The document provides a comprehensive overview of computation and information within computer science, detailing concepts such as boolean algebra, predicate logic, set relations, programming models, and complexity analysis. It elaborates on the principles of programming languages including syntax and semantics, the significance of logical connectives, and examples of boolean functions and theorems. Additionally, it touches upon tree and graph structures, program verification, and the Big O notation for assessing algorithm efficiency.

1. Overview
www.tudorgirba.com

2. computation
information information
computer

3. Computer science is the systematic study of
information and computation.

4. 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

5. 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

6. + +
a = 0 a = 1
y = 0 y = 1

7. 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

8. 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)

9. 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)

10. 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)

11. 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

12. 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

13. ∀ 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

14. 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)

15. 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

16. 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

17. 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’)

18. Reflexivity
A ⊆ A.
Anti-symmetry
A ⊆ B ∧ B ⊆ A A = B.
Transitivity
A ⊆ B ∧ B ⊆ C A ⊆ C

19. 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

20. 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).

21. 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

22. 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

23. 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

24. 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 ) )

25. 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 ∪ ...

26. 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

27. n1
n10
n2
n9
n3
n8
n4
n5 n7
n6

28. n1
n10
n2
n9
n3
n8
n4
n5 n7
n6

29. n1
n2 n3 n4
n5 n6 n7 n8 n9
n10

30. 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

31. breadth-first traversal
1
n1
2 3
n2 n3
4 5 6
n4 n5 n6

32. preorder(node)
visit(node).
preorder(node.left).
preorder(node.right). 1
n1
2 5
n2 n3
3 4 6
n4 n5 n6

33. inorder(node)
inorder(node.left).
visit(node).
inorder(node.right). 4
n1
2 6
n2 n3
1 3 5
n4 n5 n6

34. postorder (node)
postorder(node.left).
postorder(node.right).
visit(node). 6
n1
3 5
n2 n3
1 2 4
n4 n5 n6

35. 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

36. G = (V, E)
E = { {u,v} | u,v ∈ V}
a e
c d g
b f

37. 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

38. Hamiltonian path Eulerian path
a e a e
c d g c d g
b f b f

39. 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

40. 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 }

41. 1 public long factorial (int n) {
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 }

42. 1 public long factorial (int n) {
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 }

43. 1 public long factorial (int n) {
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

44. 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))

45. 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

46. 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))

47. 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))

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

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

50. NP-complete
NP, and one cannot do better

51. 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

52. program
S
{P} {Q}
precondition postcondition

53. 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)

54. 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))

55. 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.

56. 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

58. meta-meta-model
instanceOf
meta-model
instanceOf
model
representedBy
subject

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

60. - 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

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

62. 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

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

64. in g, 1937
Alan Tur

65. Imperative Functional
data + algorithms stateless + pure functions
Object-oriented Logic
objects + messages facts + rules

66. 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)

68. programming = modeling

69. 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

71. computation
information information
computer

72. computational thinking
computation
information information
computer

74. Reflection

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

76. 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()

77. computational thinking
computation
information information
computer

78. Tudor Gîrba
www.tudorgirba.com
creativecommons.org/licenses/by/3.0/