12 - Overview

Overview

www.tudorgirba.com

computation

information information
computer

Computer science is the systematic study of
information and computation.

Boolean algebra Predicate logic
1, 0, ∧, ∨, ¬ ∃, ∀

Sets Relations
U, ∅, ∈, ∪, ∩, , ⊆ AxB, aRb

Trees Graphs
node, edge, path, binary node, edge, cycle, path

Complexity Program veriﬁcation
O(g), Ω(g), θ(g) {P} S {Q}, wp(S,Q)

Models and languages Programming languages
model, meta, syntax, semantics programming = modeling

+ +

a = 0 a = 1

y = 0 y = 1

Conjunction (AND) Disjunction (OR) Negation (NOT)
a ∧ b = y a ∨ b = y ¬ a = y
0 0 0 0 0 0 0 1
0 1 0 0 1 1 1 0
1 0 0 1 0 1
1 1 1 1 1 1

Neutral elements Commutativity
a ∧ 1 = a a ∨ b = b ∨ a
a ∨ 0 = a a ∧ b = b ∧ a

Zero elements Associativity
a ∧ 0 = 0 a ∧ (b ∧ c) = (a ∧ b) ∧ c
a ∨ 1 = 1 a ∨ (b ∨ c) = (a ∨ b) ∨ c

Idempotence Distributivity
a ∧ a = a a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c)
a ∨ a = a a ∨ (b ∧ c) = (a ∨ b) ∧ (a ∨ c)

Negation DeMorgan’s
a ∧ ¬ a = 0 ¬ (a ∧ b) = (¬ a) ∨ (¬ b)
a ∨ ¬ a = 1 ¬ (a ∨ b) = (¬ a) ∧ (¬ b)

Boolean algebra Propositional logic
0, 1 True, False

boolean variables: a∈{0, 1} atomic formulas: p∈{True,False}

boolean operators: ¬ ∧ ∨ logical connectives: ¬ ∧ ∨

boolean functions: propositional formulas (propositions):

0, 1 True and False

boolean variables atomic formulas

if a is a boolean function, then ¬a is a if a is a propositional formula, then ¬a
boolean function is a propositional formula

if a and b are boolean functions, then if a and b are propositional formulas,
a∧b, a∨b, a b, a b are boolean then a∧b, a∨b, a b, a b are
functions propositional formulas

truth value of a boolean function (truth truth value of a propositional formula (truth
tables) tables)

Universal quantiﬁer ∀ : Existential quantiﬁers ∃ :

∀x: U(x): F(x) ∃x: U(x): F(x)

∀x:x∈N:x>5 ∃x:x∈N:x>5
for all natural numbers x, for some natural numbers x,
x is greater than 5 x is greater than 5

∀ i: 0≤i<N: ai=0 ∃ i: 0≤i<N: ai ≠ 0
all elements ai of array a (of length N) are at least one element of array a (of length N) is
zero not zero

∀ i: 1≤i<N: ai-1 ≤ ai ∃ i: 1≤i<N: ai-1 > ai
elements of array a are monotonically elements of array a are not monotonically
increasing increasing

∀ i,j: 0≤i<j<N: ai ≤ aj ∃ i,j: 0≤i<j<N: ai > aj
elements of array a are monotonically elements of array a are not monotonically
increasing increasing

∀ i,j: 0≤i,j<N: ai = aj ∃ i,j: 0≤i<j<N: ai ≠ aj
all elements of array a are equal not all elements of array a are equal

∀ i: 0<i<N: ai ≤ a0 ∃ i: 0<i<N: ai > a0
a0 is the greatest element of array a a0 is not the greatest element of array a

s
E xample

De Morgan’s Axiom ¬∃ :
¬(∃ x: U(x): F(x)) ∀ x: U(x): ¬ F(x)

De Morgan’s Axiom ¬∀ :
¬(∀ x: U(x): F(x)) ∃ x: U(x): ¬ F(x)

ambiguous

All elements of an array a of length N are either zero or one

(1) ∀i: 0≤i<N: ( ai=0 ∨ ai=1 )

(2) (∀i: 0≤i<N: ai=0) ∨ (∀i: 0≤i<N: ai=1)

Example

Neutral elements Commutativity
A ∩ U = A A ∪ B = B ∪ A
A ∪ ∅ = A A ∩ B = B ∩ A

Zero elements Associativity
A ∩ ∅ = ∅ A ∩ (B ∩ C) = (A ∩ B) ∩ C
A ∪ U = U A ∪ (B ∪ C) = (A ∪ B) ∪ C

Idempotence Distributivity
A ∩ A = A A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
A ∪ A = A A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)

Complement DeMorgan’s
A ∩ A’ = ∅ (A ∩ B)’ = (A’) ∪ (B’)
A ∪ A’ = U (A ∪ B)’ = (A’) ∩ (B’)

Reﬂexivity
A ⊆ A.

Anti-symmetry
A ⊆ B ∧ B ⊆ A A = B.

Transitivity
A ⊆ B ∧ B ⊆ C A ⊆ C

Cartesian product AxB

AxB={ (a,b) | a∈A and b∈B }

A × ∅ = ∅.
A × (B ∪ C) = (A × B) ∪ (A × C).
(A ∪ B) × C = (A × C) ∪ (B × C).

N-ary Relation

A1, A2, ..., An
R ⊆ A1 x A2 x...x An

Binary Relation

A 1, A 2
R ⊆ A1 x A2

(a,b) ∈ R
aRb

Reﬂexive relation
every element x of A is in relation R with itself
∀x: x∈A: xRx
Equivalent relation

Symmetric relation
if there is a relation between x and y, then there is a relation between y and x
∀x,y: x,y∈A: xRy yRx

Transitive relation
...
∀x,y,z: x,y,z∈A: (xRy ∧ yRz) xRz

Irreﬂexive relation
no element x of A is in relation R with itself
∀x: x∈A: ¬(xRx)

Antisymmetric relation
if there is a relation between x and y and one between y and x, then x equals y
∀x,y: x,y∈A: (xRy ∧ yRx) x=y

Asymmetric relation
xRy and yRx cannot hold at the same time
∀x,y: x,y∈A: xRy ¬(yRx)

Non-symmetric relation
a relation that is not symmetric
∀x,y: x,y∈A: (xRy) ∧ ¬(yRx)

Total relation
R is deﬁned on the entire A.
∀x,y: x,y∈A: xRy ∨ yRx

Acyclic relation
there are no elements with transitive closure to themselves
∀n: n∈N:
( ¬(∃x1, x2, ...,xn:
x1, x2, ...,xn∈A:
x1Rx2 ∧ x2Rx3 ∧ ... ∧ xn-1Rxn ∧ xnRx1 ) )

R ⊆ AxA
{(a,b), (b,c), (c,d)} a b c d

Transitive closure
R1 = R;
∀i:i>1:Ri = Ri-1 ∪ {(a,b) | ∃c:: (a,c)∈Ri-1 ∧ (c,b)∈Ri-1}.
Rt = ∪i≥1Ri = R1 ∪ R2 ∪ R3 ∪ ...

n1
n10
n2

n9
n3

n8
n4

n5 n7
n6

n1

n2 n3 n4

n5 n6 n7 n8 n9

n10

not a binary tree binary tree

n1 n1

n2 n3 n2 n3

n4 n5 n6 n4 n5 n6

complete binary tree

n1

n2 n3

n4 n5 n6 n7

breadth-ﬁrst traversal

1

n1

2 3

n2 n3

4 5 6

n4 n5 n6

preorder(node)
visit(node).
preorder(node.left).
preorder(node.right). 1

n1

2 5

n2 n3

3 4 6

n4 n5 n6

inorder(node)
inorder(node.left).
visit(node).
inorder(node.right). 4

n1

2 6

n2 n3

1 3 5

n4 n5 n6

postorder (node)
postorder(node.left).
postorder(node.right).
visit(node). 6

n1

3 5

n2 n3

1 2 4

n4 n5 n6

G = (V, E)
E = { {u,v} | u,v ∈ V}

a e

c d g

b f

Path Cycle

a e

c d g

b f

Path: (b, a, c); Length (b, a, c) = 2
Path: (b, d, f)
Cycle: (f, g, e, d, f); Length (f, g, e, d, f) = 4

Hamiltonian path Eulerian path

a e a e

c d g c d g

b f b f

1 public long factorial (int n) {

2 long factorial = 1;

3 int i = 1;

4 while ( i ≤ n ) {

5 factorial = factorial * i;

6 i = i + 1;

7 }

8 return factorial

9 }


2 long factorial = 1; 1

3 int i = 1; 1

4 while ( i ≤ n ) { 4

5 factorial = factorial * i; 4

6 i = i + 1; 3

7 }

8 return factorial 1

9 }


2 long factorial = 1; 1 1

3 int i = 1; 1 1

4 while ( i ≤ n ) { 4 n

5 factorial = factorial * i; 4 n

6 i = i + 1; 3 n

7 }

8 return factorial 1 1

9 }


2 long factorial = 1; 1 1

3 int i = 1; 1 1

4 while ( i ≤ n ) { 4 n

5 factorial = factorial * i; 4 n

6 i = i + 1; 3 n

7 }

8 return factorial 1 1

9 }

f(n) = 1+1+4*n+4*n+3*n+1 = 11*n+3

Big O Notation

f: N -> N
g: N -> N

f ∈ O(g)
∃ c, n0 :
(c,n0∈N) ∧ (c>0) :
(∀n : n∈N ∧ n>n0 : f(n) ≤ c*g(n))

O(1) ∈ O(log n) ∈ O(n) ∈ O(n2) ∈ O(nk) ∈ O(2n)

!"#$%$&'()("'*$+ ,'-.$/'--
O(1) Constant

O(log n) Logarithmic

O(n) Linear

O(n2) Quadratic

O(nk) Polynomial

O(kn) Exponential

Best case estimation

f: N -> N
g: N -> N

f ∈ Ω(g)
∃ c, n0 :
(c,n0∈N) ∧ (c>0) :
(∀n : n∈N ∧ n>n0 : f(n) ≥ c*g(n))

Average case estimation

f: N -> N
g: N -> N

f ∈ θ(g)
∃ c 1, c 2, n 0 :
(c1,c2,n0∈N) ∧ (c1>0) ∧ (c2>0):
(∀n : n∈N ∧ n≥n0 : c1*g(n) ≤ f(n) ≤ c2*g(n))

solvable
in O(nk) P ⊂ NP veriﬁable
in O(nk)

solvable
in O(nk) P = NP
? veriﬁable
in O(nk)

NP-complete
NP, and one cannot do better

program
S

{P} {Q}

precondition postcondition

Weakest Precondition wp(S, Q)

∀ {P} S {Q} :: P wp(S,Q)

Veriﬁcation of {P} S {Q}

1. Compute wp(S, Q)

2. Prove P wp(S, Q)

Assignment
wp(x:=A, Q) = Qx←A

Sequencing
wp(S1; S2, Q) wp(S1, wp(S2, Q))
=

Conditional
wp(if (B) then S1 else S2, Q) =
(B wp(S1, Q)) ∧ (¬B wp(S2, Q))

While loop
L = while (B) do S end
wp(L,Q) I ∧
=
∀y, ((B ∧ I) wp(S, I ∧ x < y))

∀y, ((¬B ∧ I) Q)

Loop veriﬁcation

I = property which stays true before and after every loop

0. P I;
1. I∧B wp(s, I);
2. I∧¬B Q.

meta-meta-model

instanceOf

meta-model

instanceOf

model

representedBy

subject

A language is a set of sequences of symbols that
we interpret to attribute meaning.

- digit . digit

part = {digit} x
Nonter
decimal_part “.” part
= "x" mi
Termin nal
number = [-] part [decimal_part] x y al
Sequen
x|y ce
Alterna
{ x } tiv
Iteratio e
( x ) n
Binding
[ x ]
Option
y = x.
Deﬁniti
on

Please, stand up by sitting down!
Syntactically correct sentences
do not necessarily have a meaning

A programming language is a a language for
communicating software designs.

Imperative Functional
data + algorithms stateless + pure functions

Object-oriented Logic
objects + messages facts + rules

comments functions

# Compute factorials variables
def fact(n)
if n == 0
literals
1
else
control n * fact(n-1) expression
constructs end
end

statements puts fact(ARGV[0].to_i)

computational thinking
computation

information information
computer

Class aClass = Class.forName(“java.lang.Object”);
Method m[] aClass.getDeclaredMethods();

Class aClass = Class.forName(“java.lang.Object”);
Method m[] aClass.getDeclaredMethods();

protected void java.lang.Object.finalize() throws java.lang.Throwable
public final void java.lang.Object.wait() throws java.lang.InterruptedException
public final native void java.lang.Object.wait(long) throws
java.lang.InterruptedException
public final void java.lang.Object.wait(long,int) throws
java.lang.InterruptedException
public boolean java.lang.Object.equals(java.lang.Object)
public java.lang.String java.lang.Object.toString()
public native int java.lang.Object.hashCode()
public final native java.lang.Class java.lang.Object.getClass()
protected native java.lang.Object java.lang.Object.clone() throws
java.lang.CloneNotSupportedException
private static native void java.lang.Object.registerNatives()
public final native void java.lang.Object.notify()
public final native void java.lang.Object.notifyAll()

Tudor Gîrba
www.tudorgirba.com

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12 - Overview

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