Cs3102: Theory of Computation Class 11: Moore, Mealy, and Markov Models Spring 2010 University of Virginia David Evans.
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cs3102: Theory of Computation
Class 11: Moore, Mealy, and Markov Models
Spring 2010University of VirginiaDavid Evans
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• 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; 01
1 0
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 Machineused 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 , input alphabet)
Equality Rule: two machines are the same if they compute the same function
S= {0, 1} n = 3
q1; 01
q2; 0 q3; 1
1
00
0
How many experiments is enough?
Alternate Game
Given: state machineExperiment: input -> outputWin: 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 ModelM
oore
M
achi
neM
ealy
M
achi
ne
Computing ModelM
oore
M
achi
neM
ealy
M
achi
ne
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
xx
For any Mealy Machine M, we can construct a Moore Machine M’ that performs the same function:
qa
qi
qb
xy
For any Mealy Machine M, we can construct a Moore Machine M’ that performs the same function:
qa
qi
qb
xy
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
Sneezy
0.2
0.4
0.1
0.9
0.3
0.3
0.7
0.1
1.0
Markov Model with Outputs
Happy
Grumpy
Sleepy
Sneezy
0.2
0.4
0.1
0.9
0.3
0.3
0.7
0.1
1.0
“#%#$&”
“ARRGH”
0.70.3
“Zzzzzzzz”
1.0
“achoo!”
1.0
“ho ho ho!”
“wahoowa!”
0.50.5
Markov Model Examples
a.com
b.com
c.org
d.com
1/3
1/3
1/3
1/2
1/2
1/2
1/2
Nodes: URLsLinks: hyperlinksProbabilities: 1/n number of non-self outgoing links
Pr(u) = probability ofreaching u starting from random seed states
Lawrence Page, Sergey Brin, Rajeev Motwani and Terry Winograd
Garkov
http://www.joshmillard.com/garkov/
Hidden Markov Model
Happy
Grumpy
Sleepy
Sneezy
0.2
0.4
0.1
0.9
0.3
0.3
0.7
0.1
1.0
“#%#$&”
“ARRGH”
0.70.3
“Zzzzzzzz”
1.0
“achoo!”
1.0
“ho ho ho!”
“wahoowa!”
0.50.5
From just the outputs guess the states (and machine)
Hidden Markov Model Example
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Sent Review Topics
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Hidden Markov Model
LazyWant more challenging
exam
Active Student
No Review Question
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
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