Finite state automata

Report
Chapter 2
Language and Automata Theory
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
• Languages
• Finite state automata
2
2
Languages
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}
L4 = {all possible sequences 4of a queue}
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 

*
An
n 0
where A0 = {e}, An = AAn-1
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Regular expressions
Example :
E = {a,b,g}
L1 = {e, a, abb}, L2 = {g}
L1L2 = {g, ag, abbg}
L2* = {e, g, gg, ggg, ...}
L1* = {e, a, abb, aa, aabb, abba, abbabb, ...}
L1+ L2 ={e, a, abb, g}
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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.
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Regular expressions
Examples:
L = (a+b)g*  {???}
L = (ab)*+g  {???}
L = ((a+b)g*)*
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Finite-state automata
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Definition
Definition: A finite-state automaton (FSA) is a five-tuple
(E, X, f, x0, 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
State transition diagram
Definition:
Consider an automaton (E, X, f, x0, 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}
a
a
x
g
b
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y
a
g
z
b
Sample paths:
???
b
g
Automata as language recognizers
Definition: A string u is recognized
by a finite automaton (E, X, f, x0, F)
if f(x0, u) = x where x  F.
Definition: The language recognized by a
finite-state automaton A = (E, X, f, x0, F)
is a set of strings {u: f(x0, u) x  F}.
a
L(A) = language recognized by A.
a
x
g b
Example:
aab
L(A) = ???
z
13
b
a
b
y
g
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 L1 does not contain the empty string,
L2 = L1L2 + L3 → L2 = L1*L3
L2 = L2L1 + L3 → L2 = L3L1*
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a
a
x
g b
z
b
a
b
y
g
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}.
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FSA models of queueing systems
a
Queue
a
0
d
a
1
d
Server
d
a
a
2
b
I
d
m
Queueing perspective
B
l
D
Server perspective
I = idle, B = busy
D = down
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