MA/CSSE 474 Day 08 Announcements and Summary
Announcements:
1) Nothing new. See day 6 and 7 announcements.
Main ideas from today:
1.
Minimal-state DFSM for a language: Recap of where we are so far (assume unreachable states removed): a.
x

L
y means that

z
 
* (xz

L iff yz

L) i.

L
is an equivalence relation. ii.
[w] means " the equivalence class of

L
that contains w b.
If L is a regular language, let M be any DFSM that accepts L. Then (# of states in M) ≥ (# of equivalence classes of

L
). c.
Theorem: Let L be a regular language over some alphabet

.
Then there is a DFSM M with L=L(M) and that has precisely n states where n is the number of equivalence classes of

L
.
Any other FSM that accepts L must either have more states than M or it must be identical to M except for state names. d.
We began the proof last time. We construct the prescribed DFSM directly from

L
. i.
M = (K,

,

, s, A), where:
1.
K contains one state for each equivalence class of

L
.
2.
s = [

], the equivalence class of

under

L
.
3.
A = {[x] : x

L}. [Aside: what is the union of all of these classes?]
4.

([x], a) = [xa]. (if M is in state containing x, then, after reading the next symbol, a, it
goes to state containing xa). e.
Need to show: i.
K is finite. M regular  #equ. classes of

L
≤ #states of M, which is finite. ii.

is a well-defined function. i.e. ∀ x,y ∊ Σ*,a ∊ Σ ([x]=[y] → [xa]=[ya])
work with another student to write a proof of this iii.
L = L(M). Before we prove this, we first prove this more general lemma:
1.
Lemma:
 s,t ∊ Σ*( ([

], st) ⊦
M
* ([s], t)). f.
We prove the lemma by induction on |s|. The base case is trivial:
2.
Induction step. Assume that the claim is true for strings of length k. What can we say when |s| = k+1?
Since |s|

1, we know that s = yc, where y
 
* and c
 
.
3.
We apply e.iii.1. above to the special case t= ___ to get _______________
Thus:

4.
A smaller machine M# that also accepts L does not exist.
5.
A different machine M# that also has n states and that accepts L does not exist.
6.
Myhill-Nerode Theorem: A language is regular iff the number of equivalence classes of

L
is finite. a.
Proof: Show the two directions of the implication: b.
L regular

the number of equivalence classes of

L
is finite: If L is regular, then c.
The number of equivalence classes of

L
is finite

L regular: If the cardinality of

L
is finite, then
7.
Two approaches to minimizing a given machine a.
Start with separate states and merge. b.
Start with overclustering, and then separate distinguishable states. Successively approximate.
8.
Define ≡ (a relationship on states of a DFSM M) by p≡q iff for every string w ∊ Σ*, either w takes M to an accepting state from both p and q, or w takes M to a rejecting state from both p and q. a.
In first-order logic:
9.
We construct

as the limit of a sequence of approximating equivalence Relations
 n a.
p
 0 q iff configurations (p, ε) and (q, ε) either both lead to accepting states or both lead to nonaccepting states. I.e., ________________________________________' b.
For all k >= 0, p
 k+1 q iff both of the following are true:
Can you express the meaning of
 n p
 k q
For all a ∊ Σ, δ(p, a)
 k δ(q, a) non-recursively in concise English?
10.
Example:
11.
We can put any DFSM into a canonical form that makes it easy for us to tell whether two machines are "the same"