Learning objectives :
Introduce finite state automata
Able to capture state, events and dynamic behavior of
“man-made systems”
Present logical properties
Textbook :
C. Cassandras and S. Lafortune, Introduction to Discrete
Event Systems, Springer, 2007
1
1

Plan
2
2

3
3

Definitions
Definition :
A language L , defined over an alphabet E , is a set of strings formed from events in E .
Language vs DES :
E = set of events string = sequence of events
L = set of all possible event sequences
Example :
E = { a,b,g}
L1 =
{e, a, abb}
, L2 = {all strings of length 3}
L3 = {all strings of finite length ending with a
}
4

Regular expressions
Operations
E : alphabet, A, B : languages e
= empty string, u, v, w: strings
•
Concatenation :
AB = {w : w = uv, u

A, v

B}
•
Union :
A+B = {w : w

A, or v

B}
•
Kleene closure :

A
* 
A n n

0 where A 0 = { e
}, A n = AA n-1
5

Regular expressions
Example :
E = { a,b,g}
L
L
1
1
L
=
{e, a, abb}
, L
2
2
=
{g, ag, abbg}
= { g
}
L
2
* = { e, g, gg, ggg, ...
}
L
1
* = { e, a, abb, aa, aabb, abba, abbabb, ...
}
L
1
+ L
2
=
{e, a, abb, g}
6
6

Regular expressions
Definition : A regular expression is a representation defined recursively as follows :
1. F is a regular expression denoting the empty set, e is a regular expression denoting the set { e
}, e is a regular expression denoting the set {e} for any e

E.
2. If r and s are regular expressions, then rs, (r+s), r* and s* are regular expressions.
3. There are no regular expressions other than those constructed by applying rules 1 and 2 a finite number of times
Definition : Any language that can be denoted by a regular expression is a regular language .
7

Regular expressions
Examples :
(a+b)g*  {???}
(ab)*+g  {???}
(a+b)g*)*
8

9

Definition
Definition : A finite-state automaton (FSA) is a five-tuple
(E, X, f, x
0
, F) where
•
E is a finite alphabet
•
X is a finite state set
• f is a state transition function f: X× E → X where f is a partial function and f(x, e) is sometime not defined.
• x0 is an initial state, x

X
• F is a set of final states, F

X

Definition
•
Transition function f can be extended to strings u : f(x, u)
•
An automaton is a device that generates a language according to some rules.
•
An automaton is nondeterministic if there are two possible transitions with the same event and starting from the same state.
•
Nondeterministic automaton can always be transformed into deterministic FSA by taking all possible subsets reached by events as states.
•
Nondeterministic automaton will not be considered.

Example
Definition :
Consider an automaton (E, X, f, x
0
, F)
E = { a, b, g
}
X = {x, y, z} f(x, a
) = x, f(x, b
) = f(x, g
) = z f(y, a
) = x, f(y, b
) = f(y, g
) = y f(z, b
) = z, f(z, a
) = f(z, g
) = y initial state x0 = x
Final states : F = {x, z}
Sample paths:
???
12
State transition diagram a b x a g g b y a g z b

Automata as language recognizers
Definition : A string u is recognized by a finite automaton (E, X, f, x
0
, F) if f(x
0
, u) = x where x

F.
Definition : The language recognized by a finite-state automaton A = (E, X, f, x
0
, F) is a set of strings {u: f(x
0
, u) x

F}.
L(A) = language recognized by A.
Example: aab
L(A) = ???
13 a x g b a g z a b b y g

Equivalence of FSA and regular expressions
Theorem : If a language is regular, then it can be generated by some finite-state automaton; and if it is generated by a finite-state automaton, then it is a regular language.
Deriving the language of an FSA
• define the language Ls terminating at state s;
• write one-step equations for all Ls;
•
Solve the equations.
Arden’s rule : If L
1
L
2
= L
1
L
2
+ L
3
→ L
2 does not contain the empty string,
= L
1
*L
3
L
2
= L
2
L
1
+ L
3
→ L
2
= L
3
L
1
*
14 a x g b a g z a b b y g

Properties of a FSA model
Reachability :
A state s is reachable from the initial state x0 if there exists a sequence u such that f(x0, u) = s.
Blocking-free :
For any reachable state, there exists a sequence of transitions leading to a final state.
Deadlock : a state s is called a deadlock state if there is no feasible event from it.
Livelock : a subset of states that are mutually reachable but with no event going out of the set.
Example : FSA = (0, a, 1), (1, g, 5), (1, a, 3), (1, b, 2), (2, g, 0), (3, b, 4),
(4, a, 3), (4, g, 4), x0 = 0, F = {2}.
15

a
FSA models of queueing systems
Queue
Server d a
0 1 d a d
2 a d
Queueing perspective
16
I a b
B m l
D
Server perspective
I = idle, B = busy
D = down