Deterministic Finite State Machines

Chapter 5

Languages and Machines

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Regular Languages

Regular Language

Regular Expression

Finite State Machine

L

Accepts

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Finite State Machines

Vending Machine: An FSM to accept $.50 in change

Drink is 25 centsNo more than 50 cents can be deposited

X

Other FSM examples?4

Definition of a DFSM

M = (K, , , s, A), where:

K is a finite set of states is an alphabet s K is the initial state A K is the set of accepting states, and is the transition function from

(K ) to Kstate input symbol state

Quintuple

Cartesian product

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Accepting by a DFSM

Informally, M accepts a string w iff M winds up in some element of A when it has finished reading w.

The language accepted by M, denoted L(M), is the set of all strings accepted by M.

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Configurations of DFSMs

A configuration of a DFSM M is an element of:

K *

It captures the two things that can make a difference to M’s future behavior:

• its current state

• the input that is still left to read.

The initial configuration of a DFSM M, on input w, is:

(sM, w), where sM is the start state of M 7

The Yields Relations

The yields-in-one-step relation |-M :

(q, w) |-M (q', w') iff

• w = a w' for some symbol a , and • (q, a) = q'

The relation yields, |-M*, is the reflexive, transitive closure of |-M.

C1 |-M* C2 : M can go from C1 to C2 in 0 or more steps.

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Computations Using FSMs

A computation by M is a finite sequence of configurations C0, C1, …, Cn for some n 0 such that:

• C0 is an initial configuration,

• Cn is of the form (q, ), for some state q KM,

• C0 |-M C1 |-M C2 |-M … |-M Cn.

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Accepting and Rejecting

A DFSM M accepts a string w iff:

(s, w) |-M * (q, ), for some q A.

A DFSM M rejects a string w iff:

(s, w) |-M* (q, ), for some q AM.

The language accepted by M, denoted L(M), is the set of all strings accepted by M.

Theorem: Every DFSM M, on input s, halts in |s| steps.

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An Example Computation

An FSM to accept odd integers: even odd

even

q0 q1

odd

On input 235, the configurations are:

(q0, 235) |-M (q0, 35)|-M

|-M

Thus (q0, 235) |-M* (q1, ) 11

Regular Languages

A language is regular iff it is accepted by some FSM.

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A Very Simple Example

L = {w {a, b}* : every a is immediately followed by a b}.

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A Very Simple Example

L = {w {a, b}* : every a is immediately followed by a b}.

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A Very Simple Example

L = {w {a, b}* : every a is immediately followed by a b}.

M = (K, , , s, A) = ({q0, q1, q2}, {a, b}, , q0, {q0}), where

= {((q0, a), q1), ((q0, b), q0),

((q1, a), q2), ((q1, b), q0),

((q2, a), q2), ((q2, b), q2)}15

Parity Checking

L = {w {0, 1}* : w has odd parity}.

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Parity Checking

L = {w {0, 1}* : w has odd parity}.

M = (K, , , s, A) = ({q0, q1}, {0, 1}, , q0, {q1}), where

= {((q0, 0), q0), ((q0, 1), q1),

((q1, 0), q1), ((q1, 1), q0)}

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No More Than One b

L = {w {a, b}* : w contains no more than one b}.

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No More Than One b

L = {w {a, b}* : w contains no more than one b}.

M = (K, , , s, A)

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Checking Consecutive Characters

L = {w {a, b}* :

no two consecutive characters are the same}.

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Checking Consecutive Characters

L = {w {a, b}* :

no two consecutive characters are the same}.

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Dead States

L = {w {a, b}* : every a region in w is of even length}

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Dead States

L = {w {a, b}* : every a region in w is of even length}

M = (K, , , s, A)

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Dead States

L = {w {a, b}* : every b in w is surrounded by a’s}

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The Language of Floating Point

Numbers is Regular Example strings:

+3.0, 3.0, 0.3E1, 0.3E+1, -0.3E+1, -3E8

The language is accepted by the DFSM:

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A Simple Communication Protocol

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Controlling a Soccer-Playing Robot

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A Simple Controller

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Programming FSMs

Cluster strings that share a “future”.

Let L = {w {a, b}* : w contains an even number of a’s and an odd number of b’s}

What states are needed? How many states are there?

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Even a’s Odd b’s

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Vowels in Alphabetical Order

L = {w {a - z}* : all five vowels, a, e, i, o, and u,

occur in w in alphabetical order}.

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Vowels in Alphabetical Order

L = {w {a - z}* : all five vowels, a, e, i, o, and u,

occur in w in alphabetical order}.

O

u

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Programming FSMs

L = {w {a, b}* : w does not contain the substring aab}.

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Programming FSMs

L = {w {a, b}* : w does not contain the substring aab}.

Start with a machine for L:

How must it be changed?

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Programming FSMs

L = {w {a, b}* : w does not contain the substring aab}.

Start with a machine for L:

How must it be changed?

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A Building Security System

L = {event sequences such that the alarm should sound}

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FSMs Predate Computers

The Prague Orloj, originally built in 1410

The Chinese Abacus,14 century AD

The Jacquard Loom, invented in 1801

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Finite State Representations in Software Engineering

A high-level state chart model of a digital watch.

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Making the Model Hierarchical

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The Missing Letter Language

Let = {a, b, c, d}.

Let LMissing = {w : there is a symbol ai not appearing in w}.

Try to make a DFSM for LMissing:

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