\prod_{x\in X}[-f(x),f(x)].$

Notice that the space $\prod_{x\in X}[-f(x),f(x)],$ equipped with the product
topology, is compact by Tychonoff’s Theorem. Thus $\left(f_{\alpha}\right)$
has a convergent subnet $\left(g_{\beta}\right)_{\beta\in B}$ in $\prod_{x\in
X}[-f(x),f(x)]$ to some $g.$ This means that

$g_{\beta}(x)\longrightarrow g(x)\ \text{for all }x\in X.$

According to Lemma 3, $g_{\beta}$ is $uo$-convergent to $g$ in $E.$ Since
bounded uo-convergent nets are order convergent, we have that
$g_{\beta}\overset{o}{\longrightarrow}g$. This proves that $\mathfrak{I}$ is a
$p$-compact operator.

  2. 2.

We prove now that $\mathfrak{I}$ is not sequentially $p$-compact. Let
$\left(\varphi_{n}\right)$ be a sequence in $\\{-1,\ 1\\}^{X}$ which has no
convergent subsequence (see Example 3.3.22 in [7]). This sequence is order
bounded in $E$ and every subsequence $\left(\psi_{n}\right)$ of
$\left(\varphi_{n}\right)$ does not converges in $\\{-1,\ 1\\}^{X},$ that is,
for some $x\in X,$ $\psi_{n}\left(x\right)$ diverges. According to Lemma 3,
$\left(\psi_{n}\right)$ is not uo-convergent in $E.$ Since
$\left(\psi_{n}\right)$ is order bounded it does not converge in order. This
finishes the proof.

In classical theory of Banach spaces the identity map is compact if and only
if the space is finite-dimensional. In contrast of this the situation is not
clear in general case. We already have seen an example of infinite-dimensional
space on which the identity map is $p$-compact. This question has been
investigated in [3] where the authors showed that $I_{L_{1}\left[0,1\right]}$
fails to be compact however, $I_{\ell_{1}}$ is $p$-compact. In the next
example we show that $I_{L_{\infty}\left[0,1\right]}$ is not $p$-compact,
answering a question asked in [3].

###### Example 8

The identity operator $I$ on the lattice normed space $(L_{\infty}[0,1],|\ .\
|,L_{\infty}[0,1])$ is neither $p$-compact nor sequentially $p$-compact. To
this end, consider the sequence of Rademacher function given by :

$\begin{array}[c]{ccccl}r_{n}&:&[0,1]&\longrightarrow&\mathbb{R}\\\ &&\
t&\longmapsto&sgn\left(\sin(2^{n}\pi t)\right)\end{array}$

for all $n\in\mathbb{N},$ which is order bounded since $|r_{n}|=1.$ Suppose
now that $\left(r_{n}\right)$ has an order convergent subnet
$\left(r_{n_{\alpha}}\right)_{\alpha\in\Gamma}.$ Then
$r_{n_{\alpha}}\overset{o}{\rightarrow}r$ for some $r\in
L_{\infty}\left[0,1\right].$ Let $\alpha\in A.$ For every $\beta>\alpha,$
$\int_{0}^{1}r_{n_{\alpha}}r_{n_{\beta}}d\mu=0.$ On the other hand
$\left(r_{n_{\alpha}}r_{n_{\beta}}\right)_{\beta}$ converges in order to
$r_{n_{\alpha}}r$ in $L_{\infty}\left[0,1\right]$ and then in
$L_{1}\left[0,1\right].$ Since the integral is order continuous, we deduce
that

$\int_{0}^{1}r_{n_{\alpha}}rd\mu=0.$

This equality holds for every $\alpha\in A,$ and a similar argument leads to

$\int_{0}^{1}r^{2}d\mu=0,$

which is a contradiction since $\left|r\right|=1,$ and the claim is now
proved.

##  4 Semicompact operators

The notion of semicompact operators has been introduced by Zaanen in [9] and
extended in the framework of lattice normed spaces in [3].

Let $(X,p,E)$ be a lattice normed space and $(Y,q,F)$ be an lattice normed
vector lattice. A linear operator $T:$ $X\rightarrow Y$ is called
$p$-semicompact if it maps $p$-bounded sets in $X$ to $p$-almost order bounded
sets in $Y$. We recall that a subset $B$ of $Y$ is said to be $p$-almost order
bounded if for any $w\in F_{+}$, there is $y_{w}\in Y$ such that

$q((|y|-y_{w})^{+})=q(|y|-y_{w}\wedge|y|)\leq w\ \text{for all }y\in B.$

Semicompact operators from Banach spaces to Banach lattices fail, in general,
to be compact (see [1]). This yields trivially that $p$-semicompactness does
not imply $p$-compactness. However, the converse is true in the classical case
as has been shown in Theorem 5.71 in [1]. And one can expect to extend this
result in general situation. This is already the subject of Question 4 in [3].
Unfortunately the answer is again negative. Before stating our counterexample
let us recall that every order bounded operator from a vector lattice $E$ to a
Dedekind complete vector lattice $F$ has a modulus ([1]).

###### Example 9

Let $E$ be a Dedekind complete Banach lattice with order continuous norm and
$T$ be a norm-compact operator in $\mathcal{L}(E)$ such that $T$ has no
modulus, and therefore $T$ can not be order bounded. For the existence of such
operator we refer the reader to the Krengel’s example in [1, p 277.]. Consider
now the following lattice-normed vector spaces $(E,\|.\|,\mathbb{R})$ and
$\left(E,p,\mathbb{R}^{2}\right),$ where $p(x)=\begin{pmatrix}\|x\|\\\
0\end{pmatrix}$ for all $x\in E.$ It is straightforward to prove that $T$ is
again $p$-compact operator and we claim that $T$ is not $p$-semicompact. To
this end we we will argue by contradiction and we assume that $T$ is
$p$-semicompact. Fix an element $u\in E^{+}$ and let $w=\begin{pmatrix}0\\\
1\end{pmatrix}$, then there exists $z_{w}$ such that
$p\left((T(x)-z_{w})^{+}\right)\leq w$ for all $x\in[-u,u]$, which means that
$(T(x)-z_{w})^{+}=0.$ Noting that this occurs for $x$ and $-x$ we see that

$|T(x)|\leq z_{w}\text{ for all }x\in\left[-u,u\right].$

This shows that $T$ is order bounded, a contradiction. and our proof comes to
an end.

A slight modification of the proof of Example 9 leads to a more general
result. The proof of it will be left for the reader.

###### Proposition 10

Let $\left(E,p,V\right)$ be a lattice normed space and $\left(F,q,W\right)$ a
lattice normed vector lattice. We assume that $q\left(F\right)^{d}$ is not
trivial. Then every semicompact operator
$T:\left(E,p,V\right)\longrightarrow\left(F,q,W\right)$ is $p$-bounded as an
operator from $\left(E,p,V\right)$ to $\left(F,\left|.\right|,F\right).$

##  5 $rp$-compact operators

As every $rp$-compact operator between lattice-normed spaces is $p$-compact,
the following result is an immediate consequence of Theorem 2.

###### Theorem 11

Let $(E,p,V)$ and $(F,q,W)$ be lattice-normed spaces and $T$ be in
$\mathfrak{L}(E,F)$. If $T$ is $rp$-compact then $T$ is $p$-bounded.

In the following example we will prove that sequentially $p$-compact operators
need not be $rp$-compact.

###### Example 12

Let $E$ be the Riesz space defined above. We claim that the identity operator
$I:E\rightarrow E$ is sequentially $p$-compact but fails to be $rp$-compact.
Let $\left(x_{n}\right)$ be a bounded sequence in $E,$ that is, $|x_{n}|\leq
x$ for some $x\in E^{+}.$ Write $x=\alpha+f,$ and $x_{n}=\alpha_{n}+f_{n}$
where $\alpha\in\mathbb{R}^{+}$ and $f\in F$ and $\alpha_{n}\in\mathbb{R},$
$f_{n}\in F$ for $n=1,2,...$ It is easily seen that $|\alpha_{n}|\leq\alpha$,
$|f_{n}|\leq x+\alpha$. By a standard diagonal argument there exists a
subsequence such that $f_{\varphi(n)}\left(a\right)$ converges for every
$a\in\mathbb{R}$ and $\alpha_{\varphi\left(n\right)}$ converges in
$\mathbb{R}.This$ shows that $x_{n}$ converges pointwise on $\mathbb{R}$ and
its limit is clearly in $E.$ By Lemma 3, $x_{n}\overset{uo}{\longrightarrow}x$
in $\mathbb{R}^{\mathbb{R}}$ and then $x_{n}\overset{o}{\longrightarrow}x$ in
$\mathbb{R}^{\mathbb{R}}$ as it is an order bounded sequence. Now using Lemma
27 in [2] and Lemma 4 we deduce that $x\in E.$ On the other hand, let
$\mathcal{F}$ be the collection of finite subsets of $\mathbb{R}_{+}$ ordered
by inclusion and consider the net $\left(g_{A}\right)_{A\in\mathcal{F}}$ where
$g_{\alpha}=\mathbf{1}_{\alpha}$. Then $\left(g_{\alpha}\right)$ is order
bounded in $E$ but has no convergent subnet. Since
$g_{\alpha}\uparrow\mathbf{1}_{\mathbb{R}_{+}}$ in $\mathbb{R}^{\mathbb{R}}$
and $E$ is regular in $\mathbb{R}^{\mathbb{R}},$ it follows that
$\left(g_{\alpha}\right)$ is not order convergent in $E$ and so are all its
subnets.

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