Discrete mathematical structures are widely used in machine learning. Boolean algebra, in particular, is used in many machine learning algorithms and applications. The document discusses: 1) How Boolean logic is used in neural networks, with layers representing conjunction, disjunction, etc. 2) An example of classifying patterns with perceptrons using truth tables of Boolean AND and OR operations. 3) Real-life applications that use Boolean logic and machine learning, including diagnosing diseases from biomarker data and building a prognostic classifier for neuroblastoma. Discrete mathematics concepts like Boolean algebra thus play an important role in machine learning algorithms and their applications to solve real-world problems.

1. I B.Tech. - I Semester
SCHOOL OF ENGINEERING AND TECHNOLOGY
DISCRETE MATHEMATICAL STRUCTURES
Department: CSE(AI&ML)
Uses of Discrete Mathematical
Structures(DMS) in Machine Learning

2. #GROUP MEMBERS
61. MAHESH.K
62. NIKHITHA REDDY.K
63. K.B.VEDANTH
64. REDDY SREEJA.K
65. THANMAYI.K
66. VAMSI.K
67. HEMA NANDA REDDY.K
68. ARJUN SAI.K
69. GAYATHRI.K
70. SURYA PRAKASH.M
71. MADHULIKA.M
72. SIREESHA.M

3. #PREFACE
• The fundamentals of machine learning are deeply Enrouted in discrete
Mathematics. It is introduced in machine learning to overcome the drawbacks of
the field.
• One of the major drawbacks is that machine learning
algorithms follows kind of Blackbox technique.
• Algorithms like random forest and decision trees
can describe the working but many times we don’t
get good results .These kind of drawback can be
resolved using Boolean algebra.
• Also, introduction of Boolean algebra in machine learning
made use of Boolean algebra to build sets of
understandable rules for excellent performance.
• perceptron is an algorithm for supervised learning
of binary classifiers

4. #BOOLEAN CONJUCTION
• Logical conjunction is an operator on two values typically the values of two
proposition that produces a value of true if and only if both of its operands area true
• For example:
P: Two is an even number.
Q: Two is a prime number.
• Most of the background processes in computers uses Boolean logics to process and
maintain the data.
• Differential diagnosis of pleural mesothelioma is also achieved using Logic Learning
machine
( P ∧ Q): Two is an even and a prime number

5. #BOOLEAN DISJUCTION
• The logical connectivity "Disjunction" of two propositions A and B is denoted by
(A v B). The truth value is "True" if anyone of the propositional variable A or B is
True, otherwise it is "False".
• For example:
P: Two is an even number.
Q: Two is a prime number.
• Solid modeling systems for computer aided design and machine learning offer a
variety of methods for building objects from other objects, combination by
Boolean Operations.
• Today, all modern general purpose computers perform their functions using two-
value Boolean logic.
(P v Q) : Two is an even or a prime number

6. #BOOLEAN IMPLICATION
• If premise then consequences
• In the above rule, the premise contains one or many conditions on the input
• the consequence contains output value
• condition on the premises can have different forms according to the type of input : If
variables are categorical then the input value must be in a subset If variables are
ordered then the condition is written as inequality or an interval
• Learning Algorithms via Neural Logic Networks in Machine learning is achieved
by using Boolean Implication.

7. #BOOLEAN BI-CONDITIONAL
• In a biconditional statement the output is True only if the both input values
are True or if the both input values are False .
• **For either of the input values being True or False , the value is False .
• Thus , biconditional PQ may be read by following ways :
• 1) P if and only if Q .
• 2)P is equivalent to Q.
• 3)P is necessary and sufficient condition for Q.
• 4)Q is necessary and sufficient condition for P.
• Switching Neural Networks of Machine learning domain maintains and processes
the data flow using the Boolean Multi-conditional statements in background.

8. #BACKGROUND PROCESSES USING BOOLEAN
ALGEBRA IN MACHINE LEARNING
• An example of a model where Boolean algebra is used with the layers of connection is
neural networks. In the architecture of this work, we find the first layer of the model is
containing an A/D converter that transforms the input samples into binary strings, and
then the next two layers of the network use a positive Boolean Function that solve
in an A/D converter domain the original classification problem. The
function used by the neural network in this work can be written in the
form of intelligible rules. A proper method for reconstructing the
positive Boolean function can be adapted to train the model.
They have named the model Switching Neural Network. The image
shown is a presentation of the schema of Switching Neural Networks.
• We can consider this work as a neural network with three feed-forward
layers where the first one is used for binary mapping and the next two
layers are used for expressing the positive Boolean function. Every port
in the second layer is connected only to some of the outputs leaving the
Latticizers.
Switching Neural Networks: A New Connectionist Model for Classification:

9. LEARNING ALGORITHMS VIA NEURAL LOGIC NETWORKS:
• This work is based on making a paradigm for neural networks to learn using the Boolean neural network.
Basic differential operators from the Boolean system such as conjunction, disjunction, and exclusive-OR are
used. These Basic differential operators can be combined with deep neural networks like MLP. This work
can be a witness to overcoming some of the drawbacks of the MLP for learning discrete-algorithmic tasks.
The model of this work is known as Neural Logic Network in which Neural Logic Layers based on have
been introduced using any Boolean function.
• Types of these Neural Logic Layers are as follows:
• 1)Neural conjunction layer: hold the conjunction function
from Boolean algebra.
• 2)Neural disjunction layer: holds the disjunction function
from Boolean algebra.
• 3)Neural XOR Layer: hold the XOR or exclusive OR
function from Boolean algebra.
• The image below is a comparison of MLP vs
NLN for learning Boolean functions.

10. .
• The above-given approaches are two basic works which after introduction have been
updated and used in various real-life applications. In the next section , we will
discuss the real-life application of machine learning algorithms that are using
Boolean algebra.

11. #MAJOR APPLICATIONS
• To demonstrate how perceptrons can classify the linearly separable patterns, the truth tables of
Boolean AND or OR operations can be used.
Perceptron:
• A perceptron is an artificial neuron it is the
simplest possible neural network. Neural network
are the building blocks of machine learning.
• As AND and OR gates are linearly separable the perceptron algorithm will be valid.
DEMONSTRATION OF CLASSIFICATION BY A PERCEPTRON

12. .
• Output of AND gate is 1 only if both the inputs are 1.
• The results of the operations indicate the class labels while the input points represent
the data points in the 2D data space.

13. DEMONSTRATION OF CLASSIFICATION BY A PERCEPTRON
• XOR:
• XOR, or Exclusive Or, is a binary logical operator that takes in Boolean inputs and gives out True
• if and only if the two inputs are different. This logical operator is especially useful when we want to check two conditions
that can't be simultaneously true. The following is the Truth table for XOR function
• XOR in terms of AND,OR,NOT:
XOR gate can be written as a combination of AND gates, NOT gates and OR gates in the following way:
• a XOR b = (a AND NOT b)OR(b AND NOT a)
• PERCEPTRON :
• A Perceptron is an Artificial Neuron
• It is the simplest possible Neural Network
• Neural Networks are the building blocks of Machine Learning
• The Perceptron can classify the input patterns of Boolean AND or OR operations with a single-layer architecture. But
they fail to classify the patterns of an XOR operation. To classify them correctly, lead the development of the multilayer
Perceptron.
• MULTI LAYER PERCEPTRON:
• Multi layer perceptron (MLP) is a supplement of feed forward neural network. It consists of three types of layers—the
input layer, output layer and hidden layer.

14. DIFFERENT GATES USED IN LSTM RECURRENT NEURAL
NETWORK:
• Forget Gate(f): It determines to what extent to forget the previous data.
• Input Gate(i): It determines the extent of information be written onto the Internal
Cell State.
• Input Modulation Gate(g): It is often considered as a sub-part of the input gate and
much literature on LSTM’s does not even mention it and assume it is
inside the Input gate.
• Output Gate(o): It determines what output(next Hidden State) to generate from the
current Internal Cell State.

15. #REAL LIFE APPLICATIONS
• We can see the uses of this approach, i.e. machine learning with Boolean algebra, in various fields
like medicine, financial services, and supply chain management. In this section of the article, we
will discuss some of the important and famous real-life applications that are listed below.
• Multiple osteochondromas (MO), previously known as hereditary multiple exostoses (HME), is an
autosomal dominant disease characterized by the formation of several benign cartilage-capped
bone growth defined osteochondromas or exostoses. Various clinical classifications have been
proposed but a consensus has not been reached. The aim of this study was to validate (using a
machine learning approach) an “easy to use” tool to characterize MO patients in three classes
according to the number of bone segments affected, the presence of skeletal deformities and/or
functional limitations. The proposed classification has been validated (with a highly satisfactory
mean accuracy) by analyzing 150 different variables on 289 MO patients through a Switching
Neural Network approach (a novel classification technique capable of deriving models described by
intelligible rules in if-then form). This approach allowed us to identify ankle valgism, Madelung
deformity and limitation of the hip extra-rotation as “tags” of the three clinical classes. In
conclusion, the proposed classification provides an efficient system to characterize this rare disease
and is able to define homogeneous cohorts of patients to investigate MO pathogenesis.
Validation of a new multiple osteochondromas classification through
Switching Neural Networks

17. DIFFERENTIAL DIAGNOSIS OF PLEURAL MESOTHELIOMA
USING LOGIC LEARNING MACHINE
• Tumour markers are standard tools for the differential diagnosis of cancer. However, the occurrence of nonspecific symptoms and different
malignancies involving the same cancer site may lead to a high proportion of misclassifications.
• Classification accuracy can be improved by combining information from different markers using standard data mining techniques, like Decision
Tree (DT), Artificial Neural Network (ANN), and k-Nearest Neighbour (KNN) classifier. Unfortunately, each method suffers from some
unavoidable limitations. DT, in general, tends to show a low classification performance, whereas ANN and KNN produce a "black-box"
classification that does not provide biological information useful for clinical purposes.
• Logic Learning Machine (LLM) is an innovative method of supervised data analysis capable of building classifiers described by a set of
intelligible rules including simple conditions in their antecedent part. It is essentially an efficient implementation of the Switching Neural
Network model and reaches excellent classification accuracy while keeping low the computational demand.
• LLM was applied to data from a consecutive cohort of 169 patients admitted for diagnosis to two pulmonary departments in Northern Italy from
2009 to 2011. Patients included 52 malignant pleural mesotheliomas (MPM), 62 pleural metastases (MTX) from other tumours and 55 benign
diseases (BD) associated with pleurisies. Concentration of three tumour markers (CEA, CYFRA 21-1 and SMRP) was measured in the pleural
fluid of each patient and a cytological examination was also carried out.
• The performance of LLM and that of three competing methods (DT, KNN and ANN) was assessed by leave-one-out cross-validation.
METHODS

18. Mesothelamia condition
Cure Algorithm using Machine Learning

19. USE OF ATTRIBUTE DRIVEN INCREMENTAL DISCRETIZATION
AND LOGIC LEARNING MACHINE TO BUILD A PROGNOSTIC
CLASSIFIER FOR NEUROBLASTOMA PATIENTS
• Now, we will discuss some of the important and famous real life applications as
listed below
• This is applied to make a prognostic classifier for neuroblastoma patients
• Neuroblastoma is type of cancer that is mainly discovered in the small gland
• In basic , this classifier consists of 9 rules utilising mainly two conditions of the
relative expression of 11 probe set algorithm and applied to microarray data and
patients classification

20. Analysis and prediction of state of children in future using
Machine Learning
Background processing of analytics using
Boolean Algebra

21. DISCRETE MATHEMATICS PLAY A MAJOR ROLE
IN OUR REAL LIFE APPLICATIONS AS WE HAD
DISCUSSED NOW