CPS420 Discrete Structures
3.2A Deterministic Finite State
Automata
© Sophie Quigley
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2
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3.2 Deterministic Finite State Automata
Table of Contents
Part A: Example and Definitions
Part B: Languages accepted by DFAs
Next Topic: Non-Deterministic Finite Automata
3
3.2 Deterministic Finite State Automata
Table of Contents
Part A: Example and Definitions
➢Vending Machine Example
• DFA Definition
• Next-state Function
• Transition Diagrams
• Tabular Representations
Part B: Languages accepted by DFAs
Next Topic: Non-Deterministic Finite Automata
4
5
Example
A vending machine
accepts quarters (Q)
and loonies (L)
• It does not make
change.
• A bottle of water
costs $2 and a soft
drink $1.50.
Build a diagram
showing what can be
purchased.
6
Example
0
A vending machine
accepts quarters (Q)
and loonies (L)
• It does not make
change.
• A bottle of water
costs $2 and a soft
drink $1.50.
Build a diagram
showing what can be
purchased.
• The machine has a starting state shown as
receiving an arrow.
7
Example
0
A vending machine
accepts quarters (Q)
and loonies (L)
• It does not make
change.
• A bottle of water
costs $2 and a soft
drink $1.50.
Build a diagram
showing what can
be purchased.
• This starting state is labelled with a 0
which is the total amount of money
entered into the machine so far
8
Example
0
A vending machine
accepts quarters (Q)
and loonies (L)
• It does not make
change.
• A bottle of water
costs $2 and a soft
drink $1.50.
Build a diagram
showing what can
be purchased.
Q
0.25
• When a quarter (Q) is added, the machine
now has a balance of $0.25
9
Example
0
A vending machine
accepts quarters (Q)
and loonies (L)
• It does not make
change.
• A bottle of water
costs $2 and a soft
drink $1.50.
Use a diagram to
show what can be
purchased.
Q
0.25
• The action of putting in a quarter is shown
as an arrow labelled with “Q” for quarter.
10
Example
0
A vending machine
accepts quarters (Q)
and loonies (L)
• It does not make
change.
• A bottle of water
costs $2 and a soft
drink $1.50.
Build a diagram
showing what can
be purchased.
Q
0.25
Q
0.50
Q
0.75
Q
1.00
Soft
Drink
Q
Q
1.25
• After 6 quarters have been entered, the
customer can buy a soft drink.
11
Example
0
A vending machine
accepts quarters (Q)
and loonies (L)
• It does not make
change.
• A bottle of water
costs $2 and a soft
drink $1.50.
Build a diagram
showing what can
be purchased.
Q
0.25
Q
0.50
Q
0.75
Q
1.00
Soft
Drink
Q
Q
1.25
• The customer can stop entering money at
this point, and the soft Drink state is
called a “final” state.
12
Example
0
A vending machine
accepts quarters (Q)
and loonies (L)
• It does not make
change.
• A bottle of water
costs $2 and a soft
drink $1.50.
Build a diagram
showing what can
be purchased.
Q
0.25
Q
0.50
Q
0.75
Q
1.00
Q
Q
Water
1.75
Soft
Drink
Q
Q
1.25
• The customer can continue entering
quarters in order to buy a bottle of water.
13
Example
L
0
A vending machine
accepts quarters (Q)
and loonies (L)
• It does not make
change.
• A bottle of water
costs $2 and a soft
drink $1.50.
Build a diagram
showing what can
be purchased.
Q
Q
0.25
L
0.50
Q
L
0.75
Q
L
1.00
Q
Q
Water
1.75
Soft
Drink
Q
Q
1.25
L=Loonie
Q=quarter
• When loonies are entered, the customer
fast tracks through the states.
L
14
Example
L
0
A vending machine
accepts quarters (Q)
and loonies (L)
• It does not make
change.
• A bottle of water
costs $2 and a soft
drink $1.50.
Build a diagram
showing what can
be purchased.
Q
Q
0.25
L
0.50
Q
L
0.75
Q
L
1.00
Q
Q
Water
1.75
Soft
Drink
Q
Q
L=Loonie
Q=quarter
• Try to avoid crossing arrows
1.25
L
15
Example
L
0
A vending machine
accepts quarters (Q)
and loonies (L)
• It does not make
change.
• A bottle of water
costs $2 and a soft
drink $1.50.
Build a diagram
showing what can
be purchased.
Q
Q
0.25
L
0.50
Q
L
0.75
Q
L
1.00
Q
Q
Water
L=Loonie
Q=quarter
1.75
Soft
Drink
Q
Q
1.25
L
?
L
16
Example
L
0
A vending machine
accepts quarters (Q)
and loonies (L)
• It does not make
change.
• A bottle of water
costs $2 and a soft
drink $1.50.
Build a diagram
showing what can
be purchased.
Q
Q
0.25
L
0.50
Q
L
0.75
Q
L
1.00
Q
Q
Water
L=Loonie
Q=quarter
1.75
Soft
Drink
Q
Q
1.25
L
3.2 Deterministic Finite State Automata
Table of Contents
Part A: Example and Definitions
✓Vending Machine Example
➢DFA Definition
• Next-state Function
• Transition Diagrams
• Tabular Representations
Part B: Languages accepted by DFAs
Next Topic: Non-Deterministic Finite Automata
17
18
DFA Definition
A Deterministic Finite State Automaton (DFA) A, also called a Finite
State Automaton (FSA or FA) consists of 5 objects:
• A set I called the input alphabet, of input symbols
• A set S of states the automaton can be in
• A designated state s0 called the initial state
• A designated set of states F  S called the set of accepting states, or
final states
• A next-state function N: S× I → S that associates a “next-state” to
each ordered pair consisting of a “current state” and “current input”.
For each state s in S and input symbol m in I, N(s,m) is called the state
to which A goes if m is input to A when A is in state s
19
DFA Definition
L
0
A Deterministic
Finite State
Automaton (DFA),
also called a Finite
State Automaton (FA)
A
consists of 5 objects:
Q
Q
0.25
L
0.50
Q
L
0.75
Q
L
1.00
Q
Q
Water
1.75
Soft
Drink
Q
Q
1.25
L=Loonie
Q=quarter
1. A set I called the input alphabet, of
input symbols
I = {Q, L}
L
20
DFA Definition
L
0
A Deterministic
Finite State
Automaton (DFA),
also called a Finite
State Automaton (FA)
A
consists of 5 objects:
Q
Q
0.25
L
0.50
Q
L
0.75
Q
L
1.00
Q
Q
Water
1.75
Soft
Drink
Q
Q
L
1.25
L=Loonie
Q=quarter
2. A set S of states the automaton can be in
S = {0, 0.25, 0.50, 0.75, 1.00, 1.25,
Soft Drink, 1.75, Water}
21
DFA Definition
L
0
A Deterministic
Finite State
Automaton (DFA),
also called a Finite
State Automaton (FA)
A
consists of 5 objects:
Q
Q
0.25
L
0.50
Q
L
0.75
Q
L
1.00
Q
Q
Water
1.75
Soft
Drink
Q
Q
1.25
L=Loonie
Q=quarter
3. A designated state s0 called the initial
state
s0 = 0
L
22
DFA Definition
L
0
A Deterministic
Finite State
Automaton (DFA),
also called a Finite
State Automaton (FA)
A
consists of 5 objects:
Q
Q
0.25
L
0.50
Q
L
0.75
Q
L
1.00
Q
Q
Water
1.75
Soft
Drink
Q
Q
1.25
L=Loonie
Q=quarter
4. A designated set of states F  S called
the set of accepting states, or final
states
F = {Soft Drink, Water}
L
23
DFA Definition
L
0
A Deterministic
Finite State
Automaton (DFA),
also called a Finite
State Automaton (FA)
A
consists of 5 objects:
Q
Q
0.25
L
0.50
Q
L
0.75
Q
L
1.00
Q
Q
Water
1.75
Soft
Drink
Q
Q
1.25
L=Loonie
Q=quarter
5. A next-state function defining the
operations of the automaton
L
3.2 Deterministic Finite State Automata
Table of Contents
Part A: Example and Definitions
✓Vending Machine Example
✓DFA Definition
➢Next-state Function
• Transition Diagrams
• Tabular Representations
Part B: Languages accepted by DFAs
Next Topic: Non-Deterministic Finite Automata
24
25
Next-state Function
L
0
The next-state function N
is defined as follows:
• N: S×I → S that
associates a “next-state”
to each ordered pair
consisting of a “current
state” and “current
input”.
• For each state s in S and
input symbol m in I,
N(s,m) is called the state
to which A goes if m is
input to A when A is in
state s.
Q
Q
0.25
L
0.50
Q
L
0.75
Q
L
1.00
Q
Q
Water
L=Loonie
Q=quarter
1.75
Soft
Drink
Q
Q
1.25
L
Current State
Next-state Function
Next State
L
0
The next-state function N
is defined as follows:
• N: S×I → S that
associates a “next-state”
to each ordered pair
consisting of a “current
state” and “current
input”.
• For each state s in S and
input symbol m in I,
N(s,m) is called the state
to which A goes if m is
input to A when A is in
state s.
Current Input
Q
Q
0.25
L
0.50
Q
L
0.75
Q
L
1.00
Q
Q
Water
L=Loonie
Q=quarter
1.75
Soft
Drink
Q
Q
1.25
L
26
27
Next-state Function
L
0
The next-state function N
is defined as follows:
• N: S×I → S that
associates a “next-state”
to each ordered pair
consisting of a “current
state” and “current
input”.
• For each state s in S and
input symbol m in I,
N(s,m) is called the state
to which A goes if m is
input to A when A is in
state s.
Q
Q
0.25
L
0.50
Q
L
0.75
Q
L
1.00
Q
Q
Water
L=Loonie
Q=quarter
1.75
Soft
Drink
Q
Q
1.25
L
28
Next-state Function
L
Q
0
The next-state function N
is defined as follows:
• N: S×I → S that
associates a “next-state”
to each ordered pair
consisting of a “current
state” and “current
input”.
• For each state s in S and
input symbol m in I,
N(s,m) is called the state
to which A goes if m is
input to A when A is in
state s.
Q
0.25
0.50
L
Q
L
0.75
Q
L
1.00
Q
Q
Water
1.75
Soft
Drink
L=Loonie
Q=quarter
N(0.50,Q) = 0.75
∈S ∈I
∈S
Q
Q
1.25
L
3.2 Deterministic Finite State Automata
Table of Contents
Part A: Example and Definitions
✓Vending Machine Example
✓DFA Definition
✓Next-state Function
➢Transition Diagrams
• Tabular Representations
Part B: Languages accepted by DFAs
Next Topic: Non-Deterministic Finite Automata
29
30
Transition Diagrams
The operation of an FA is commonly described by a diagram called a
(state-)transition diagram.
In a transition diagram:
• states are represented by circles, and accepting states by double
circles.
• There is one arrow that points to the initial state and other arrows
between states as follows: There is an arrow from state s to state t
labeled m (where ∈ I) iff N(s,m)=t.
31
non-accepting state
Transition Diagram
L
0
The operation of an
FSA is commonly
described by a
diagram called a
(state-) transition
diagram.
In a transition
diagram:
Q
Q
0.25
L
0.50
Q
L
0.75
Q
L
1.00
Q
Q
Water
L=Loonie
Q=quarter
1.75
Soft
Drink
Q
Q
L
1.25
accepting state
• states are represented by circles,
• and accepting states by double circles.
32
Transition Diagram
L
0
The operation of an
FSA is commonly
described by a
diagram called a
(state-) transition
diagram.
In a transition
diagram:
Q
Q
0.25
L
0.50
Q
L
0.75
Q
L
1.00
Q
Q
Water
1.75
Soft
Drink
Q
Q
1.25
L=Loonie
Q=quarter
• There is one arrow that points to the
initial state
L
33
Transition Diagram
L
0
The operation of an
FSA is commonly
described by a
diagram called a
(state-) transition
diagram.
In a transition
diagram:
Q
Q
0.25
L
0.50
Q
L
0.75
Q
L
1.00
Q
Q
Water
1.75
Soft
Drink
Q
Q
L
1.25
L=Loonie
Q=quarter
• and other arrows between states as
follows: There is an arrow from state s to
state t labeled m (∈I) iff N(s,m)=t.
3.2 Deterministic Finite State Automata
Table of Contents
Part A: Example and Definitions
✓Vending Machine Example
✓DFA Definition
✓Next-state Function
✓Transition Diagrams
➢Tabular Representations
Part B: Languages accepted by DFAs
Next Topic: Non-Deterministic Finite Automata
34
35
Next-state tables
• The next-state table is a tabular representation of the next-state
function.
• In the annotated next-state table, the initial state is indicated by an
arrow and the accepting states by double circles.
36
Next-state Table
I = {input symbols}
Q
L
0
0.25
1.00
0.25
0.50
1.25
0.50
0.75
Soft Dr.
0.75
1.00
1.75
1.00
1.25
Water
1.25
Soft Dr.
Soft Dr.
1.75
1.75
Water
Water
S = {states}
L
0
Q
Q
0.25
L
0.50
Q
L
0.75
Q
L
1.00
Q
Q
Water
1.75
Soft
Drink
Q
Q
1.25
L=Loonie
Q=quarter
The next-state table is a tabular
representation of the next-state function
N[state,input] = N(state,input)
L
Annotated Nextstate Table
S0 = Initial state
Q
L
→0
0.25
1.00
0.25
0.50
1.25
0.50
0.75
Soft Dr.
0.75
1.00
1.75
1.00
1.25
Water
1.25
Soft Dr.
Soft Dr.
1.75
1.75
Water
Water
F= {Final states}
37
L
0
Q
Q
0.25
L
0.50
Q
L
0.75
Q
L
1.00
Q
Q
Water
1.75
Soft
Drink
Q
Q
L
1.25
L=Loonie
Q=quarter
In the annotated next-state table, the initial
state is indicated by an arrow and the
accepting states by double circles.
Annotated Nextstate Table
Q
L
→0
0.25
1.00
0.25
0.50
1.25
0.50
0.75
Soft Dr.
0.75
1.00
1.75
1.00
1.25
Water
1.25
Soft Dr.
Soft Dr.
1.75
1.75
Water
Water
38
L
0
Q
Q
0.25
L
0.50
Q
L
0.75
Q
L
1.00
Q
Q
Water
1.75
Soft
Drink
Q
Q
L
1.25
L=Loonie
Q=quarter
The annotated next-state table defines the
automaton completely, precisely, and
unambiguously.
39
Programming a DFA
Q
L
→0
0.25
1.00
0.25
0.50
1.25
0.50
0.75
Soft Dr.
0.75
1.00
1.75
1.00
1.25
Water
1.25
Soft Dr.
Soft Dr.
1.75
1.75
Water
Water
Current state = initial state
While current state is not final
• read current input
• current state = N[current state,
current input]
40
Programming a DFA
Q
L
→0
0.25
1.00
0.25
0.50
1.25
0.50
0.75
Soft Dr.
0.75
1.00
1.75
1.00
1.25
Water
1.25
Soft Dr.
Error
Soft Dr.
1.75
Error
1.75
Water
Error
Water
Error
Error
Current state = initial state
While current state is not final
• read current input
• current state = N[current state,
current input]
41
Next Topics
The next set of slides will cover languages accepted by deterministic
finite state automata.