1. Formal Modeling Concepts
Quick Review
Akram Salah
ISSR 2013

2. Modeling What is a Model?
A Model is a purposeful abstract representation of reality
Purpose
You may have more than one model for the same reality for different purpose
Abstract
Does not represent instances
Schematic description
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3. Models Textual Visual Formal Pseuducode, structured English
Flow charts, UML, ERD
Formal
Mathematics, logic, graphs
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4. Models Enhance better understanding Higher level of thinking
Important in design
Better communication
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5. Sets
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6. Sets A set is a collection (group) of elements Tuple (List)
Notation { }, {a, x, book} {s | student (s)}
Ordering
Repetition
Explicit/ Implicit
Cardinality (Function)
F : Empty Set
Tuple (List)
Notation( )
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7. Relationships Membership: Subset: Equivalence:
(I: Element & Set, O: Boolean)
Subset:
(I: Two Sets, O: Boolean)
Equivalence:
Note: Subset & Equivalence
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8. Operations Union: Intersection Difference (I: Two Sets, O: a Set)
Commutative
Intersection
Difference
Not Commutative
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9. Operations Cartesian Product S x R = {(s, r) | s e S & r e R}
Cardinality
Order
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10. Relation A relation, R, between two sets S and T
Meaningful association
A set of pairs, one from S and one from T
R = {(s, t) | s e S & t e T}
A subset from Cartesian Product
Properties
Order or Arity
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11. Functions F: X -> Y or y = f (x). Domain X Co-domain Y
Image of x under f is y
Range
A relation with unique images
Cardinality as a function
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12. Graph theory
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13. Graphs
A graph G is a construct of two finite sets, V={v1, v2, …, vn} of vertices and a set E={e1, e2, …, em} of edges.
Each edge is a pair of vertices from V: ei=(vj, vk), where vj and vk e V.
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14. Graphs An edge can be directed or undirected
A directed graph is a type of graph in which each edge (vi , vj) is directed, otherwise it is undirected.
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15. Graphs
A path, from v1 to vk in a graph G(V, E) is a sequence of vertices (v1, v2, …, vk-1, vk) such that {(v1, v2), (v2, v3), …, (vk-1, vk)} exist in E.
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16. Graphs A cycle in a graph G is a path (vi, …, vi)
Cyclic graph is a graph in which there is any cycle, otherwise it is acyclic.
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17. Graphs
Connected graph is a graph in which there is a path from every vertix to all other vertices, otherwise it is disconnected.
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18. Trees A tree is a connected acyclic graph. Root (out edges)
Leaves (in edges)
Height (length of the path from root to a leaf)
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19. Logic
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20. Logic
Logic is the branch of science that studies the relationship between premises (Assumptions) & results (Conclusion).
Symbolic Logic
Well-Formed-Formulae
Propositional
Predicate
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21. Propositional Logic Atomic Formula is a proposition.
A proposition is a statement that can be either true or false, but not both.
P: It is hot
Q: The air condition is on
R: Lights are on
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22. Propositional Logic WFF in Propositional Logic:
A proposition is a formula (Atomic Formula)
If F is a formula, then ~F (not F) is also a formula
If F & G are Formulae, then
F and G (Conjunction) is a formula
F or G (Disjunction) is a formula
If F then G is a formula
F iff G is a formula (if and only if)
Nothing Else is a formula
Note: Algebraic rules & DeMorgan’s
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23. Evaluation of a Formula
A Proposition has a value (True or False)
A formula has a values ( true or false) under each of its interpretations.
An interpretation of a formula is the result of each of its atoms by a true or false.
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24. Evaluation of a Formula
A formula is said to be valid if it is true under all its interpretations, otherwise it is invalid.
A formula is said to be inconsistent if it is false under all its interpretations, otherwise it is consistent.
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25. Predicate Logic
A predicate is a logic statement that has an argument. Usually the predicate applies to the argument(s).
Student (John): John is a student
Has-book(John): John has book
Likes(John, Mary): John Likes Mary
Student(x) has x as a variable
When x is substituted with a value, it is called an interpretation of F
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26. Predicate Logic WFF in Predicate Logic
A predicate is a formula (Atomic Formula)
If F is a formula, then ~F (not F) is also a formula
If F & G are Formulae, then
F and G (Conjunction) is a formula
F or G (Disjunction) is a formula
If F then G is a formula
F iff G is a formula
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27. Predicate Logic
If F is a formula that contains a variable x defined over a domain D, then
(“For All” x) F is evaluated as true only if it is true for each x e D.
(“There Exists” x) F is evaluated as true if at least one value of x e D makes F true.
Nothing Else is a formula
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28. Notes on Quantifiers For All & There Exists are the quantifiers.
A quantifier has a scope.
If all the variables in a formula are quantified, it is called bound, otherwise it is loose.
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29. Evaluation of Formula
The same definitions of Validity and Consistency apply in general.
“For all x” F(x) is true if F is true for all values of x e D
“There Exists x” F(x) is true if any value of x e D makes F true
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30. Logical Consequence
Given a set of formulae, G, F is a logical consequence from G, G |= F, if we can prove F using a subset of G.
{F u G} is consistent
{~F u G} is inconsistent
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31. Proof Procedure
The path from the assumption to the result is called the proof procedure.
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