# AUTOMATA - The Myhill-Nerode Theorem - KUMAR 7, undefined for Automata. Uttar Pradesh Technical University (UPTU)

## Automata

Description: Lecture slides on:The Myhill-Nerode Theorem, Myhill-Nerode Relations, Relations b/t DFAs and Myhill-Nerode relations, isomorphism, bisimulation,Autobisimulation,
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Transparency No. 10-1
Formal Language
and Automata Theory
Chapter 10
The Myhill-Nerode Theorem
(lecture 15,16 and B)
The Myhill-Nerode theorem
Transparency No. 10-2
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
Transparency No. 10-3
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
Transparency No. 10-4
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