Module #16 – Finite State Machines
Modeling Computation
Rosen 5th ed., ch. 11
Ref: Wikipedia
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Module #16 – Finite State Machines
Finite State Machines (FSM)
Model machines (or components in a
computer) using a particular structure
Classification


FSM with no output: determine whether the input
is accepted (recognized) or not
FSM with output: generate output from the given
input
Representations (p.752)
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
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State diagram
State transition table
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Module #16 – Finite State Machines
FSM with Output
Definition: M = (S, I, O, f, g, s0)
S: finite set of states
I, O: input/output alphabets
f: transition function, f: S  I  S
g: output function
s0: initial state

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Mealy machine: g: SI O
Moore machine:g: S O
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Module #16 – Finite State Machines
Simple Example (p.753)
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Module #16 – Finite State Machines
Generating Output
Input string: x = x1x2…xk
s1 = f(s0, x1), s2 = f(s1, x2), …, sk = f(sk-1, xk)
These state transition produces output y =
y1y2…yk
y1 = g(s0, x1), y2 = g(s1, x2), …, yk = g(sk-1, xk)
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Module #16 – Finite State Machines
Another Example
start
Input: 101011
Output: 001000
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Module #16 – Finite State Machines
Ex: vending machines
[description]
State table
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Module #16 – Finite State Machines
Vending Machine: state diagram
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Module #16 – Finite State Machines
Ex: Unit Delay Machine
Input: x1x2…xk; Output: 0x1x2…xk-1
state
s0
s1
s2
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f
0
s2
s2
s2
g
1
s1
s1
s1
0
0
1
0
Example & how the machine is
designed (p.755)
1
0
1
0
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Module #16 – Finite State Machines
Ex: FSM for Addition
Design:


s0: to remember
that the previous
carry is 0
s1: to remember
that the previous
carry is 1
[example]
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Module #16 – Finite State Machines
FSM with No Output
Language recognition; design and
implementation of compilers

A string is recognized IFF it takes the
starting state to one of the final states
Def: Finite-state Automata: do not
produce output, but have a set of final
states
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Ex: automaton to recognize
“nice”
Module #16 – Finite State Machines
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Module #16 – Finite State Machines
Finite-State Automata
M = (S, I, f, s0, F)



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S: finite set of states
I: input alphabets
f: transition function, f: S  I  S
s0: initial state
F: final states, subset of S (indicated by
double circles)
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Module #16 – Finite State Machines
Example: construct the state
diagram
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Module #16 – Finite State Machines
Recognizing Language
A string x is said to be recognized by
the machine M = (S, I, f, s0, F) if f(s0, x)
is a state in F
The language recognized (accepted) by
the machine M, denoted by L(M), is the
set of all strings accepted by M.
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Module #16 – Finite State Machines
Deterministic
finite automaton
Example
The following example is of a DFA M,
with a binary alphabet, which
determines if the input contains an
even number of 0s
Start
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Module #16 – Finite State Machines
Transformations from/to
State Diagram
It is possible to draw a state diagram from the table.
Draw the circles to represent the states given.
For each of the states, scan across the corresponding
row and draw an arrow to the destination state(s).
There can be multiple arrows for an input character if
the automaton is an NFA.
Designate a state as the start state. The start state is
given in the formal definition of the automaton.
Designate one or more states as accept state. This is
also given in the formal definition.
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Module #16 – Finite State Machines
Example
L(M1) = {1n | n = 0,1,2,…}
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Module #16 – Finite State Machines
Example (cont)
L(M2) = {1, 01}
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Module #16 – Finite State Machines
Example (cont)
L(M3) = {0n,0n10x | n = 0,1,2,…, x is
any string}
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