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}
students have trouble with the notation (x,f(x)) to denote functions (they are used to y = f(x)). Answers: 1 (yes), 2(yes), 3(both), 4 (S1 only)
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 f:M  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. NPC’s 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. NPC’s 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 ~S
Chase E S D 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?
“current state”
“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)
of the computation
We are going to define automata indicating for a state s and some input data d, which is the state that will be reached
Transition: ((s,d), s’) s d s’
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
This is a crucial issue: understanding that the mathematical definition of determinism uses functions and nondeterminism uses relations

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:
Finite automata use a constant amount of memory
Later in the course we will be able to prove formally that there is no finite automata that can solve 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!
no! (or yes!)
Given an input the automaton answers “yes” if its computation ends in a favorable state. Otherwise the automaton answers “no” 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: q q’ a
Indicates:
((q,a),q’) is a transition in 
The start state is marked with:
>
The favorable states are marked with:

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)
Answer: yes, for every pair (state,symbol) there is one and only one transition
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 , wi+1) = ri+1
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
“such that”

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
Language accepted is the set of all words except those containing two consecutive a’s
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 1’s}
1.31 Let A be a regular language. Show that:
AR = {wR | w is in A} is also regular