# Data Distribution Dynamics in Real-World WiFi-Based Patient Activity
Monitoring for Home Healthcare

Mahathir Monjur1, Jia Liu3, Jingye Xu2, Yuntong Zhang2, Xiaomeng Wang3,
Chengdong Li4, Hyejin Park4,
Wei Wang2, Karl Shieh5, Sirajum Munir6, Jing Wang4, Lixin Song3, Shahriar
Nirjon1
{mahathir<EMAIL_ADDRESS>{cli7<EMAIL_ADDRESS>{jingye.xu,
yuntong.zhang<EMAIL_ADDRESS>
{liuj14, wangx14<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
1Department of Computer Science, University of North Carolina at Chapel Hill,
NC
2Department of Computer Science, University of Texas at San Antonio, TX
3School of Nursing, University of Texas Health Science Center at San Antonio,
TX
4College of Nursing, Florida State University, FL
5School of Nursing, University of North Carolina at Chapel Hill, NC
6Bosch Research and Technology Center, Pittsburgh, PA

###### Abstract

This paper examines the application of WiFi signals for real-world monitoring
of daily activities in home healthcare scenarios. While the state-of-the-art
of WiFi-based activity recognition is promising in lab environments,
challenges arise in real-world settings due to environmental, subject, and
system configuration variables, affecting accuracy and adaptability. The
research involved deploying systems in various settings and analyzing data
shifts. It aims to guide realistic development of robust, context-aware WiFi
sensing systems for elderly care. The findings suggest a shift in WiFi-based
activity sensing, bridging the gap between academic research and practical
applications, enhancing life quality through technology.

###### Index Terms:

WiFi sensing; activity recognition; real-world dataset shift.

##  I Introduction

WiFi signals, a common household technology, can be repurposed for detecting
and classifying daily activities in innovative ways. These signals, emitted by
routers, permeate rooms and interact with objects and people within. When a
person moves or performs activities, they disrupt the WiFi signal pattern.
Sensing systems detect these disruptions, interpreting them as specific
activities like walking or sitting. By analyzing signal strength changes and
patterns, the system can accurately identify different movements. This
technology, therefore, transforms ordinary WiFi into a tool for monitoring
daily life unobtrusively.

In healthcare domain, WiFi sensing is gaining traction due to its ability to
monitor various activities and health-related events without physical contact
or the need for wearable devices [1, 10, 7, 2, 4]. This technology can detect
critical events like falls, sleep disturbances, wandering behavior,
respiratory disorders, and abnormal cardiac activity, particularly in
vulnerable populations such as the elderly. Its appeal lies in its ease of
operation indoors and the comfort it provides to individuals being monitored,
as it is less intrusive compared to wearables [9], RGBD sensors [8], camera
images [6], or acoustic solutions [5].

Unfortunately, while WiFi sensing in healthcare shows promise in academic
studies, it encounters substantial difficulties transitioning from the
controlled settings of a lab to the complexities of real-world environments.
In laboratories, conditions are ideal, with minimal interference and
predictable variables. However, in uncontrolled environments like homes, where
patients reside for home healthcare or aging in place, numerous unpredictable
factors come into play. Factors such as varying WiFi signal strengths,
different home layouts, presence of multiple individuals, and diverse daily
activities create a complex environment. This complexity often leads to
decreased accuracy and reliability of WiFi sensing systems. They struggle to
adapt to the dynamic and individualized nature of home settings, where
conditions differ significantly from the lab. This gap between lab-controlled
conditions and real-world variability is a critical limitation that needs
addressing for effective deployment in healthcare scenarios.

In this paper, we share our experience from a multi-year study where we
deployed WiFi sensing systems in 8 different environments with 16 participants
involved. These environments encompass a variety of settings such as offices,
kitchens, laboratories, and living room areas, while the participant
demographic is diverse, including males and females, young and elderly, and
both healthy and weak patients. Additionally, before these deployments, we had
trained and tested the system in various lab environments. Our deployments
varied in duration, ranging from a few hours to a maximum of two weeks of
continuous monitoring. In each deployment, we recorded the ground truth using
cameras and/or wearable devices. From the data we collected, we observed
significant shifts in data distribution due to multiple factors. We summarize
our preliminary findings through a comprehensive study involving:

  * •

We analyze various factors such as individual presence, environmental
conditions, and time intervals affecting WiFi signal variability. This
detailed analysis aids in understanding and quantifying the contribution of
each factor to the overall shift in data distribution.

  * •

Our rigorously evaluate existing data adaptation techniques, assessing their
effectiveness beyond the controlled settings of a lab.

  * •

Utilizing machine learning and signal processing algorithms, we aim to uncover
hidden patterns within the chaotic data, providing insights into the
mechanisms driving dataset shift.

This research will significantly advance elderly care monitoring by developing
adaptive sensing systems, leading to a paradigm shift in WiFi-based activity
sensing. It aims to unify academic and industrial efforts, enhancing quality
of life through innovative and context-aware technology.

##  II AURA System

This paper is part of a multi-year NIH project, _“AURA,”_ , which enhances
voice assistants like Alexa with WiFi sensing for home healthcare. Figure 1
shows AURA’s use in detecting activities, enabling smarter, context-aware
Alexa interactions.

Figure 1: User’s activity-aware conversational assistant.

An AURA-enabled voice assistant can monitor users and their environment
passively. This means it operates without needing frequent user interactions,
which can be bothersome, or requiring the user to carry extra devices such as
GPS or health trackers, which are often forgotten or inconvenient.
Importantly, the AURA system avoids using cameras to respect privacy, a
crucial factor in home environments where both patients and their families are
present. WiFi sensing, unlike cameras, functions in darkness and can penetrate
walls, enabling AURA to extend its reach beyond the confines of a single room.
However, for calibration and ground truth recording purposes, cameras were
temporarily employed during the initial hour of our study.

Figure 2: Architecture of AURA.

The architecture of AURA is depicted in Figure 2. AURA offers four interaction
interfaces for patients and caregivers: WiFi (RF), voice, web, mobile. Data
from these interfaces are standardized by adapters before entering the AURA
runtime. This design ensures easy extension with new interactions without
altering the core system. The AURA runtime monitors patient status, logs data,
offers APIs, and manages AURA apps execution. The number and types of apps
vary with the deployment scenario. Each AURA app monitors patient needs
specific to their health application, such as hydration and sleep for post-
treatment CBC patients. Using AURA’s APIs, these apps convert WiFi sensing
data, like movement frequency and drinking gestures, into clinically relevant
information.

In this paper, we focus solely on AURA’s WiFi data collection, analysis, and
activity classifier modules.

Figure 3: Processed WiFi CSI for (a) no activity, (b) fall, (c) jogging, and
(d) squat.

##  III Dataset

###  III-A Study Design

In this study, we conducted several recording sessions involving 16
participants. Before each session commenced, we briefed participants on the
activities they were to perform and informed them about our data handling
procedures, including anonymization and publication policies. We also obtained
written consent from all participants.

Each session involved collecting a paired dataset consisting of image
sequences (RGB and depth) and WiFi CSI. To capture the RGB and depth images,
we used an Intel D435 depth camera, set to $1280\times 720$ resolution and 10
fps. For CSI data, we employed a pair of ASUS RT-AC86U routers. One router was
configured to transmit packets at 200 packets per second rate on a single
spatial channel. The other router was linked to an NVIDIA Jetson Nano computer
to receive the packets. Both routers were set to monitor mode using the Nexmon
framework [3]. To ensure synchronization, the collection of WiFi CSI data and
camera image data was initiated concurrently via the jetson nano computer and
each camera image frame was timestamped. The WiFi packets also contained
microsecond-level time information, which was later used to align each camera
and WiFi CSI data pair.

We collected paired image and CSI data from 16 participants across 8 distinct
environments. During each session, we continuously recorded camera and WiFi
CSI data while participants engaged in various activities. These activities
were annotated based on their start and end times, corresponding to specific
image frames, which enabled us to extract the relevant WiFi CSI packets. These
sessions resulted in a comprehensive dataset of 606 minutes or 1̃0 hours. The
participant group comprised 7 males and 9 females. In addition to the paired
data, we also gathered WiFi CSI data specific to different human activities
(with labels) from 3 participants in 5 different environments.

###  III-B Data Processing

After collecting raw WiFi CSI data, we arrange the packets by the number of
antennas and then apply fft on the raw data. This results in a spectrogram
with the shape of $(N\times N_{ant})\times M$, where $N$ is number of packets,
$N_{ant}$ is number of antenna used from the receiver router which can be $1$
or $4$, and $M$ is the number of sub-channels. Although our setup was
configured for collecting data on $80$ MHz bandwidth and $256$ sub-channels,
Nexmon allows CSI collection of $M=242$ data-channels while the other $14$
channels is used as control sub-channels. Hence we remove the additional
channels with no information. Examples of processed CSI data for different
activities are shown in Figure 3.

###  III-C Patient Activity Monitoring

In our dataset, participants engaged in a diverse array of activities such as
walking, eating, drinking, light arm exercises, jogging, squat, jumping,
falling, sitting down, standing up, and remaining seated or standing with
minimal movement. These activities were continuously recorded using both
cameras and WiFi routers using single antenna. The sessions were meticulously
labeled post-recording, with the start and end image frames for each activity
sequence being marked to create a comprehensive metadata file. Additionally,
we have augmented our original dataset from patients with labeled CSI data,
encompassing 4 channels, for three extra participants across five different
environments. A detailed statistics of the paired image and WiFi dataset is
shown in I, where we list each unique environment, and the number of male and
female participants, number of image frames, and number of CSI packets
collected within that environment. In total, we collected 232,440 image frames
and 7,436,828 CSI packets. This enriched dataset provides a robust foundation
for detailed activity recognition research and analysis. The dataset and
relevant scripts can be found here: https://github.com/Monjur-Mahathir/AURA-
DNN.git.

TABLE I: Dataset Statistics Environment | # of participants | # of images | # of CSI packets
---|---|---|---
Living Room-1 | 1 Male | 21,779 | 399,687
Living Room-2 | 1 Female | 11,991 | 429,607
Living Room-3 | 1 Male | 20,210 | 458,948
Living Room-4 | 1 Female | 11,000 | 413,181
Living Room-5 | 1 Male | 7,707 | 389,985
Living Room-6 | 1 Male, 1 Female | 20,986 | 1,065,504
Kitchen-7 | 4 Female, 3 Male | 92,878 | 3,838,007
Office-8 | 2 Female | 45,889 | 441,909
Figure 4: Data shift due to environment change. Figure 5: Data shift due to
activity from different people.

##  IV Dataset Analysis

###  IV-A Data Analysis with t-SNE Embedding

To analyze the effectiveness of the WiFi CSI data collected from different
patient’s home, we used t-Distributed stochastic neighbor embedding (t-SNE),
which is a powerful technique for visualizing high-dimensional data by
reducing it to lower-dimensional spaces, while maintaining the local structure
of the data. It starts by converting the distances between data points in
high-dimensional space into conditional probabilities that represent
similarities. The similarity of datapoint $x_{j}$ to datapoint $x_{i}$ is
given by a conditional probability $p_{j|i}$, which is computed as:

$p_{j|i}=\frac{\exp(-||x_{i}-x_{j}||^{2}/2\sigma_{i}^{2})}{\sum_{k\neq
i}\exp(-||x_{i}-x_{k}||^{2}/2\sigma_{i}^{2})}$

where $||x_{i}-x_{j}||$ is the Euclidean distance between $x_{i}$ and $x_{j}$,
and $\sigma_{i}$ is the variance of the Gaussian that is centered on datapoint
$x_{i}$. In the low-dimensional space, a similar probability $q_{j|i}$ is
calculated, often using a Student’s t-distribution. The aim of t-SNE is to
minimize the divergence between these two probability distributions, typically
using the Kullback-Leibler divergence as a cost function. This minimization is
usually achieved through gradient descent. The result is a map that reveals
the intrinsic structure of the data in a way that is easily interpretable,
often revealing clusters and relationships that are not apparent in the high-
dimensional space. In Figure 6, we can see that the raw CSI data from a
session forms several differentiable clusters for different activities.

Figure 6: t-SNE embedding of WiFi CSI data

###  IV-B Dataset Shift

Figure 7: Data shift due to temporal variation.

####  IV-B1 Dataset Shift from Environment

To evaluate the adaptability of WiFi CSI data in patient activity monitoring
across new environments, we partition our dataset into training and testing
subsets. Notably, the testing set comprises data from three entirely new
environments: an office, a lab, and a living room. After training our DNN with
the training set, we test its performance on this unseen data. As depicted in
Figure 4, the model’s accuracy notably decreases (ranging between 13%-54%)
when applied to these new environments. This drop in performance is
particularly pronounced in the complex and object-filled environment of the
living room, while it is less severe in more controlled settings like the lab,
which has fewer objects.

We further analyze the shift in performance by comparing the softmax
probability distributions of the outputs on the training set,
$P_{train}(y|X)$, and the testing set from the unseen environments,
$P_{test}(y|X)$. This comparison reveals significant differences in the
model’s behavior between familiar and unfamiliar settings. In addition, a
t-SNE visualization, which projects the high-dimensional feature space
generated by the DNN, $Z=f_{\theta}(X)$, into a two-dimensional space for
enhanced interpretability, demonstrates that the data from these three
distinct environments form three separate, clearly identifiable clusters. This
clustering, represented in the t-SNE plot, is indicative of the varying
environmental impacts on the model’s performance.

####  IV-B2 Dataset Shift from People

In our study to assess dataset shift in CSI data for activity recognition
involving unseen participants, we conduct an experiment where three
individuals from a single environment are intentionally excluded from the
training dataset, while data from two different individuals from the same
environment are retained in the training set. The results show a moderate
accuracy degradation of 12-20%, a contrast to the more substantial drops
observed in Section IV-B1 for completely unseen environments. Further
analysis, including the probability distribution and t-SNE plots presented in
Figure 5, reveals a relatively minor data shift when different participants
were involved in a familiar environmental setting, underscoring the impact of
environmental familiarity on the model’s performance in activity recognition
tasks.

####  IV-B3 Dataset Shift from Time

In our investigation to determine the temporal stability of WiFi CSI data and
its influence on dataset distribution, we conduct an analysis involving three
distinct sessions. We strategically partition the data from these sessions,
incorporating the first half into the training set and the latter half into
the testing set. This approach aims to observe potential changes in the
probability distribution over time. The results, as depicted in Figure 7,
indicates a minimal impact of dataset shift over time, provided that both the
environment and the participants remain consistent within the training set.
This finding suggests that temporal variations do not significantly affect the
dataset distribution for activity recognition tasks when the environmental and
participant variables remain constant, thereby affirming the temporal
robustness of the CSI data under controlled conditions.

##  V Activity Recognition

###  V-A Model Description

To facilitate the deployment of our DNN on resource-constrained NVIDIA Jetson
Nano, we engineer a compact and computationally efficient DNN architecture.
This architecture integrates a sequence of three convolutional layers, each
succeeded by a batch normalization layer and a max pooling operation.
Following the convolutional stack, the processed feature set is input into
three stacked bidirectional LSTM layers, designed to capture temporal
dependencies effectively. The output at the final timestep of these LSTM
layers is then channeled into a dense layer, which is coupled with a softmax
activation function to facilitate multi-class classification. The DNN’s total
size is approximately 6.761MB, encompassing around 1.77 million parameters.

In our baseline experiment, the DNN was trained over 500 epochs using an Adam
optimizer with a learning rate of 0.001 and a batch size of 16, with the
dataset divided into training and testing sets in an 80-20% split. To address
domain-specific effects, we extended our analysis to include two domain
adaptation strategies: the application of a Gradient Reversal Layer (GRL) and
a few-shot adaptation technique for a new domain. For the few-shot approach, 5
samples from each activity class were used to fine-tune the pre-trained model
for domain adaptation. In the case of the GRL, we modified the DNN
architecture to incorporate an additional dense layer following the LSTM
output for domain classification, with the GRL applied before this layer to
ensure the model prioritizes class-related features and disregards
environment-specific features.

###  V-B Results

In our experimentation, the developed model, tailored for a 10-class
classification task, attained a notable overall accuracy of 82% across various
domains, as delineated in the confusion matrix presented in Figure 8, where
the rows represent the true class and the columns represent the predicted
class. The model demonstrated superior performance in accurately identifying
instances of no activity (categorized as ’empty’) and the ’walking’ activity,
which is characterized by significant bodily movements. However, the confusion
matrix reveals a specific challenge: approximately 30% of instances involving
’arm exercise’ were misclassified as ’clapping.’ This misidentification is
attributable to the similarity in hand gestures between these two activities.
Furthermore, the model’s ability to discern ’sitting down’ activities was
suboptimal, largely due to the motion resemblance to ’standing up’ activities.

Figure 8: Confusion matrix of activity recognition across all domains.

When tested in novel environmental contexts, our model achieves an accuracy of
only $42\%$, demonstrating a marked degradation. We show the confusion matrix
in Figure 9. A key observation from this analysis is the model’s inability to
accurately classify subtle activities, such as ’drinking,’ frequently
categorizing them as ’no activity.’ Such behavior underscores the model’s
current limitation in adapting to unfamiliar environments and highlights the
necessity for advanced domain adaptation methodologies to enhance its
generalization capabilities across diverse environmental conditions.

Figure 9: Confusion matrix of activity recognition for unseen domains.

In our final set of observations, we noted that the same model, when augmented
with an additional domain classification branch preceded by a Gradient
Reversal Layer (GRL), attained an improved accuracy of 63% on datasets derived
from the same previously unseen environment. This enhancement is visually
represented in the confusion matrix shown in Figure 10. A notable shift in the
confusion matrix towards a more diagonal pattern was observed, indicating a
significant improvement in the model’s ability to correctly classify
activities, including those involving minute movements.

Figure 10: Confusion matrix of activity recognition for unseen domains using
domain classification strategy.

###  V-C Effect of Environment

In our research, as detailed in Section IV-B1, we evaluated our model’s
performance across datasets from three unseen environments and compared it
with datasets from environments encountered during the training phase. The
accuracies were assessed for the base network (vanila), the enhanced network
equipped with an additional domain classification branch and a GRL, and
through the implementation of k-shot fine-tuning. Our findings indicate that
both domain adaptation strategies yielded improvements in model accuracy for
unseen environments, with enhancements ranging between 2% to 27%, as shown in
Figure 11. The most significant accuracy increase was observed in the context
of an unseen living room environment. Despite these advancements, a notable
accuracy gap of 10-30% persists between the performance in unseen and seen
environments. This discrepancy underscores the necessity for more
sophisticated approaches to further bridge this divide, suggesting potential
avenues for future research in enhancing the model’s adaptability and
generalization capabilities across varying environmental contexts.

Figure 11: Comparison of different training algorithms acrooss seen and unseen
domains.

###  V-D Effect of People

Our analysis extends to examining the model’s accuracy in recognizing
activities performed by different individuals within the same environment, as
depicted in Figure 12. The results indicate that while the model’s performance
degradation is less pronounced when encountering data from unseen individuals,
as compared to entirely new environments, there is still a noticeable accuracy
drop of 12-15% when utilizing the standard (vanilla) training algorithm.
However, the implementation of domain adaptation techniques has shown to
improve this decline in performance. Specifically, the application of domain
adaptation strategies has effectively narrowed the accuracy disparity,
reducing it to a range of 2-10%.

Figure 12: Comparison of different training algorithms acrooss seen and unseen
people.

##  VI Conclusion and Future Work

In this paper, we presented the outcomes of our multi-year study, where we
deployed WiFi sensing systems in eight distinct environments with 16
participants, after thorough preliminary testing in lab settings. The
deployments, ranging from a few hours to two weeks, were meticulously recorded
using cameras and wearable devices for accurate ground truth comparison. Our
analysis highlighted substantial shifts in data distribution, attributable to
variables like individual presence, environmental conditions, and time
intervals. We also showed that traditional domain adaptation techniques for
handling data shift is not sufficient to fully remove the effect of
environment in WiFi sensing. In future, we intend to explore hyper network
based approaches to parameterize the environment and model the estimated shift
in CSI feature for activity recognition, which can provide a pathway towards
developing adaptive WiFi-based sensing systems, particularly in enhancing
elderly care.

## References

  * [1] Z. Chen, L. Zhang, C. Jiang, Z. Cao, and W. Cui.  WiFi CSI based passive human activity recognition using attention based blstm.  IEEE Transactions on Mobile Computing, 18(11):2714–2724, 2018.
  * [2] J. Ding and Y. Wang.  Wifi csi-based human activity recognition using deep recurrent neural network.  IEEE Access, 7:174257–174269, 2019.
  * [3] M. Schulz, D. Wegemer, and M. Hollick.  Nexmon: Build your own wi-fi testbeds with low-level mac and phy-access using firmware patches on off-the-shelf mobile devices.  In Proceedings of the 11th Workshop on Wireless Network Testbeds, Experimental evaluation & CHaracterization, pages 59–66, 2017.
  * [4] S. Shang, Q. Luo, J. Zhao, R. Xue, W. Sun, and N. Bao.  LSTM-CNN network for human activity recognition using wifi csi data.  In Journal of Physics: Conference Series, volume 1883, page 012139. IOP Publishing, 2021.
  * [5] J. M. Sim, Y. Lee, and O. Kwon.  Acoustic sensor based recognition of human activity in everyday life for smart home services.  International Journal of Distributed Sensor Networks, 11(9):679123, 2015.
  * [6] T.-H. Tan, M. Gochoo, S.-C. Huang, Y.-H. Liu, S.-H. Liu, and Y.-F. Huang.  Multi-resident activity recognition in a smart home using RGB activity image and DCNN.  IEEE Sensors Journal, 18(23):9718–9727, 2018.
  * [7] W. Wang, A. X. Liu, M. Shahzad, K. Ling, and S. Lu.  Understanding and modeling of WiFi signal based human activity recognition.  In Proceedings of the 21st annual international conference on mobile computing and networking, pages 65–76, 2015.
  * [8] C. Zhang and Y. Tian.  RGB-D camera-based daily living activity recognition.  Journal of computer vision and image processing, 2(4):12, 2012.
  * [9] W. Zhuang, Y. Chen, J. Su, B. Wang, and C. Gao.  Design of human activity recognition algorithms based on a single wearable IMU sensor.  International Journal of Sensor Networks, 30(3):193–206, 2019.
  * [10] H. Zou, Y. Zhou, J. Yang, W. Gu, L. Xie, and C. Spanos.  Multiple kernel representation learning for WiFi-based human activity recognition.  In Proceedings of the 16th IEEE International Conference on Machine Learning and Applications (ICMLA ’17), pages 268–274, Cancun, Mexico, Dec. 2017. IEEE.


# On rough mereology and VC-dimension in treatment of decision prediction for
open world decision systems

Lech T. Polkowski
Department of Mathematics and Informatics
University of Warmia and Mazury in Olsztyn Poland
e-mail<EMAIL_ADDRESS>
###### Abstract

Given a raw knowledge in the form of a data table/a decision system, one is
facing two possible venues. One, to treat the system as closed, i.e., its
universe does not admit new objects, or, to the contrary, its universe is open
on admittance of new objects. In particular, one may obtain new objects whose
sets of values of features are new to the system. In this case the problem is
to assign a decision value to any such new object. This problem is somehow
resolved in the rough set theory, e.g., on the basis of similarity of the
value set of a new object to value sets of objects already assigned a decision
value. It is crucial for online learning when each new object must have a
predicted decision value. There is a vast literature on various methods for
decision prediction for new yet unseen object, we mention a few works below.

The approach we propose is founded in the theory of rough mereology and it
requires a theory of sets/concepts, and, we root our theory in classical set
theory of Syllogistic within which we recall the theory of parts known as
Mereology. Then, we recall our theory of Rough Mereology along with the theory
of weight assignment to the Tarski algebra of Mereology. This allows us to
introduce the notion of a part to a degree.

Our choice of Syllogistic as the theory of concepts is motivated not only by
simplicity of this theory and its completeness but also by our remembrance of
Professor Helena Rasiowa who studied in years 1939-1943 at Clandestine Warsaw
University with Jan Łukasiewicz, the great authority on Aristotle’s
Syllogistic.

Once we have defined basics of Mereology and rough Mereology, we recall our
theory of weight assignment to elements of the Boolean algebra defined within
Mereology and this allows us to define the relation of parts to the degree and
we apply this notion in a procedure to select a decision for new yet unseen
objects. In selecting a plausible candidate which would pass its decision
value to the new object, we employ the notion of Vapnik - Chervonenkis
dimension in order to select at the first stage the candidate with the largest
VC-dimension of the family of its $\varepsilon$-components for some choice of
$\varepsilon$.

Our text is, accordingly, divided into sections which are, respectively:

1\. The section on basic notions of the rough set theory.

2\. The section on Syllogistic.

3\. The section on Mereology.

4\. The section on weight assignment.

5\. The section on Rough Mereology.

6\. The section on our idea for the treatment of new objects in the Open World
Approach (OWA).

7\. On the problem of mistakes.

In memory of Professor Helena Rasiowa (1917-1994)

###### Keywords 1

Syllogistic, Mereology, Rough Mereology, Rough Set Theory, decision systems,
Open World assumption, Vapnik-Chervonenkis dimension.

##  1 The Rough Set Theory: Basic notions

We begin with a decision system $\mathcal{D}$ = $(O,F,d,V,V_{d})$ in which $O$
is a set of objects, $F$ is a set of features, $d\notin F$ is the decision
feature, $V$ is the set of feature values, and, $V_{d}$ is the set of decision
values. The set $F$ consists of subjective features of our free choice. We
assume the set $V_{d}$ to be a convex set in a finite dimensional vector
space, in particular it may be the interval in the real line.

We adopt the latter as our standard model. Rough Set Theory due to Pawlak [24]
aims at description of concepts - subsets of the set $O$ and a fortiori at
assigning to decision values their descriptions in terms of values of
subjective features. Expressions which do this work are decision rules, their
sets forming decision algorithms, particular cases of Datalog cf. Abiteboul et
al. [1]. The premiss of a decision rule is a conjunct of descriptors of the
form $\bigwedge_{i}(f_{i},v_{i})$ where $f_{i}$ is a subjective feature and
$v_{i}$ its value. The conclusion is a descriptor of the form $(d,v)$ where
$v$ is a decision value, features $f_{i}$ run over a subset $R$ of the set
$F_{1}$ and the decision rule in question is of the form $r:\bigwedge_{f\in
R}(f,v_{f})\rightarrow(d,v)$. The arrow symbol $\rightarrow$ is used only in
the context of decision rules, otherwise and in general we use the horseshoe
symbol $\supset$.

Concepts definable as subsets of the set of objects $O$ are unions of
indiscernibility classes of the form $IND_{H}(o)=\\{o^{\prime}\in
O:f(o)=f(o^{\prime})\\}$ for $o\in O$ and $f\in H$ for $H\subset F_{1}$; other
concepts are non-definable (rough) and they may be defined only approximately
by Jordan - type approximations by definable sets.

The interested reader may consult for rough set theory proceedings of the
annual series of IJCRS conferences (LNCS, Springer-Verlag) and issues of
Transactions on Rough Sets [41]; an account of foundations may be found in
Pawlak [24], [25], Pawlak et al. [26], Polkowski [28], or, Komorowski et al.
[12]. Among works dedicated to prognosis of decision values for new yet unseen
objects we may mention Chen et al., [4], Düntsch and Gediga [7], Udristoiu et
al. [42] and Wu et al. [45]. Rough set theory-based analysis and approximation
of decision is the subject in Pawlak-Słowiński [27], Yao [46], Greco et al.
[8], Jian et al. [11], Polkowski and Skowron [33], [34], Pal et al.[23],
Polkowski and Artiemjew [32], among many others.

##  2 Syllogistic in a nutshell

We choose as our theory of collective terms the Aristotle Syllogistic
presented originally in Analytics and translated and edited, e.g., in Ross
[36] and we borrow our description of Syllogistic from Łukasiewicz [17]. The
reason for this choice is given in Abstract. We use the expression ’term’ as
it is used in analysis of the Aristotle’s system of Syllogistic in [17].
Syllogistic involves terms in premisses in which one term is predicated of the
other term like in the expression ’Europeans are predicated of all Poles’
where ’Poles’ is the subject and ’Europeans’ is the predicate. The expression
’All’ makes the premiss into a universal one. Other qualifiers to premisses
are ’No’, ’Some’. ’No’ qualifies a premiss as universal and ’Some’ qualifies a
premiss as particular. Let us note that all predicates are mass expressions
like ’man’, ’Pole’, ’human being’. Some divergent opinions concern subjects,
Łukasiewicz takes the view that subjects should also be collective terms, some
other authors like Smith [39] allow singular terms as subjects as in the
syllogism: ’If all men are mortal and Socrates is a man then Socrates is
mortal’. We make use of this possibility later on.

Analysis by Łukasiewicz led to the formulation of expressions of Syllogistic
as syllogisms rendered as implications like ’If All Poles are Europeans and
All Warsawians are Poles then All Warsawians are Europeans’. Abstract forms of
syllogisms like ’* If All a is b and All b is c then All a is c’ are moods
divided into four Figures. The form of a mood is not uniquely adopted in the
literature. We follow the version by Łukasiewicz with the subject before the
predicate, some authors, e.g., Smith [39] place the predicate before the
subject. In some places in those moods ’All’ is replaced by ’Some’ or ’No’,
or, ’Some … is Not …’ . Aristotle used in validity proofs methods of reduction
of moods to some ’obvious’ moods, see [17]. Medieval scholars gave moods names
like ’Barbara’ for the mood ∗ above and that name system hid methods of
reduction cf. Lagerlund [13]. One may also use Eulerian circles (alias Venn
diagrams). Aristotle established validity of 24 moods out of possible 256.
Those 24 moods provide a complete description of relations among mass
collections of things. We introduce Syllogistic in the axiomatic form due to
Łukasiewicz [17].

The formal notation for premisses is:

$Aab$: All $a$ are $b$.

$Iab$: Some $a$ is $b$.

$Oab$: Some $a$ is Not $b$.

$Eab$: No $a$ is $b$.

Constants $A,I$ come from the first two wovels in ’affirmo’ and constants
$E,O$ are the vowels in ’nego’. Premisses are made into moods of Syllogistic
written down for short by means of sentential sentence-making functors of
conjunction $\wedge$ and implication $\supset$.

###### Definition 1

[A logical form of a mood] For $X,Y,Z\in\\{A,I,O,E\\}$, a mood is of the form

$Xam\wedge Ymb\supset Zab$ (1)

The deductive scheme for selection of all valid moods was proposed in
Łukasiewicz [17]. We recall that scheme.

Axiom schemes

(T1) $Aaa$.

(T2) $Iaa$.

(T3) $Amb\wedge Aam\supset Aab$.

(T4) $Amb\wedge Ima\supset Iab$.

Inference rules

(D) Detachment. (S) Substitution of a sentential variable or a term for
equiform occurrences of a variable.

(R) Replacement by equivalents: $(R1):\ Oab\equiv\neg Aab$, $(R2):\
Eab\equiv\neg Iab$.

Derivations of moods require some laws of contradiction, sub-alternation or
conversion.

Exemplary rules for derivations are [17]:

(L1) The law of contradiction: $Eab\supset\neg Iab$: if No a is b, then it is
not true that Some a is b.

Proof: 1\. In the theorem $p\supset p$ of sentential calculus, substitute
$p/\neg Iab$ and apply the replacement rule (R2).

(L2) The law of sub-alternation: $Aab\supset Iab$: if All a is b, then Some a
is b.

Proof: 1\. In the theorem $((p\wedge q)\supset r)\supset(q\supset(p\supset
r))$ of the sentential calculus, substitute $p/Aab$, $q/Iaa$, $r/Iab$. The
obtained formula is $\phi$.

2\. In (T4), substitute $m/a$, let the obtained formula be $\psi$.

3\. Detach $\psi$ from $\phi$, call the obtained formula $\chi$.

4\. Detach (T2) from $\chi$.

(L3) The law of conversion: $Iab\supset Iba$: if Some a is b, then Some b is
a.

Proof: 1\. In the theorem $((p\wedge q)\supset r)\supset(p\supset(q\supset
r))$ of the sentential calculus, substitute $p/Aaa$, $q/Iab$, $r/Iba$, call
the obtained formula $\phi$.

2\. In (T4), substitute $m/a$, $a/b$, $b/a$, call the obtained formula $\psi$.

3\. Detach $\psi$ from $\phi$, call the obtained formula $\chi$.

4\. Detach (T1) from $\chi$.

(L4) The law of conversion: $Aab\supset Iba$ follows from (L2) and (L3) by the
hypothetical syllogism.

We may now include after [17] an exemplary proof of the mood Barbari:
$Amb\wedge Aam\supset Iab$.

Proof of Barbari: 1. In the theorem $((p\wedge q)\supset r)\supset((s\supset
q)\supset((p\wedge s)\supset r)$ of the sentential calculus, substitute
$p/Amb$, $q/Ima$, $r/Iab$, $s/Aam$. Let the result be $\phi$.

2\. Detach (T4) from $\phi$ and call the result $\psi$.

3\. IN (L4), substitute $b/m$. The result is Barbari.

In analogous veins, one obtains 22 valid moods which along with (T3) (Barbara)
and (T4) (Datisi) make up for all 24 valid moods of Syllogistic. This leaves
232 moods verified by Aristotle (loc.cit) as invalid.

Łukasiewicz, see [17], produced an axiomatic system for rejection of moods.
Its axiom schemes are as follows; the asterisk means: the formula is rejected.

  1. 1.

F1 $*\ Acb\wedge Aab\supset Iac$.

  2. 2.

F2 $*\ Ecb\wedge Eab\supset Iac$.

Rejection rules are the following.

(F3) Rejection by detachment: if the implication $p\supset q$ is accepted and
$q$ is rejected, then $p$ is to be rejected.

(F4) Rejection by substitution: if $q$ is obtained by a valid substitution
from $p$ and $q$ is rejected, then $p$ is to be rejected.

Although it is traditional to list moods divided into Figures, yet we list all
valid moods by the conclusions. We only list premisses for the given
conclusion. After the name of a mood, we give the negation of the mood, for a
further purpose.

(M1) For the conclusion $Aab$.

(M1.1) $Amb\wedge Aam$. [Barbara] $Amb\wedge Aam\wedge Oab$.

(M2) For the conclusion $Iab$.

(M2.1) $Amb,Ima$. [Datisi] $Amb\wedge Ima\wedge Eab$.

(M2.2) $Amb,Aam$. [Barbari] $Amb\wedge Aam\wedge Eab$.

(M2.3) $Amb,Iam$. [Darii] $Amb\wedge Iam\wedge Eab$.

(M2.4) $Amb,Ama$. [Darapti] $Amb\wedge Ama\wedge Eba$.

(M2.5) $Imb,Ama$. [Disamis] $Imb\wedge Ama\wedge Eba$.

(M2.6) $Abm,Ama$. [Bamalip] $Abm\wedge Ama\wedge Eba$.

(M2.7) $Ibm,Ama$. [Dimatis] $Ibm\wedge Ama\wedge Eba$.

(M3) For the conclusion $Eab$.

(M3.1) $Emb,Aam$. [Celarent] $Emb\wedge Aam\wedge Iab$.

(M3.2) $Ebm,Aam$. [Cesare] $Ebm\wedge Aam\wedge Iab$.

(M3.3) $Abm,Eam$. [Camestres] $Abm\wedge Eam\wedge Iab$.

(M3.4) $Abm,Ema$. [Calemes] $Abm\wedge Ema\wedge Iab$.

(M4) For the conclusion $Oab$.

(M4.1) $Emb,Aam$. [Celaront] $Emb\wedge Aam\wedge Aab$.

(M4.2) $Emb,Iam$. [Ferio] $Emb\wedge Iam\wedge Aab$.

(M4.3) $Ebm,Aam$. [Cesaro] $Ebm\wedge Aam\wedge Aab$.

(M4.4) $Abm,Eam$. [Camestrop] $Abm\wedge Eam\wedge Aab$.

(M4.5) $Ebm,Iam$. [Festino] $Ebm\wedge Iam\wedge Aab$.

(M4.6) $Abm,Oam$. [Baroco] $Abm\wedge Oam\wedge Aab$.

(M4.7) $Emb,Ama$. [Felapton] $Emb\wedge Ama\wedge Aab$.

(M4.8) $Omb,Ama$. [Bocardo] $Omb\wedge Ama\wedge Aab$.

(M4.9) $Emb,Ima$. [Ferison] $Emb\wedge Ima\wedge Aab$.

(M4.10) $Abm,Ema$. [Camelop] $Abm\wedge Ema\wedge Aab$.

(M4.11) $Ebm,Ama$. [Fesapo] $Ebm\wedge Ama\wedge Aab$.

(M4.12) $Ebm,Ima$. [Fresison] $Ebm\wedge Ima\wedge Aab$.

Completeness of Syllogistic was proved in Słupecki [37], [38]. The fact that
terms may be interpreted as sets/concepts, prompted authors to transfer the
mechanisms of Syllogistic into set structures of modern logic, see Corcoran
[6], Andrade-Becerra [2], Moss [22]. In this setting completeness of
Syllogistic was proved, e.g., in Corcoran [5] and in Moss [22] who also
considers some fragments of Syllogistic and gives Henkin-style proofs of
completeness of those fragments.

Regarding Syllogistic as a formal system, we may hint at yet another proof of
completeness. We include its idea.

###### Definition 2

[Consistency] A provable formula $\phi$ in an axiomatic system $\mathcal{A}$
is consistent if and only if its negation $\neg\phi$ is not provable.

###### Definition 3

[Models for moods] A model for a mood of Syllogistic is a set of 3 Eulerian
circles each of them representing the respective premiss or conclusion.

###### Definition 4

[Completeness] An axiomatic system is complete if each valid formula is
provable.

###### Theorem 1

An axiomatic system is complete if and only if each consistent formula is true
in some model.

###### Proof 1

Suppose a formula $\phi$ is valid but not provable. Thus, the formula
$\neg\neg\phi$ is not provable and it follows that the formula $\neg\phi$ is
consistent, hence, $\neg\phi$ has a model, contrary to the validity of $\phi$.

###### Theorem 2

The axiomatic system of Syllogistic is complete and decidable.

###### Proof 2

Each of moods from (M1.1) to (M4.12) has its negation non-provable, hence,
each of these moods is consistent. Each of these moods has an appropriate
model of three Eulerian circles which proves validity of all them. This
concludes proof of completeness of the system of Syllogistic. As for
decidability, it follows by the finite model property of Syllogistic: each
valid mood has a finite model.

We need now an account of Mereology.

##  3 Mereology in a nutshell

We outline the theory of Mereology due to Leśniewski [15]. We set this
rendition within Syllogistic theory of terms. It is well known that the
primitive notion of a part is definable in Syllogistic, see, e.g., Vlasits
[44]. Many authors understand part as containing also the notion of the whole
yet the original scheme in [15] defined part as the proper part distinct from
the whole reserving for eventual part or the whole the term of an ingredient.
We call it the component. Consistency of Mereology was established in Lejewski
[14].

We express the term ’proper part’ by the formula $Pab$ read ’a is a proper
part of b’:

$Pab=_{df}Aab\wedge Oba$ (2)

It is accepted that Syllogistic can be extended by formulas with sequences of
nested operators like
$\alpha_{1}\supset(\alpha_{2}\supset(...\supset(\alpha_{n}\supset\beta)...)$
see, e.g., Lagerlund [13]; we will occasionally make use of this possibility.
The equivalent definition of the proper part will run as follows.

###### Definition 5

[Constant of proper part] $Pab=_{df}Aam\supset(Amb\supset(Pab\supset Obm))$

This definition conforms to the Leśniewski manner of defining: a definition is
a formula, with definiendum and definiens in it, which is assumed to be true
and in that way it reveals the sense of definiendum. Properties of $P$ follow.

###### Theorem 3

The following are true.

1\. It is not true that $Paa$, because $Oaa$ is false.

2\. $Pab\wedge Pbc\supset Pac$, because by the valid syllogism $Aab\wedge
Abc\supset Aac$, and def. 2 we obtain $Aac$; by def.2, we have $Oba$ and
$Ocb$; was $\neg Oca$ we would have $Aca$, hence $Acb$, hence, $\neg Obc$, a
contradiction.

3\. $Pab\supset\neg Pba$; by (1) and (2).

We next define the constant $=$ of identity.

$=ab=_{df}Aab\wedge Aba$ (3)

By properties of $A$, $=$ satisfies
$=aa,=ab\supset=ba,=ab\wedge=bc\supset=ac$.

The constant $C$ of a component is defined next.

$Cab=_{df}Pab\vee=ab$ (4)

###### Theorem 4

The following are properties of components.

1\. $Caa$.

2\. $Cab\wedge Cba\supset=ab$.

3\. $Cab\wedge Cbc\supset Cac$.

###### Proof 3

1\. follows by definition. 2. Of all possibilities:
$(Pab\vee=ab)\wedge(Pba\vee=ba)$ only $=ab\wedge=ba$ is true. 3. Follows on
lines similar to those of 2.

The mereological relation of overlap may be expressed by the constant $OV$.

###### Definition 6

[The constant of overlapping] $OVab=_{df}Iab$.

###### Theorem 5

$OVab\supset OVba\wedge OVba\supset OVab$.

###### Proof 4

Because of the equivalence $Iba$ if and only if $Iab$.

We have rendered basic notions of Mereology. The next set of notions are those
of a topological tint.

###### Definition 7

[The constant ’exterior’ EX] The term $EXab$: ’a is exterior to b’ is defined
as $EXab=_{df}Eab$.

###### Theorem 6

$EXab$ if and only if $EXba$.

###### Proof 5

The thesis follows from the fact that $Eab$ is equivalent to $Eba$.

We are closing on the notion of an interior. First, we define the notion of
relative exterior $Re_{b}am$ of $a$ to $m$ relative to $b$.

###### Definition 8

[The constant ’relative exterior’ Re] $Re_{b}am=_{df}Pab\wedge Pmb\wedge
EXam$.

###### Theorem 7

$Re_{b}am$ if and only if $Re_{b}ma$.

###### Proof 6

It follows from the equivalence of $EXam$ and $EXma$.

###### Definition 9

[Axiom A]. $((Cma\supset Inb\wedge Cnb)\supset OVmn)\supset Cab$. In reading:
[if ($m$ is any component of $a$, then Some $n$ is $b$ and this $n$ is a
component of $b$), then $m$ and $n$ overlap], then $a$ is a component of $b$.

Mereology of Leśniewski contains a discussion of the notion of a class as a
collective term. The idea is that classes are defined for collections of terms
making them into singular terms. It is here that we allow singular subjects in
formulae of Syllogistic.

###### Definition 10

[The constant ’Class’] For a collection $B$ of terms, the class of $B$,
$ClaB$: read ’a is the class of B’ is defined by the two following
requirements:

1\. $b\ is\ B\supset Cba$.

2\. $Cba\supset(IcB\wedge OVbc)$.

As shown in Tarski [40], within any mereological universe there exists the
structure of a complete Boolean algebra without the null element. Its
operations are defined as follows.

###### Definition 11

[The Boolean algebra in mereological universe] Its operations are defined as
follows.

1\. $a+b=Cl\\{t:Cta\vee Ctb\\}$.

2\. $a\cdot b=Cl\\{t:Cta\wedge Ctb\\}$.

3\. $Cl\\{all\ terms\\}=V$.

4\. $-a=Cl\\{t:EXta\\}$.

It is known that the Tarski algebra is complete due to class existence. $V$ is
the universe of all terms.

The implication $\hookrightarrow$ is defined as

###### Definition 12

[The mereological implication] $x\hookrightarrow y=-x+y$. $x\hookrightarrow y$
is valid if and only if $-x+y=V$.

The implication $\hookrightarrow$ satisfies the standard properties valid in
Boolean algebras:

1\. $(x\hookrightarrow y)\hookrightarrow((y\hookrightarrow
z)\hookrightarrow(x\hookrightarrow z))$.

2\. $x\cdot y)\hookrightarrow x$.

3\. $(x\cdot y)\hookrightarrow(y\cdot x)$.

4\. $(x\cdot(x\hookrightarrow y))\hookrightarrow(y\cdot(y\hookrightarrow x))$.

5\. $(x\hookrightarrow(y\hookrightarrow z))\hookrightarrow((x\cdot
y)\hookrightarrow z)$.

6\. $((x\cdot y)\hookrightarrow
z)\hookrightarrow(x\hookrightarrow(y\hookrightarrow z))$.

7\. $(x\hookrightarrow(y\hookrightarrow z))\hookrightarrow(((y\hookrightarrow
x)\hookrightarrow z)\hookrightarrow z)$.

One checks that the value of each formula among 1-7 is $V$, i.e., each of them
is valid.

##  4 The weighted Tarski algebra

We endow the Tarski algebra with a weight function $m$ (called a mass
assignment in Polkowski [29]) with values in the interval $(0,1]$. The
function $m$ should conform to the following requirements.

###### Definition 13

[The weight function] Concerning the function $m$:

1\. $((m(x)=1)\supset(x=V))\wedge((x=V)\supset(m(x)=1))$.

2\. $(x\hookrightarrow y)\supset((m(y)=m(x)+m((-x)\cdot y))$.

3\. $m(x)>0$ for each thing $x\in U$.

We extend $U$ by adding the empty term $e$ and declaring:

4\. $m(e)=0$.

From axiom schemes (1)-(4), we infer the following properties of the function
$m$ cf. [29].

###### Theorem 8

The function $m$ satisfies the following properties:

(m1) $(Cxy\supset x\hookrightarrow y)\wedge(x\hookrightarrow y\supset Cxy)$.

(m2) $(Cxy\supset[(x\cdot y=x)\wedge(x\cdot y=x\supset Cxy)]$.

(m3) $(x=y)\supset((m(x)=m(y))$.

(m4) $m(x+y)=(m(x)+m((-x)\cdot y))$.

(m5) $(x\cdot y=e)\supset(m(x+y)=m(x)+m(y))$.

(m6) $m(x)+m(-x)=1$.

(m7) $m(y)=m(y\cdot x)+m(y\cdot(-x))$.

(m8) $m(x+y)=m(x)+m(y)-m(x\cdot y)$.

(m9) $Cxy\supset(m(x)\leq m(y))$.

(m10) $(m(x)+m(y)=m(x+y))\supset(x\cdot y=e)$.

(m11) $Cxy\supset(x\hookrightarrow y=1)$.

(m12) $Cxy\supset(x\cdot(-y)=e)$.

(m13) $Cyx\supset(m(x\hookrightarrow y)=1-m(x)+m(y))$.

(m14) $m(x\hookrightarrow y)=1-m(x-y).$

Let us observe that (m13) replicates the Łukasiewicz implication cf.
Łukasiewicz and Tarski [19]. We now apply the notion of weight toward
definitions of uncertain terms.

##  5 Rough Mereology: A theory of uncertain terms

In this section, we develop this topic going in the footsteps of Łukasiewicz
[18] who attached weights to unary formulae over a finite universe. Our
version yields an abstract form.

The weight function induces a measure of part to a degree called also rough
inclusion cf. [30] denoted $P_{r}xy$ (read: x is a part to degree r of y).

###### Definition 14

[Part to a degree] $P_{r}xy\supset(\frac{m(x\cdot
y)}{m(x)})=r)\wedge((\frac{m(x\cdot y)}{m(x)}=r)\supset P_{r}xy)$.

###### Theorem 9

Definition 14 yields the following properties of rough inclusions cf.[30]:

1\. $P_{1}ab$ if and only if $Cab$.

2\. $P_{1}ab\wedge P_{r}ca\subset\exists s\geq r.P_{s}cb$.

We impose on approximate mereology the structure of many-valued logic. We
begin with the [0,1]-lattice of Łukasiewicz and its operators. A bit of
introduction is in order. The crucial object to begin with is a t-norm cf.
Menger [20], also Hajek [9] or Polkowski [31].

###### Definition 15

[t-norm] A t-norm $T:[0,1]\rightarrow[0,1]$ is a function conforming to
conditions:

1\. $T(x,y)=T(y,x)$.

2\. $T(x,T(y,z))=T(T(x,y),z)$.

3\. $x_{1}\leq x_{2}\supset T(x_{1},y)\leq T(x_{2},y)$.

4\. $T(x,0)=0,T(x,1)=x$.

We apply $T_{L}$, the Łukasiewicz t-norm.

###### Definition 16

[the Łukasiewicz t-norm] The Łukasiewicz t-norm is
$T_{L}(x,y)=max\\{0,x+y-1\\}$; the derived operators are: strong conjunction
$x\&y=T_{L}(x,y)$, conjunction $x\wedge y=min\\{x,y\\}$, disjunction $x\vee
y=max\\{x,y\\}$, strong disjunction
$x\underline{\vee}y=S_{L}(x,y)=min\\{1,x+y\\}$ (it is $T_{L}$-co-norm),
$x\supset_{L}y=min\\{1,x-y+1\\}$, negation $-_{L}x=1-x$.

We apply these operators to terms of rough mereology; the semantics we propose
is as follows. As semantic operators, we adopt the universal valid for each
t-norm formulae; t-norm $T$ induces the strong conjunction $\&=T$ and its
residuum $\supset_{T}$.

Other operators are defined via formulae:

  1. 1.

$\neg p\equiv p\supset 0$, where 0 denotes the value zero and at the same time
serves as the constant.

  2. 2.

$p\wedge q\equiv p\&(p\supset q)$.

  3. 3.

$p\vee q\equiv[(p\supset q)\supset q]\wedge[(q\supset p)\supset p]$.

  4. 4.

$p\underline{\vee}q\equiv\neg(\neg p\&\neg q)$.

These operators bear on our terms as follows, where $T$ stands for $T_{L}$
(clearly, these formulae are universal for any t-norm $T$):

1\. $P_{r}ca\wedge P_{s}cb\supset P_{max\\{r,s\\}}c(a+b)$ (the sum).

2\. $P_{r}ca\wedge P_{s}cb\supset P_{r\underline{\vee}s}c(a\oplus b)$ (the
strong sum).

3\. $P_{r}ca\wedge P_{s}cb\supset P_{min\\{r,s\\}}c(a\cdot b)$ (the product).

4\. $P_{r}ca\wedge P_{s}cb\supset P_{T(r,s)}c(a\odot b)$ (the strong product).

5\. $P_{r}ca\wedge P_{s}cb\supset P_{T(r,s)}c(a\supset_{X}b)$ (the
implication).

6\. $\neg P_{r}ca\supset P_{-_{T}r}c-a$ (the negation).

##  6 Applications to uncertain knowledge: Approximate decisions forecasting
in Open World setting

We consider a decision system $\mathcal{D}$ = $(O,F,d,V,V_{d})$ cf. sect.1
under the Open World Assumption which means that the set $O$ admits new
objects whose sets of feature values have been as of yet unseen.

The problem is to ascertain the decision prediction ability of the system
$\mathcal{D}$. Value and decision prediction are important problems in vital
areas of engineering, business, and, artificial intelligence, and machine
learning, see, e.g., Mohri [21], Cesa-Bianchi-Lugosi [3], Ray et al.[35] as
examples from among many others. We propose an approach applicable to decision
systems from the rough set theory point of view which uses CV dimension and
similarity measure induced from rough mereology. The scenario we envision is
the following.

Predicting agents: Objects in the decision system $\mathcal{D}$.

To be predicted: decision values for new yet unseen objects with their values
of features in the feature set $F$.

The expert giving the exact value of decision.

We are close to the setting of online learning cf. [21].

Our protocol

1\. We consider a new object $\omega$ with $F$-value set
$A=\\{\omega_{f}=f(\omega):f\in F\\}$.

2\. For each object $o\in O$, we define the touching set
$T_{A}(o,\omega)=\\{\omega_{f}:\omega_{f}=f(o)\\}$.

3\. We propose to apply to our problem the complexity measure of families of
concepts (in our case sets) known as the Vapnik - Chervonenkis dimension,
denoted VCdim, [43], see also, e.g., Mohri et al. [21]. To the best of our
knowledge it is the first application of complexity measure to the task of
decision’s value assignment in rough set theory. We now include the
appropriate definitions.

###### Definition 17

[Shattering] We say that a family of sets $\mathcal{F}$ shatters a non-empty
set $S$ if and only if for each subset $T$ of $S$ there exists
$f\in{\mathcal{F}}$ with the property that $f\cap S=T$.

###### Definition 18

[Vapnik-Chervonenkis dimension] In the notation of Def.17, if there exists a
natural number $m$ with the property that $\mathcal{F}$ shatters a set of
cardinality $m$ and it doesn’t shatter any set of cardinality $m+1$ then
VCdim($\mathcal{F})=m$; else VCdim($\mathcal{F})=\infty$.

4\. We establish a parameter $\varepsilon\in[0,1]$ and we define the notion of
an $\varepsilon$-component.

###### Definition 19

[$\varepsilon$-component] A set C is an $\varepsilon$-component of a set $X$,
when the relation $P_{\varepsilon}CX$ holds with respect to the weight
assignment $m(Z)=\frac{|Z|}{|A|}$ for $Z\subseteq A$ and inclusion degree
$\mu(X,Y)=\frac{|X\cap Y|}{|X|}$.

5\. For each $o\in O$, we denote by the symbol
${\mathcal{C}}_{\varepsilon}(o)$ the set of $\varepsilon$-components in $A$ of
the set $T_{A}(o,\omega)$.

6\. By the symbol VC(o) for $o\in O$, we denote the Vapnik-Chervonenkis
dimension of the collection of sets ${\mathcal{C}}_{\varepsilon}(o)$. The
symbol VC∗ will denote the maximal value of VC(o).

7\. We now consider the case of consistent decision system $\mathcal{D}$. In
this case, each indiscernibility class $[o]_{F}=\\{o^{\prime}:\forall f\in
F.f(o)=f(o^{\prime})\\}$ is contained in the decision class
$[o]_{d}=\\{o^{\prime}:d(o)=d(o^{\prime})\\}$.

8\. We select a parameter $\delta$, being a positive natural number, and for
each $o\in O$, we define the prediction of decision for $\omega$ as
$\hat{d}(o)\in N(d(o),\lfloor\delta\cdot\frac{VC(o)}{VC^{*}}\rfloor$),
$N(o,r)$ meaning the neighborhood of $o$ of radius $r$ in $V_{d}$. It follows
that when an object misses $\omega$, i.e., $T_{A}(o,\omega)=\emptyset$, its
prediction is its decision value.

9\. We now input the expert decision $d^{exp}$, and we say that $d^{exp}$ is
agreeable with $\hat{d}(o)$ if $d^{exp}\in
N(d(o),\lfloor\delta\cdot\frac{VC(o)}{VC^{*}}\rfloor)$ in which case we assign
the reward $R(o)$ of $1$ to $o$, otherwise, the reward $R(o)$ is $0$.

10\. We say that the system $\mathcal{D}$ $\varepsilon,\delta$ \-
approximately predicted the decision value on a set $\Omega$
=$\\{\omega_{i}:i=1,2,\ldots,\omega_{n}\\}$ if $min_{i}\sum_{o}R_{i}(o)\geq
1$, where $R_{i}(o)$ is the reward of $o\in O$ in the trial of $\omega_{i}$.
This criterion means that in each trial for any $\omega_{i}$ there is an
object $o_{i}$ with $R(o_{i})=1$.

###### Remark 1

1\. Our approach is based on the idea of reasoning by analogy, hence, we
adhere to the view that the larger the touching set, the better prediction
ability of the object. This assumption may not necessarily turn to be sound in
which case we may simply regard objects as experts predicting the decision as
their own decision plus a real expert giving the correct decision (i.e., the
reliable case) and apply the Halving Algorithm cf. [21] for it.

2\. We follow he general ideas of online learning, in particular one may trace
in our approach some echoes of Weighted Majority Algorithm by Littlestone and
Warmuth [16], cf. [21].

3\. In case of a non-consistent decision system $\mathcal{D}$, we may define
indiscernibility with respect to the set $F\cup\\{d\\}$ of features and made
it into a consistent one.

4\. Parameters $\varepsilon,\delta$ may be tried in various configurations,
e.g., instead of $\varepsilon$ \- components, one may consider all $\eta$ \-
components for $\eta\geq\varepsilon$ which may be important in the discrete
case and changing $\delta$ towards $0$ will lead towards the Halving
Algorithm.

5\. The usage of neighborhoods stresses the approximate character of the
algorithm and clearly improves the predicting ability of the system, however,
the degree of approximation depends on our choice of $\delta$ which may be
arbitrarily small. Let us observe that VC-dimension of the family
${\mathcal{N}}=\\{(N(d(o),\lfloor\delta\cdot\frac{VC(o)}{VC^{*}}\rfloor)\\}$
is at most $2\cdot\delta$ and the loss is less than $2\cdot\delta$.

6\. Concerning the parameter $\varepsilon$, let us observe that:

6.1 $\varepsilon=0$. Then, the shattered set is $|A\setminus T_{A}(o)|$,
hence, $VC*=max|A\setminus T_{A}(o)|$ = $|A|-min_{o}|T_{A}(o)|$ which prefers
small touching sets and the acceptable loss for larger touching sets becomes
of order of $\delta\cdot\frac{|A\setminus T_{A}(o)|}{VC^{*}}$.

6.2 $\varepsilon=1$. Then, $VC(o)=|T_{A}(o)|$, thus,
$VC^{*}=max_{o}|T_{A}(o)|$, hence, the decisive role in assigning neighborhood
sizes is played by cardinalities of touching sets. The loss error is at most
$\delta$ and it is set to values respectively smaller for less numerous
touching sets: the worse agent, the more restrictive condition for its
prediction.

6.3 For $\varepsilon\in(0,1)$, we need to weigh the influence of cardinalities
of sets $A,T_{A}(o)$. It follows that to satisfy the condition for
$\varepsilon$-component, a set $Q$ as a candidate should satisfy the
inequality

$|Q|\leq
max\\{\lfloor\frac{|T_{A}(o)|}{\varepsilon}\rfloor,\lceil\frac{|A\setminus
T_{A}(o)|}{1-\varepsilon}\rceil\\}$

(we have applied floor and ceiling as the quotients in the formula need not be
integers). Accordingly $VCdim(T_{A}(o))\leq|Q|$.

As elementary analysis shows, an optimal $\varepsilon$ is
$\Theta(sqrt[\frac{|T_{A}(o)|}{|T_{A}(o)|\setminus|A\setminus T_{A}(o)|}])$ if
the condition

$|T_{A}(o)|>|A\setminus T_{A}(o)|$

is satisfied. It follows that in this case our approach prefers larger
touching sets which is consistent with intuition.

7\. In case when there are for the given $\omega$ more than one $o\in O$ with
the reward $R(o)=1$, we select as the decision pointed to by the assembly of
$O$ as the $\hat{d}(\omega)$ = $\hat{d}(o)^{*}$ with the minimal loss
$|d^{exp}-\hat{d}(o)|$; in case this minimum is obtained for more than one
$o$, the decision is taken according to a chosen strategy, e.g., the random
tie resolution. Due to convexity of the value set $V_{d}$, we may apply the
idea of weighted prediction and define the predicted value as

$\hat{d}(\omega)=\sum_{o}\frac{\hat{d}(o)\cdot\frac{VC(o)}{VC^{*}}}{\sum_{o}\frac{VC(o)}{VC^{*}}}.$

8\. The factor $\frac{VC(o)}{VC^{*}}$ in the radius definition may be regarded
as the counterpart of the weights in the Weighted Average Prediction protocol,
cf. [3].

9\. The notion of regret, see Hannan [10], cf. [21], [3], for the single trial
with $\omega$, may be defined as

$|d^{exp}-\hat{d}(\omega)|-min_{o}\\{|d^{exp}-\hat{d}(o)|\\}.$

##  7 The mistakes problem

By a mistake we understand the event when an object $o$ fails to satisfy the
condition $d^{exp}\in N(d(o),\lfloor\delta\cdot\frac{VC(o)}{VC^{*}}\rfloor)$.
We denote by the symbol $\mathcal{H}$ the collection
$\\{N(d(o),\lfloor\delta\cdot\frac{VC(o)}{VC^{*}}\rfloor):o\in O\\}$. As
$\bigcup{\mathcal{H}}$ covers the decision set $V_{d}$, there must be at least
one object $o_{m}\in O$ which is mistake-free. Hence the number $Mist$ of
mistakes is at most $|O|-1$.

We assume that we are in the supervised learning mode and the supervisor
informs each object $o$ whether its decision has been correct or not. We
consider a sequence of weighted votes. In each vote, the incorrect agents are
dismissed and the remaining objects have their neighborhoods radii diminished
by a learning rate $\eta$ and these objects select new decision values
$\hat{d}^{1}(o)$. The procedure repeats until the empty set is obtained in
which case the fore last set is the set localizing the true value of decision.

Conclusion We have presented classical concept theories Syllogistic and
Mereology as preludes to the theory of Rough Mereology. Using notions of Rough
Mereology, we have defined $\varepsilon$-components in order to allow us to
define representatives for new objects coming to the decision system from the
outside world and in consequence to approximate the assignment of a decision
value to each such object. Further research may be empirical, for instance
regarding the new object as coming from the leave-one-out procedure for $O$.

## References

  * [1] Abiteboul, S., Hull, R., Vianu, V.: Foundations of Databases. Addison-Wesley, 1995.
  * [2] Andrade, E. J., Becerra, E.: Corcoran’s Aristotelian syllogistic as a subsystem of first-order logic. Revista Colombiana de Matematic$\acute{a}$s 41, 67-80, 2007\.
  * [3] Cesa-Bianchi, N., Lugosi, G.: Prediction, Learning and Games. Cambridge Univ. Press, Cambridge & New York, 2006.
  * [4] Chen, D. et al.: Uncertainty learning of rough set-based prediction under a holistic framework. Information Sciences 463–464, 129-151, 2018.
  * [5] Corcoran, J.: Completeness of an ancient logic. Journal of Symbolic Logic 37(4), 696-702, 1972.
  * [6] Corcoran, J.: A mathematical model of Aristotle’s Syllogistic. Archiv für Geschichte der Philosophie 55 (2):191-219 (1973).
  * [7] Düntsch, I., Gediga, M.: Uncertainty measures of rough set prediction. Artificial Intelligence, 2018.
  * [8] Greco, S., Matarazzo, B., Słowi nski, R.: Rough sets theory for multicriteria decision analysis. European J. Oper. Research 129(1), 1-47, 2001. DOI: 10.1016/S0377-2217(00)00167-3.
  * [9] H$\acute{a}$jek, P. Metamathematis of Fuzzy Logic. Springer Science + Business B. V., Dordrecht 1988.
  * [10] Hannan, J.: Approximation to Bayes risk in repeated plays. Contributions to the Theory of Games, 3, 97-139, 1957.
  * [11] Jian,L., Liu, S., Lin, Y.: Hybrid Rough Sets and Applications in Uncertain Decision-Making. CRC Press, Taylor and Francis, Boca Raton, FL, US, 2011.
  * [12] Komorowski, J., Polkowski, L., Skowron, A.: Rough Sets: A Tutorial. https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi= 77b8bf21ac4ba3cbe89553f7341d484e7dd43a3c
  * [13] Lagerlund, H.: ”Medieval Theories of the Syllogism”, The Stanford Encyclopedia of Philosophy (Summer 2022 Edition), Edward N. Zalta (ed.), URL = ¡https://plato.stanford.edu/archives/sum2022/entries/medieval-syllogism/¿
  * [14] Lejewski C.: Consistency of Leśniewski’s Mereology. Journal of Symbolic Logic 34(3), 321–328, 1969.
  * [15] Leśniewski, S.: Foundations of the General Theory of Sets I (In Polish: Podstawy Ogólnej Teorii Mnogości I. Polish Scientific Circle in Moscow. Moscow: The Popławski printing house, 1916. Cf. Surma et al. eds.: Leśniewski’s Collected Works. Nijhoff Intl. Philosophy Series 44/1. Springer Netherlands, 1992.
  * [16] Littlestone, N., Warmuth, M. K.: The weighted majority algorithm. Information and Computation, 108(2), 212-261, 1994.
  * [17] Łukasiewicz, J.: Aristotle’s Syllogistic from the Standpoint of Modern Formal Logic, 2nd.ed., Oxford, Clarendon Press, 1957.
  * [18] Łukasiewicz. J.: Die Logischen Grundlagen der Wahrscheinlichtkeitsrechnung. Drukarnia Polska (Polish Printing House). Kraków, 1913. Cf. Logical foundations of probabability, in L. Borkowski ed.: Jan. Łukasiewicz. Selected Works. North Holland, Polish Sci. Publ (PWN), Amsterdam Warsaw, 1-63, 1970.
  * [19] Łukasiewicz, J., Tarski, A.: Untersuchungen uber den Aussagenkalk uls. C. R. Soc. Sci. Lettr. Varsovie 23, 130-152, 1930.
  * [20] Menger, K.: Statistical metrics. Proc. Natl. Acad. Sci. USA 28, 535-537, 1942.
  * [21] Mohri, M., Rostamizadeh, A., Talwalkar, A.: Foundations of Machine Learning. MIT Press, Cambridge MA, 2012.
  * [22] Moss, L.: Completeness theorems for syllogistic fragments. https://iulg.sitehost.iu.edu/moss/uwe.pdf
  * [23] Pal, S., Polkowski, L., Skowron, A.: Rough-neuro Computing: Techniques for Computing with Words. Springer-Verlag, New York, 2003.
  * [24] Pawlak, Z.:Rough sets, International Journal of Computer and Information Sciences, 11, 341-356, 1982
  * [25] Pawlak, Z.: Rough Sets: Theoretical Aspects of Reasoning about Data. Kluwer, Dordrecht, 1991.
  * [26] Pawlak, Z., Polkowski, L., Skowron, A.: Rough Sets. Encyclopedia of Database Technologies and Applications, 575-580, 2005.
  * [27] Pawlak, Z., Słowiński, R.: Decision analysis using rough sets. Int. Trans. Opl. Res. 1(1), 107-114, 1994.
  * [28] Polkowski, L.: Rough Sets. Mathematical Foundations. Physica Verlag/Springer - Verlag, Heidelberg - New York, 2002.
  * [29] Polkowski, L.: Introducing mass-based rough mereology in a mereological universe with relations to fuzzy logics and a generalization of the Łukasiewicz logical foundations of probability. Fundam. Informaticae 166(3): 227-249 (2019).
  * [30] Polkowski, L.: Approximate Reasoning by Parts. An Introduction to Rough Mereology. Springer-Verlag Berlin Heidelberg. Intelligent Systems Reference Library 20, 2011.
  * [31] Polkowski, L.: Logic. Reference Book for Computer Scientists. ISRL 245. Springer Nature Switzerland AG, 2023.
  * [32] Polkowski, L., Artiemjew, P.: Granular Computing in Decision Approximation. An Application of Rough Mereology. ISRL 77, Springer International Publishing, Cham, Switzerland, 2015.
  * [33] Polkowski, L., Skowron, A. (eds.): Rough Sets in Knowledge Discovery 1: Methodology and Applications. Physica-Verlag, Heidelberg, 1998.
  * [34] Polkowski, L., Skowron, A. (eds.): Rough Sets in Knowledge Discovery 2: Applications, Case Studies, Software Systems. Physica-Verlag, Heidelberg, 1998.
  * [35] Ray, A., Phoha, S., Sarkar, S.: Behavior prediction for decision and control in cognitive autonomous systems. New Mathematics and Natural Computation 9(3), 2013.
  * [36] Ross, W. D., (ed.): Aristotle’s Prior and Posterior Analytics. Oxford, Clarendon Press 1951.
  * [37] Słupecki, J.: Z badan nad sylogistyka̧ Arystotelesa (Investigation on Aristotle’s Syllogistic, in Polish). Travaux de la Soci$\acute{e}$t$\acute{e}$ des Sciences et des Lettres de Wroclaw, S$\acute{e}$rie B, 9. Wroclaw (1948).
  * [38] Słupecki, J.: On Aristotelian syllogistic. Studia Philosophica 4 (1949-1950), pp. 275-300.
  * [39] Smith, R.: ”Aristotle’s Logic”, The Stanford Encyclopedia of Philosophy (Winter 2022 Edition), Zalta, E. N., Nodelman, U. (eds.), URL = ¡https://plato.stanford.edu/archives/win2022/entries/aristotle-logic/¿.
  * [40] Tarski, A.: Zur Grundlegung der Booleschen algebra I. Fundam. Math. 24, 177-198, 1935.
  * [41] Transactions on Rough Sets. Part of the series Lecture Notes in Computer Science. Springer - Verlag. https://www.springer.com/series/7151.
  * [42] Udristoiu, A. et al.: Predicting Students’ Results Using Rough Sets Theory. In: LNCS 8669: Proc. Int. Conf. IDEAL 2014, Salamanca, Spain, 2014.
  * [43] Vapnik, V. N., Chervonenkis, A. Ya.: On the Uniform Convergence of Relative Frequencies of Events to Their Probabilities (in Russian). Dokl. Akad. Nauk. 181 (4): 781, 1968. Eng. transl. (Steckler, B.transl.): On the Uniform Convergence of Relative Frequencies of Events to Their Probabilities. Theory of Probability and Its Applications 16 (2): 264, 1971. doi:10.1137/1116025.
  * [44] Vlasits, J.: Mereology in Aristotle’s Assertoric Syllogistic. History and Philosophy of Logic 40(1), 1-11, 2019.
  * [45] Wu, S. et al.: A Classification Prediction Method using Rough Set and Decision Tree. In: Proc. 8th Int. Conf. Computing and Artificial Intelligence ICCAI ’22, 552–557, 2022 .https://doi.org/10.1145/3532213.3532297
  * [46] Yao, Y.: Decision-theoretic rough set models. In: Proc. RSKT 2007, 1-12, 2007.