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USE OF MATHEMATICAL
MODELS FOR
SPECIFICATION AND
VALIDATION
SEMESTER PROJECT

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TABLE OF CONTENTS
Introduction to mathematical models
Uses of mathematical models
Introduction to specification
Validation in software engineering
Examples of mathematical models used for specification
and validation
Z specification
Z specification for library management system
Petri nets
Petri nets real-life examples
Sets
Uses of sets
Examples of uses of sets in validation and specification
Conclusions
References
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INTRODUCTION TO MATHEMATICAL
MODELS
 A mathematical model is a representation of a system or process
using mathematical concepts and equations
Its purpose is to capture the essential features of a system in a
simplified form
Mathematical models can take many forms, ranging from simple
equations to complex systems of equations and computer simulations

USES OF
MATHEMATICAL
MODELS
Predictive modeling
System optimization
Risk analysis
Quality assurance

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In software engineering, specifications refer to the detailed requirements
and design of a software system.
They describe what the system should do, how it should behave, and what
it should look like.
Specifications are used to guide the development of a software system and
are typically created at the beginning of the software development process.
They serve as a reference for the developers and help to ensure that the
system is developed according to the desired requirements and
specifications.
INTRODUCTION TO SPECIFICATION

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VALIDATION IN SOFTWARE ENGINEERING
In software engineering, validation refers
to the process of evaluating a software
system or component during or at the end
of the development process to determine
whether it satisfies the specified
requirements.
Validation is typically carried out through
testing, which involves executing the
system and comparing its actual behavior
to the expected behavior specified in the
requirements.

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Finite state machines:
Petri nets
Process algebras:
Model checking.
Stochastic models
Queueing models
Differential equation models
EXAMPLES OF MATHEMATICAL MODELS
USED FOR SPECIFICATION AND VALIDATION

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Z specification is a formal specification language that is used to specify the
behavior of software systems and other systems that can be described using
mathematical concepts. It is based on set theory and predicate logic and allows
the specification of systems in a precise, formal manner.
Z SPECIFICATION

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Z SPECIFICATION FOR INVENTORY SYSTEM
Here is an example of an inventory system. A developer wants to monitor his business
of stock, if the stock is available, deliver the goods to the customer and bill the invoice.
The system will perform the following tasks:
Read files from the database.
 Login. If login success, the following activities can be done:
 Monitor Stock
 Update Stock
 Get Order
 Generate Invoice
 View Order

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Petri nets are a type of mathematical model used to represent the behavior of
systems that involve concurrency and communication.
 They are named after the mathematician Carl Adam Petri, who developed the
model in the 1960s.
Petri nets consist of two types of elements: places and transitions. Places
represent locations where tokens (representing events or tasks) can be stored,
and transitions represent actions or processes that consume and produce
tokens.
Petri nets are typically represented visually as a directed graph, with places
represented as circles and transitions represented as rectangles.
PETRI NETS

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Manufacturing systems
Computer networks
Biological systems
Transportation systems
PETRI NETS REAL LIFE EXAMPLES

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PETRI NETS REAL LIFE EXAMPLES OF
VIDEO GAME INDUSTRY

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PETRI NETS REAL LIFE EXAMPLES OF
ORDER PROCESS

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SETS
In software engineering, sets are collections of objects that are grouped together
based on some common characteristic. Sets are a fundamental concept in
mathematics and are often used in software engineering to represent and analyze
collections of data.

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Representing collections of data.
Defining relationships between data.
Specifying constraints and requirements.
Defining the set of all possible input values for a system.
Specifying the set of all valid responses from a system.
Defining the set of all possible states that a system can be in.
USES
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EXAMPLE 1:
. If we are developing a software system for an online shopping website. The system
needs to be able to handle a wide range of product categories, such as electronics,
clothing, and home goods.
One way to represent the different product categories in the system is to use a set. The
set could be defined as follows:
Product Category = {electronics, clothing, home goods}.
The set could be used in the specification and validation of the system in several ways.
EXAMPLES OF USES OF SETS IN VALIDATION
AND SPECIFICATION
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EXAMPLE 2:
 For Specification:
Sets can be used for the specification by defining a set of possible states that a system can be in. For example, a
traffic light system might have a set of states that includes "red“, "yellow," and "green." This set of states can be used
to specify the behavior of the traffic light system.
Sets can also be used to define the allowed inputs and outputs for a system. For example, a vending machine might
have a set of allowable input coins and a set of allowable output products. This can be used to specify the behavior
of the vending machine and ensure that it only accepts valid input and provides valid output.
 For Validation:
In validation, sets can be used to check that the system is behaving correctly.
For example, if the traffic light system is in the "red" state, it should not allow any vehicles to pass through the
intersection. If the vending machine is accepting input coins, it should only accept coins that are in the set of
allowable input coins. If the system is not behaving as specified, it can be considered invalid.
EXAMPLES OF USES OF SETS IN VALIDATION
AND SPECIFICATION
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In proposed document, we have explained mathematical models used
in formal methods for validation and specification. The mathematical
models explained are Z-schemas, Petri-nets, and sets.
CONCLUSIONS

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Zhang, W., Mohamed, A.R. and Ong, W.J., 2020. Z‐Schema‐Photokatalysesysteme für die
Kohlendioxidreduktion: Wo stehen wir heute?. Angewandte Chemie, 132(51), pp.23092-23115.
Groves L. Refinement and the Z schema calculus. Electronic Notes in Theoretical Computer
Science. 2002 Nov 1;70(3):70-93.
https://slideplayer.com/slide/2301995/
Peterson, James L. "Petri nets." ACM Computing Surveys (CSUR) 9, no. 3 (1977): 223-252.
Murata, T., 1989. Petri nets: Properties, analysis, and applications. Proceedings of the
IEEE, 77(4), pp.541-580.
REFRENCES

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