Homomorphisms

1
Chapter 6

• Homomorphisms
• (lecture 10)

Transparency No. 6-1
2
Homomorphisms (the definition)

• Def A homomorphism is a mapping h S ? G s.t.
  for all x,y in S
• h(xy) h(x) h(y) (10-1)
• h(e) e (10.2)
• Note (10.2) is not needed since h(e ) h(e e )
  h(e ) h(e )
• gt h(e ) h(e )h(e ) gt h(e ) 0
  gt h(e ) e .
• Property 1 Any homomorphism hS ? G is
  uniquely determined by its values on S.
• I.e., if g,h S ?G are morphisms s.t. g(a)h(a)
  for all a ? S, then g h (I.e., g(x) h(x) for
  all x ? S).
• Pf if x x1x2xn gt
• h(x) h(x1x2xn) h(x1)h(x2)h(xn)
  g(x1)g(xn) g(x1x2xn) g(x).
• Conclusion To specify a homomorphism h S?G
  completely, it suffices to specify its values on
  S.

3
More about homomorphisms

• h S ? G a homomorphim A a subset of S B
  a subset of G
• h(A) def h(x) x ? A ? G is the image of A
  under h.
• h-1(B) def x h(x) ? B ? S is the preimage
  of B under h.
• Theorem 10.1
• h S?G a homomorphism B a regular language
  over G
• gt h-1(B) is regular too.
• Pf M(Q, G ,d,s,F) a DFA (or NFA) s.t. L(M)
  B
• gt let M (Q, S ,d,s,F) where d(q,a)
  D(q,h(a)) for all a in S.
• Property of M
• D(q,x) D(q,h(x)) for all q ? Q and x ? S
  ----- (10.3)
• by simple induction on x, left as an
  exercise.
• ?x?S,x?L(M) iff D(s,x)?F iff D(s,h(x)) ?F iff
  h(x)? L(M) B
• h-1(B) L(M) is regular. QED

4
Example

• ha,b ? 0,1 s.t. h(a) 01 and h(b) 10.
• B L(M)
• M
• gt M ?
• view a and b as macro instructions on M
  consisting of 01 and 10,respectively

5
Example

• ha,b ? 0,1 s.t. h(a) 01 and h(b) 10.
• gt M ? (view a and b as macro instructions
  on M
• consisting of 01 and
  10,respectively.)

6
Theorem 10.2

• h S ? G a homoporphism A ? S a regular
  set
• gt h(A) is regular.
• Let a be any reg. expr. S.t. L(a ) A.
• Let a be the reg. expr. obtained from a by
  replacing every letter a of S in a with the
  string h(a).
• eg if h(a) 000, h(b) 11, a (ab)ab gt a
  (00011)00011
• a can be defined inductively as follows _____
  (exercise)
• Lemma 10.4 for any reg. expr. b over S. L(b)
  h(L(b)) --- (10.4)
• Hence L(a) h(L(a)) h(A) is regular.
• To prove 10.4, we require the lemma
• for all C,D ? S and any family of subsets Ci ?
  S, i ? I,
• h(CD) h(C) h(D) --- (10.5)
• h(U i ? I Ci) U i ? I h(Ci) --- (10.6)
• gt can be proved directly from definition.

7
Proof of 10.2

• pf by induction on b
• case 1 b a ? S L(a) L(h(a)) h(a)
  h(L(a))
• b ? gt L(?) L(?) h()
  h(L(?)).
• b e gt L(e) L(e) e h(e)
  h(L(e)).
• Ind. cases
• 1. L((bg)) L(b g) L(b) U L(g)
  h(L(b)) U h(L(g))
• h(L(b ) U L(r)) h(L(b
  r))
• 2. L((b g)) L(b g) L(b )L(g) h(L(b
  )) h(L(g))
• h(L(b )L(g)) h(L(bg))
• 3. L((b)) L((b)) U k ? 0 (L(b))k U
  k ? 0 (h(L(b)))k
• U k?0 h(L(b)k) h( U k ?
  0 L(b)k) h( L(b))
• h(L(b)).

8
Example

• EX
• h0,1 -gt a,b s.t. h(0) ab and h(1)
  ba.
• A L(M)
• M
• gt h(A) is accepted by M ?
• (Like the previous case, view 0,1 as macro
  instructions consisting of ab and ba,
  respectively. )

9
Example

• EX
• h0,1 -gt a,b s.t. h(0) ab and h(1)
  ba.
• A L(M)
• M
• gt h(A) is accepted by M ?
• (View 0,1 as macro instructions consisting of
  ab and ba, respectively, but this time replace
  every macro instruction by its content instead of
  traversing its content. )

• ab
• 1 ba

1 ba
0 ab
1 ba
10
Quiz

• Q1 Is it true that h(A) is regular gt A is
  regular ?
• Notes
• 1. h(A) B does not mean h-1(B) A.
• Hence Q1 is not a consequence of Theorem
  10.1.
• 2. counter example
• Let A anbn n ? 0
• h a,b -gt a,b with h(a)h(b) a
• gt h(A) a2n n ? 0 B is regular, but A
  can be shown
• to be not regular.
• 3. h-1(B) ______ ? A.
• ans x ? a,b x 2n .
• Exercise Given a homomorphism hS ? G and a FA
  M, find a FA M s.t. L(M) h (L(M)).

• It is possible that h(A1)h(A2)B

11
Automata with e-transiiton

• Let M (Q,S,d,S,F) be any NFA with e-transition.
• Now define a NFA Me from M as follows
• Me (Q,SUe, d',S,F)
• is a usual NFA over the alphabet SUe where
• d' is almost the same as d except that d'(p, e)
  d (p, e) for all p ?Q.
• Lemma q0 a1 q1 a2 q2 an qn (n ? 0) is a path
  from q0 to qn in M iff q0 b1 q1 b2 q2 bn
  qn is a path from q0 to qn in Me, where bk (ak
  e ? e ak).
• pf Simple induction on n.

12

• Corollary x ? L(M) iff ? y ? L(Me) s.t. h(y)
  x, where h is a morphism with h(e) e and h(a)
  a for all a ? S.
• ( iff x ? h(L(Me)) )
• pf x?L(M) iff
• ? a path from an initial state to a final state
  in M q0 a1 q1 a2 q2 an qn with x a1a2an ,
  iff
• ? a path from an initial state to a final state
  in Me
• q0 b1 q1 b2 q2 bn qn with y b1b2bn and
  h(y) x
• iff ? y ? L(Me) and h(y) x
• Theorem Languages accepted by NFA with
  e-transition are regular.
• pf If A is a language accepted by an NFA with
  e-transition machine M, then, by previous
  corollary, A L(M) h( L(Me)) . But since L(Me)
  is regular, L(M) thus is regular too.

13
Application of homomorphisms Haming distance

• x, y two bit strings, A a set of bit strings k
  ? 0 any number.
• H(x,y) of positions at which they differ
  if x y, or
• infinite if x ? y.
• H(x,A) min y ? A H(x,y)
• Nk(A) x H(x,A) ? k is the set of
  strings of Haming distance at most k from A.
• Example
• 1. N0(000) 000 ,
• 2. N1(000) 100,010,001
• 3. N2(000) ?
• 4. N2(001, 11) 0,13 - 110 U 0,12.
• Ex3Hw2 if A is regular, then so is Nk(A) for any
  k ? 0.

14
Applications of homomorphisms

• S 0,1 S x S 00, 01,10,11
  
• define top, btm S2 -gt S with
• top( ) x and btm( ) y.
• top and btm can be extended to homomorphism (S x
  S ) -gt S
• eg
• top( ) 0001 and btm(
  ) 1011
• Dk def x ? (SxS ) x contains no more than k
  occurrences of
• (0,1) or (1,0) is certainly
  regular (why?)
• x ? (S x S ) H(top(x), btm(x)) ?
  k .

15
Applications of homomorphisms (contd)

• Now let A be any regular set over S
• gt Nk(A) top (btm-1(A) ? Dk) where
• btm-1(A) is the set of (x,y) with y ? A,
• Dk is the set of (x,y) with H(x,y) ? k
• btm-1(A) ? Dk is the set of (x,y) with y ? A and
  H(x,y) ? k
• top (btm-1(A) ? Dk) is the set of x s.t. y ? A
  with H(x,y) ? k the set of x with H(x,A) ? k
  Nk(A).
• Hence Nk(A) is regular if A is regular.
• Exercises Let M be a FA accepting a language A
  over 0,1.
• 1. Construct two machines to accept N0(A) and
  N1(A),
• respectively.
• 2. How to construct a machine to accept Nk(A)
  for any given k gt 1 ?