cs3102: Theory of Computation
Class 11:
Moore, Mealy, and Markov Models
Spring 2010
University of Virginia
David Evans
Menu
• Exam Review
• Variations on DFAs:
– Moore Machine: states produce output
– Mealy Machine: edges produce output
– Markov Model: transitions have probabilities
Moore Machine
Edward Moore, Gedanken-experiments on Sequential Machines, 1956.
http://people.mokk.bme.hu/~kornai/MatNyelv/moore_1956.pdf
Moore Machine Example
q0; 1
0
q1; 0
1
1 0
“Power” of a Machine
Power of a DFA, NFA, DPDA, NPDA/CFG:
Set of languages it can recognize/produce.
Power of a Moore Machine:
Set of functions it can perform.
Language Function
Set of strings Set of <string, string>
(input/output) pairs
Formal Definition Computing ModelDFA Moore
Computing ModelDFA Moore
Moore’s Experiments
Okay...guess the machine!
You LOSE!
q0; 1
0
q1; 0
1
10
q2; 0 q3; 0
0
1
q6; 1 q5; 0 q4; 0
1
11
0
0
01
0
You always lose.
Sometimes “you” win...
Lorenz Cipher Machine
used by Nazi high
command: links between
conquered capitals
Machine determined by
Bill Tutte (1941) from
intercepted messages
Colossus
Bletchley Park, 1943
Bletchley Park, 2004 (rebuilt)
Decoded 63 million letters in Nazi
command messages
Learned German troop locations to plan
D-Day (knew the deception was working)
Arguably, the first electronic,
digital, programmable computer.
A More Fair Game
Reveal: n, maximum number of states in the
machine (and Σ, input alphabet)
Equality Rule: two machines are the same if
they compute the same function
Σ= {0, 1} n = 3
q1; 0
1q2; 0 q3; 1
1
00
0
How many experiments is enough?
Alternate Game
Given: state machine
Experiment: input -> output
Win: guess what state the machine started in
Moore proved for some machines where all states are distinguishable,
it is impossible to know the starting state from one experiment.
Mealy MachineGeorge Mealy, A Method for Synthesizing Sequential Circuits, 1955
q0
0; 1
q1
1; 0
1; 0 0; 1
Computing Model
Mo
ore
Ma
chin
eM
ea
ly
Ma
chin
e
Computing Model
Mo
ore
Ma
chin
eM
ea
ly
Ma
chin
e
Which is more powerful?
Mealy
Moore
For any Moore Machine M, we can construct a
Mealy Machine M’ that performs the same
function:
qa; z
qi; x
qb; y
For any Moore Machine M, we can construct a
Mealy Machine M’ that performs the same
function:
qa; z
qi; x
qb; y
qa
qi
qb
x
x
For any Mealy Machine M, we can construct a
Moore Machine M’ that performs the same
function:
qa
qi
qb
x
y
For any Mealy Machine M, we can construct a
Moore Machine M’ that performs the same
function:
qa
qi
qb
x
y
qa
qi1; x
qb
qi2; y
Both have all the same outgoing
transitions as qi
Equally Powerful
Mealy
Moore
(Moore may need more needs more states)
Are they good models?
q0
0; 1
q1
1; 01; 0
0; 1
Markov Model
Andrey Markov, 1856-1922
Happy
Grumpy
Sleepy
Sneezy0.9
0.3
0.3
0.7
0.1
1.0
Markov Model with Outputs
Happy
Grumpy
Sleepy
Sneezy0.9
0.3
0.3
0.7
0.1
1.0
“#%#$&”
“ARRGH”
0.70.3
“Zzzzzzzz”
1.0
“achoo!”
1.0
“ho ho ho!”
“wahoowa!”
0.5
0.5
Markov Model Examples
a.com
b.com
c.org
d.com 1/2
1/2
1/2
1/2
Nodes: URLs
Links: hyperlinks
Probabilities: 1/n number of non-
self outgoing links
Pr(u) = probability of
reaching u starting from
random seed states
Lawrence Page, Sergey Brin, Rajeev Motwani and Terry Winograd
Garkov
http://www.joshmillard.com/garkov/
Hidden Markov Model
Happy
Grumpy
Sleepy
Sneezy0.9
0.3
0.3
0.7
0.1
1.0
“#%#$&”
“ARRGH”
0.70.3
“Zzzzzzzz”
1.0
“achoo!”
1.0
“ho ho ho!”
“wahoowa!”
0.5
0.5
From just the outputs guess the states (and machine)
Hidden Markov Model Example
No Review
Question
No Review
Question
No Review
Question
No Review
Question
No Review
Question
No Review
Question
No Review
Question
No Review
Question
No Review
Question
No Review
Question
No Review
Question
No Review
Question
No Review
Question
No Review
Question
No Review
Question
No Review
Question
No Review
Question
No Review
Question
No Review
Question
No Review
Question
No Review
Question
No Review
Question
No Review
Question
No Review
Question
No Review
Question
No Review
Question
No Review
Question
No Review
Question
No Review
Question
No Review
Question
No Review
Question
No Review
Question
No Review
Question
No Review
Question
No Review
Question
No Review
Question
No Review
Question
No Review
Question
No Review
Question
No Review
Question
No Review
Question
No Review
Question
No Review
Question
No Review
Question
Sent Review
Topics
Sent Review
Topics
No Review
Question
No Review
Question
No Review
Question
No Review
Question
No Review
Question
No Review
Question
No Review
Question
No Review
Question
No Review
Question
No Review
Question
No Review
Question
No Review
Question
No Review
Question
No Review
Question
No Review
Question
No Review
Question
No Review
Question
No Review
Question
No Review
Question
No Review
Question
No Review
Question
No Review
Question
No Review
Question
No Review
Question
No Review
Question
No Review
Question
No Review
Question
No Review
Question
No Review
Question
No Review
Question
No Review
Question
No Review
Question
No Review
Question
No Review
Question
No Review
Question
No Review
Question
No Review
Question
No Review
Question
No Review
Question
No Review
Question
No Review
Question
No Review
Question
No Review
Question
No Review
Question
No Review
Question
No Review
Question
No Review
Question
No Review
Question
No Review
Question
No Review
Question
No Review
Question
No Review
Question
No Review
Question
No Review
QuestionHidden Markov Model
LazyWant more
challenging
exam
Active
Student
No Review
Question
No Review
Question
Sent Review
Topics
Sent Review
Topics
1.00.11.0 0.9
Hidden Markov Model
A A
A K
7 2
Raise Call Fold
0.6
0.4
…
0.9
0.08
0.02
Opponent Raises
0.8
Flop: 222
Return PS3front of room
A-D E-K
L-R S-Z
Top Related