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Formal Language

and Automata Theory

Chapter 10

The Myhill-Nerode Theorem

(lecture 15,16 and B)

The Myhill-Nerode theorem

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Isomorphism of DFAs

M = (QM,S,dM,sM,FM), N = (QN,S, dN,sN,FN): two DFAs

M and N are said to be isomorphic if there is a bijection f:QM->

QNs.t.

f(sM) = sN

f(dM(p,a)) = dN(f(p),a) for all p QM, a S

p FMiff f(p) FN.

I.e., M and N are essentially the same machine up to renaming

of states.

facts:

1. Isomorphic DFAs accept the same set.

2. if M and N are any two DFAs w/o inaccessible states

accepting the same set, then the quotient automata M/and

N/ are isomorphic

3. The DFA obtained by the collapsing algorithm (lec. 14) is

the minimal DFA for the set it accepts, and this DFA is

unique up to isomorphism.

The Myhill-Nerode theorem

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Myhill-Nerode Relations

R: a regular set, M=(Q, S, d,s,F): a DFA for R w/o inaccessible

states.

M induces an equivalence relation Mon S* defined by

x My iff D(s,x) = D(s,y).

i.e., two strings x and y are equivalent iff it is

indistinguishable by running M (i.e., by running x and y,

respectively, from the initial state of M.)

Properties of M:

0. Mis an equivalence relation on S*.

(cf: is an equivalence relation on states)

1. Mis a right congruence relation on S*: i.e., for any x,y

S* and a S, x My => xa Mya.

pf: if x My => D(s,xa) = d(D(s,x),a) = d(D(s,y),a) = D(s, ya)

=> xa Mya.

The Myhill-Nerode theorem

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Properties of the Myhill-Nerode relations

Properties of M:

2. Mrefines R. I.e., for any x,y S*,

x My => x R iff y R

pf: x R iff D(s,x) F iff D(s,y) F iff y R.

Property 2 means that every M-class has either all its

elements in R or none of its elements in R. Hence R is a

union of some M-classes.

3. It is of finite index, i.e., it has only finitely many

equivalence classes.

(i.e., the set { [x]M| x S*}

is finite.

pf: x My iff D(s,x) = D(s,y) = q

for some q Q. Since there

are only |Q| states, hence

S* has |Q| M-classes

S*R

M-classes