Cs3102: Theory of Computation Class 11: Moore, Mealy, and Markov Models Spring 2010 University of Virginia David Evans.

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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Hidden Markov Model

LazyWant more challenging

exam

Active Student

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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