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,

Moore Machine Example q0; 1 0 q1;

“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. LanguageFunction Set of stringsSet of (input/output) pairs

Formal Definition

Computing Model DFAMoore

Computing Model DFAMoore

Moore’s Experiments

Okay...guess the machine!

You LOSE! q0; 1 0 q1; q2; 0q3; q6; 1 q5; 0q4;

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 , input alphabet) Equality Rule: two machines are the same if they compute the same function

 = {0, 1} n = 3

q1; 0 1 q2; 0q3;

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 Machine George Mealy, A Method for Synthesizing Sequential Circuits, 1955 q0 0; 1 q1 1; 0 0; 1

Computing Model Moore Machine Mealy Machine

Computing Model Moore Machine Mealy Machine

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; 0 0; 1

Markov Model Andrey Markov, Happy Grumpy Sleepy Sneezy

Markov Model with Outputs Happy Grumpy Sleepy Sneezy “#%#$&” “ARRGH” “Zzzzzzzz” 1.0 “achoo!” 1.0 “ho ho ho!” “wahoowa!” 0.5

Markov Model Examples a.com b.com c.org d.com 1/3 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

Hidden Markov Model Happy Grumpy Sleepy Sneezy “#%#$&” “ARRGH” “Zzzzzzzz” 1.0 “achoo!” 1.0 “ho ho ho!” “wahoowa!” 0.5 From just the outputs guess the states (and machine)

Hidden Markov Model Example No Review Question Sent Review Topics No Review Question

Hidden Markov Model Lazy Want more challenging exam Active Student No Review Question Sent Review Topics

Hidden Markov Model A A K 7 2 Raise Call Fold … Opponent Raises 0.8 Flop: 222

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