Menu
cs3102: Theory of Computation
Class 11:
Moore, Mealy, and Markov Models
Spring 2010
University of Virginia
David Evans
• 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
0
q0; 1
1
“Power” of a Machine
Power of a DFA, NFA, DPDA, NPDA/CFG:
Set of languages it can recognize/produce.
0
Power of a Moore Machine:
1
Set of functions it can perform.
q1; 0
Language
Set of strings
Function
Set of <string, string>
(input/output) pairs
Formal Definition
Computing Model
DFA
Computing Model
DFA
Moore
Moore’s Experiments
Moore
Okay...guess the machine!
You LOSE!
0
q0; 1
1
0
0
0
q1; 0
q2; 0
1
1
0
q6;
q3; 0
1
1
0
1
q5; 0
1
q4; 0
1
You always lose.
0
Sometimes “you” win...
Lorenz Cipher Machine
used by Nazi high
command: links between
conquered capitals
Colossus
Arguably, the first electronic,
digital, programmable computer.
Machine determined by
Bill Tutte (1941) from
intercepted messages
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
Bletchley Park, 1943
Decoded 63 million letters in Nazi
command messages
Learned German troop locations to plan
D-Day (knew the deception was working)
Bletchley Park, 2004 (rebuilt)
Σ= {0, 1} n = 3
0
0
q1; 0
q2; 0
1
q3; 1
1
0
Alternate Game
How many experiments is enough?
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.
George Mealy, A Method for Synthesizing Sequential Circuits, 1955
0; 1
q0
1; 0
Computing Model
Moore
Machine
Mealy Machine
1; 0
q1
Mealy
Machine
0; 1
Moore
Machine
Computing Model
Which is more powerful?
Moore
Mealy
Machine
Mealy
For any Moore Machine M, we can construct a
Mealy Machine M’ that performs the same
function:
For any Moore Machine M, we can construct a
Mealy Machine M’ that performs the same
function:
qb; y
qa; z
qb
qb; y
qa
qa; z
x
x
qi; x
qi; x
qi
For any Mealy Machine M, we can construct a
Moore Machine M’ that performs the same
function:
For any Mealy Machine M, we can construct a
Moore Machine M’ that performs the same
function:
qb
qb
qb
qa
qa
qa
y
y
x
x
qi
qi
qi1; x
qi2; y
Both have all the same outgoing
transitions as qi
Equally Powerful
Are they good models?
Moore
0; 1
Mealy
q0
0; 1
1; 0
1; 0
q1
(Moore may need more needs more states)
Markov Model
Markov Model with Outputs
1.0
1.0
0.3
0.3
0.7
0.3
0.7
0.7
Happy
Happy
0.3
0.3
Sleepy
0.5
Sleepy
1.0
0.5
“ho ho ho!”
Sneezy
0.1
“ARRGH”
Grumpy
Grumpy
Sneezy
0.9
0.1
0.9
Andrey Markov, 1856-1922
“wahoowa!”
“#%#$&”
“Zzzzzzzz”
1.0
“achoo!”
Markov Model Examples
Nodes: URLs
Links: hyperlinks
Probabilities: 1/n number of nonself outgoing links
Lawrence Page, Sergey Brin, Rajeev Motwani and Terry Winograd
b.com
a.com
1/2
c.org
1/2
Pr(u) = probability of
reaching u starting from
random seed states
d.com
1/2
1/2
Hidden Markov Model
Garkov
1.0
0.3
“ARRGH”
Grumpy
0.7
0.3
0.7
Happy
0.3
Sleepy
0.5
1.0
0.5
“ho ho ho!”
Sneezy
0.1
http://www.joshmillard.com/garkov/
“wahoowa!”
“#%#$&”
0.9
1.0
“achoo!”
From just the outputs guess the states (and machine)
“Zzzzzzzz”
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
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
Hidden Markov Model
Want more
challenging
exam
Lazy
0.9
1.0
No Review
Question
Hidden Markov Model
Active
Student
0.1
1.0
Sent Review
Topics
Return PS3
front of room
AA
Flop: 222
Opponent Raises
0.4
0.6
0.8
AK
A-D
E-K
L-R
S-Z
…
0.9
72
0.02
0.08
Raise
Call
Fold