Finite Automaton Variants
Finite Automaton Variants
Mohamed M. El Wakil
[email protected]
1
Agenda
• Finite State Automaton/Machine (FSA)
Finite State Automaton/Machine (FSA)
– State transition tables
– Non‐deterministic Finite State Automaton(NFA)
Non deterministic Finite State Automaton(NFA)
– Two‐way Deterministic Finite Automaton (2FA)
• Finite State Transducer
Fi it St t T
d
– Mealy machine
– Moore machine
2
Finite State Automaton/Machine
Finite State Automaton/Machine
• It is a model of computation that consists of:
– States
• Start/Initial state
• Acceptance states
• Other states
h
– Transitions • Moving from a state to another
– Input string
Inp t string
• Finite set of symbols
– Actions
•
•
•
•
Entry action
Entry
action
Exit action
Transition action
Input action
p
3
Recognition/Acceptance
• A
A FSA is said to recognize, or accept
FSA is said to recognize or accept an input an input
string, if it ends at an acceptance state, and the whole input string has been processed
the whole input string has been processed.
4
Example
• This machine has 7 states, 8 transitions, and 0 actions.
• State “Start” is the start state. • If the input is the word If the input is the word “nice”
nice , this machine this machine
will end at the state “Success”, otherwise it will end at the “Error”
will end at the Error state.
state
5
Transitions
• A
A transition is a move from a state A to another state B transition is a move from a state A to another state B
based on the current state (i.e. A), and the current symbol of the input string. • Transitions can be viewed as a function with two inputs (current state, and current input symbol), and one
(current state, and current input symbol), and one output (next state). For example: – (S1, “n”) Æ S2
– (S1, (S1 “not
not_n
n”)) ÆS6
ÆS6
• When a symbol is input to a transition function, it is said to be consumed.
• Symbols
Symbols in the input string are consumed from left to in the input string are consumed from left to
right.
6
State Transition Table
State Transition Table
• Transition
Transition functions are expressed via state functions are expressed via state
transition tables.
• State transition tables can be either one State transition tables can be either one
dimension or two dimensions.
• This machine accepts
– 1111000, 1010, and 000
1111000 1010 and 000
• but, does not accept
– 1001, 111, or 001
1001 111 or 001
7
State Transition Tables
• One dimension table:
Input
Current State
Next State
0
P
Q
0
Q
R
0
R
S
1
P P
1
Q R
1
R ‐
• Two dimension tables:
Next
Current
P
Q
R
S
Input
State
0
1
P
1
0
‐
‐
P
Q
P
Q
‐
‐
{0,1}
‐
Q
R
R
R
‐
‐
‐
0
R
S
‐
8
Mathematically speaking
Mathematically speaking
• A finite state machine is quintuple (Σ,S,s0,δ,F):
A finite state machine is quintuple (Σ S s0 δ F):
– Σ: Finite non‐empty set of symbols (Input).
– S: Finite, non‐empty set of states (States).
S: Finite non empty set of states (States)
– S0: Initial state S0 belongs to S (Start state). – F: Set of final states, a subset of S.
F S t f fi l t t
b t fS
– δ: Transition function (Transitions)
• δ: S x Σ Æ
δ S ΣÆS
9
Non‐deterministic Finite State Machine (NFA)
h (
)
• The FSA is deterministic, If
The FSA is deterministic If
– for each pair of state and input symbol there is one and only one transition to a next state
one and only one transition to a next state.
• The FSA is non‐deterministic, if
The FSA is non deterministic if
– it is allowed for a pair of state and input symbol to have more than one possible next state or a
have more than one possible next state, or a transition can take place without consuming input symbol (ε
y
( transitions))
10
NFA Example 1
NFA Example 1
• This
This machine is non
machine is non‐deterministic
deterministic as for the as for the
pair (P,0), there is more than one possible next state (P and Q)
state (P, and Q)
Input
State
0
1
P
P,Q
P
Q
R
R
R
S
‐
11
NFA Example 2
NFA Example 2
• This
This machine is non
machine is non‐deterministic
deterministic it can move it can move
from state Q to state R, without consuming any input
any input.
Input
State
0
1
ε
P
Q
P
‐
Q
‐
R
R
R
S
‐
‐
12
Deterministic Vs. Non‐deterministic Finite State Automaton/Machine
/
h
• NFA
–
–
–
–
Easier to design
Harder to implement
Sl
Slower
Smaller
• DFA
–
–
–
–
Easier to implement
Harder to design
Faster
Bigger
13
Two‐way
Two
way Finite Automaton
Finite Automaton
• In DFA, input can be read only once from left to right.
, p
y
g
• In 2DFA, input can be read back and forth with no limit on how many times an input symbol can be read.
• A 2DFA is a generalization of the DFA.
A 2DFA i
li ti
f th DFA
• Transition
Transition functions in 2DFA, outputs a new state, and functions in 2DFA outputs a new state and
a direction (left, right, or stand still)
14
DFA Vs. 2DFA
DFA Vs. 2DFA
• 2DFA can solve any problem solvable by DFA.
2DFA can solve any problem solvable by DFA
– Why?
• Also
Also, problems solvable by a 2DFA, can be problems solvable by a 2DFA can be
solved by a DFA1. • But, 2DFA is – much simpler, and
– more realistic.
1M. O. Rabin and D. Scott, “Finite automata and their decision problems”, IBM Journal of Research and Development, 3 (1959), 114–125.
15
Finite State Transducers (FST)
Finite State Transducers (FST)
• A
A FSA can be viewed as a function that maps an FSA can be viewed as a function that maps an
input string into the set {0,1}, {True, False}, {Accept Reject}
{Accept, Reject}… • A
A FST may generate a new string (output string) FST
i (
i )
while consuming the input string.
• A FST is a FSA with actions producing output!
p
g
p
16
FST Example
FST Example
• This
This machine outputs the one
machine outputs the one’ss complement complement
of its input, while it is recognizing this input.
• For input 000, it outputs 111
• For input 1010, it outputs 0101 17
FST variants
FST variants
• A
A FST, may be non
FST may be non‐deterministic
deterministic, if it can if it can
produce more than one output string for a given input string
given input string.
• A
A FST is either a Mealy machine, or a Moore FST is either a Mealy machine or a Moore
machine.
18
Mealy Machines
Mealy Machines
• A
A Mealy machine is finite state transducer that Mealy machine is finite state transducer that
produces an output for each transition (i.e has input actions)
input actions)
• Example: E
l
19
Moore Machines
Moore Machines
• A
A Moore machine is finite state transducer Moore machine is finite state transducer
that produces an output for each state it enters (i e has entry actions)
enters (i.e. has entry actions)
• Example:
E
l
– We have to add a new state X
20
Mathematically speaking
Mathematically speaking
• A finite state transducer is a sextuple (Σ,
A finite state transducer is a sextuple (Σ Γ,S,S0,δ,
Γ S S0 δ ω):
–
–
–
–
–
Σ: Finite non‐empty set of symbols (Input).
Γ: Finite non‐empty
Γ: Finite non
empty set of symbols (Output).
set of symbols (Output).
S: Finite, non‐empty set of states (States).
S0: Initial state S0 belongs to S
S0: Initial state S0 belongs to S (Start state). (Start state).
δ: Transition function (Transitions)
• δ: S x Σ Æ S
– ω: Output function
• ω: S x Σ Æ Γ (Mealy machine)
• ω: S Æ Γ
(Moore machine)
21