Solution

1
Solution

• Prove by induction the following statement
• If there is an edge connecting every pair of
  nodes in a graph G and n is the number of nodes
  in G, then the number of edges in G is n(n-1)/2

2
Finite Automata (Deterministic)(Chapter 1)
Note once again you are strongly advised to read
the covered chapters from the book carefully.
Particularly the examples!
3
Reminder Functions vs Relations
Let P p p is a person M m m is a
male
S1 (m,p) m is in M, p is in P and m is the
father of p S2 (m,p) m is in M, p is in P
and m is an ancestor of p

• True or false S1 ? M ? P
• True or false S2 ? M ? P
• Is either S1 or S2 a relation in M ? P?
• Is either S1 or S2 a function fM ? P?

4
You have seen Finite Automata before!

• Write a program that indicates if a given input
  text contains the string Britney Spears (or
  Justin Timberlake)
• Check this video

5
NPCs Behavior is Modeled through Finite State
Machines (I)

• State an activity performed by an avatar
• Event something that happens in the game world
  that makes state change

Enemy on sight
Attack
Patrol
6
NPCs Behavior is Modeled through Finite State
Machines (II)

• States
• Attack
• Chase
• Spawn
• Wander
• Events
• E see an enemy
• S hear a sound
• D die

Attack
E
D
E
Wander
E
E
Spawn
D
7
Deterministic Automata (Informal)
Key questions if a automaton is confronted with
a certain state where a choice must be made, 1.
are all the alternatives transitions known?,
and 2. given some input data, is it known which
transition the machine will make?
new state
If the answer to both of these questions is
yes, the automaton is said to be deterministic
transition
8
Nondeterministic Automata (Informal)
If the answer to any of these questions is no,
the automaton is said to be nondeterministic

• That is, either
• some transitions are unknown, or
• given some input data, the machine can make more
  than one transition

9
Deterministic Automata (Informal)
We are going to define automata indicating for a
state s and some input data d, which is the state
that will be reached
Let Q be the set of all states and ? be the set
of all input data. Then, the set of transitions
is a subset of
(Q ? ?) ? Q
10
Determinism, Nondeterminism, Relations and
Functions
The set of transitions defining an automaton is a
subset of (Q ? ?) ? Q
If the automaton is deterministic, should the set
of transitions be a relation or a function?

• Deterministic automata Since for each pair
  (s,d) there should be one and only one s, the
  set of transitions must be a function
• Nondeterministic automata Since for each pair
  (s,d) there might not be any s or there might be
  more than one s, the set of transitions must be
  a relation

11
Finite Automata

• Problem 1 Design a computer program that given a
  sequence of numbers a1, a2, , an returns their
  sum a1 a2 an

• Problem 2 Design a computer program that given a
  sequence of numbers a1, a2, , an returns the
  list in the inverted order an, , a2, a1

• How many memory units are needed for a program to
  execute
• problem 1
• Problem 2

1 n
12
Deterministic Finite Automaton (finite automaton)

• A deterministic finite automaton (DFA) or finite
  automaton is a 5-tuple (Q,?,?,s,F) where
• Q is a finite set of elements called states
• ? is a finite input alphabet
• ? is a transition function, (Q ?) Q (or ?
  (Q ?) ? Q)
• s ? Q called the start state
• F ? Q called the favorable states

As in yes! (or no!) The input word is (not) in
the language recognized by the automaton
Constant!

a1
a2
The fact that ? is a function makes the automaton
deterministic
13
Finite State Diagram

• A finite state diagram is a graphic
  representation for a finite automaton

• A finite state diagram is a directed graph, where
  nodes represent elements in Q (i.e., states) and
  arrows are characters in ? such that

14
Example 1
a,b
b
b
b
a
a
r
r
s
q
q

a
a
Formally, this automaton (Q,?,?,s,F) is defined
as

• ? 6 transitions
• ((s,b), s)
• ((s,a), q)
• ((q,a), r)
• ((q,b), s)
• ((r,a), r)
• ((r,b), r)

Q s,q,r
Is this automaton deterministic?
? a,b
s start state
F r
15
Kleene Star

• Given a set B of characters, B denote the
  set of all strings made of elements in B.
• Typically we take ? for all words that can be
  given as input for a DFA (Q,?,?,s,F)
• If ? 0,1, what is ??
• If ? a,b, what is ??
• If ? a,b,c, , z, what is ??

the empty word, e, is always in ?
16
Formal Definition of Computation

• Given an finite automaton M (Q,?,?,s,F), and let
  
• w w1w2 wn, where each wi is in ?
  (i.e., w in ?)
• M accepts w if there is a sequence of states
• r0 , r1, r2 , rn in Q such that
• r0 is the start state of M
• ?(ri , wi1) ri1
• rn in F
• Language recognized by M w in ? M accepts w
• Given a language A, with A ? ?, the language A
  is regular if a finite automata exists
  recognizing A

17
Example
a
b
b
b
a
a
r
r
s
q
q

a
a
b

• Is abb accepted by this automaton?
• Is abba accepted by this automaton?
• What is the language recognized by this automaton?

18
Example 2
a
b
b
b
a
a
r
r
s
q
q

a
a
b
What is the language recognized by this automaton?
19
Homework Friday Sept. 7

• 1.6 a, f, j
• 1.31
• 1.36

20
Sample Problems

• 1.6 b Let ? 0,1, Show that the following
  language over ? is regular
• w w contains at least three 1s
• 1.31 Let A be a regular language. Show that
• AR wR w is in A is also regular