CE308_2_Regular_Languages.pdf - Computing Theory(CE 308...

Computing
Theory
±CE ²³!´
Gül BOZTOK ALGINµ PhD
Fall ¶³·"

.

Finite Automata
Finite automata are good models for computers with an extremely limited
amount of memory.
An automaton is an abstract computing device. Note that a “device” need
not even be a physical hardware!
Such computers lie at the heart of various electromechanical devices.
e.g. The controller for an automatic door, a dishwasher, an electronic
thermostat, etc.

Finite Automata
A finite automaton is informally
±
a state diagram that comprehensively
captures all possible states and transitions that a machine can take while
responding to a stream or sequence of input symbols.
It is recognizer for “Regular Languages”.
+)eterministic +inite (&utomata ±+)+(&°
The machine can exist in only one state at any given time.
2Non²deterministic +inite (&utomata ±2N+(&°
The machine can exist in multiple states at the same time.

The Chomsky Hierarchy [¸]
Hierarchy of classes of formal languages.

Finite Automata
7Some (&pplications%!
䚔 Software for designing and checking the behavior of digital circuits
䚔 Lexical analyzer of a typical compiler
䚔 Software for scanning large bodies of text (e.g., web pages) for pattern
finding
䚔 Software for verifying systems of all types that have a finite number of
states e.g., stock market transaction, communication/network protocol

Finite Automata
A simple Example: The controller for an automatic door

Finite Automata
Finite automata
䚔 Their probabilistic counterpart are Markov chains*.
䚔 Are used to recognize patterns in data,
ŷ in speech processing,
ŷ in optical character recognition.
䚔 Markov chains have been used to model and predict price changes in
financial markets.
*
A process satisfies the Markov property if one can make predictions for the future of the process
based solely on its present state; that is, conditional on the present state of the system, its future
and past states are independent.

Example Finite Automata
On/Off Switch
Recognition
modeling of the
word “ then ”

A Finite Automaton M
·
The state diagram of a finite automaton called M
1
with three states.
7States : q
1
, q
2
, q
3
8Transitions : The arrows going from one state to another
The start state : q
1
(
indicated by the arrow pointing at it from nowhere
)
The accept ±final° state : q
2
(
indicated with a double circle
)
! Note that if the start state is also an accept state, M
1
accepts the empty string ε.
Input , output (accept or reject for now).
Which input strings will 22M
²
accept%$%% Which ones will it reject%$%%

Finite Automata _ Formal Definition
1
δ specifies exactly one next state for each possible combination of a state and an input symbol.
2
Zero accept states is allowable.

A Finite Automaton M
·
M
1
= (Q, Σ, δ, q
1
, F)
1. Q = {q
1
, q
2
, q
3
}
2. Σ = {0, 1}
3. δ :
4. q
1
: start state
5. F = {q
2
}

Finite Automata _ Formal Definition
Let '&''A