Common Mistakes

(1)

• Please give details of your calculation. A direct answer without explanation is not counted.

• Your answers must be in English.

• Please carefully read problem statements.

• During the exam you are not allowed to borrow others’ class notes.

• Try to work on easier questions first.

• Please draw all the diagrams clearly.

Problem 1 (20 pts)

Consider the following two-tape TM:

q₀

start q₁ q₂

q₃

q₄ q_a

t → R t → R

0 → R t → 0, R

t → L t → L

0 → R 0 → L

t → L 0 → L 0 → R t → R

t → 0, R 0 → R

t → R t → R

Assume the initial configuration is t000000 in the 1st tape and t t . . . in the 2nd tape. Run the TM and give the sequence of configurations.

Answer

q₀t 000000

q₀t t t t t tt ⇒

tq₁000000

tq₁t t t t t t ⇒

t0q₁00000

t0q₁t t t tt ⇒

t00q₁0000

t00q₁t t t t ⇒

t000q₁000 t000q₁t tt

⇒

t0000q₁00 t0000q₁t t ⇒

t00000q₁0 t00000q₁t ⇒

t000000q₁t t000000q₁t ⇒

t00000q₂0t t00000q₂0t ⇒

t000000q₃t t0000q₃00t ⇒

t00000q₂0t t000q₂000t

⇒

t000000q₃t t00q₃0000t ⇒

t00000q₂0t t0q₂00000t ⇒

t000000q₃t tq₃000000t ⇒

t00000q₂0t q₂t 000000t ⇒

t000000q₄t tq₄000000t ⇒

(2)

t0000000q₄t t0q₄00000t ⇒

t00000000q₄t t00q₄0000t ⇒

t000000000q₄t t000q₄000t ⇒

t0000000000q₄t t0000q₄00t ⇒

t00000000000q₄t t00000q₄0t

⇒

t000000000000q₄t t000000q₄t ⇒

t000000000000 t q_at t000000 t q_at

Problem 2 (40 pts)

Construct a standard TM to recognize the following language:

{0ⁿ#1ⁿ#0ⁿ|n ≥ 0},

where Σ = {0, 1}. The # states ≤ 9 (including q_accept and q_reject). You need to show the formal definition of the TM, but you can draw a state diagram instead of giving a formal definition of the transition function δ. You also need to explain the meaning of each edge.

Answer

T M = {Q, Σ, Γ, δ, q₁, q_a, q_r}:

Q = {q₁, q₂, q₃, q₄, q₅, q₆, q₇, q_a, q_r} Σ = {0, 1, #}

Γ = {0, 1, t, #, #1, #2, x, y, z}

δ is defined as the following state diagram:

(3)

q₁ start

q₂ q₃

q4

q5

q6

q₇ q_a q_r

x → R

0 → x, R

0 → R

# → R

y → R

1 → y, R

1 → R

# → R y → R

0 → y, L 0 → L

1 → L y → L

# → L

# → #₁, R y → z, R

# → #₂, R t → L z → L

#₂ → L

#₁ → R

z → L

All edges not specified go to the reject state.

That is, everytime we encounter a ‘0’ before ‘#’, we need to make sure there are a ‘1’ and a

‘0’ after the ‘#’. The first ‘0’ is then replaced by ‘x’, and the ‘1’ and ‘0’ after ‘#’ are replaced by ‘y’. If there is no more ‘0’ before #, we need to check if the replaced string is “x ∗ #y ∗ #y∗”.

Meaning of each edges:

• (q₁, q₂): replace a 0 before # by x.

• (q₂, q₂): before #, only 0 can be found.

• (q₂, q₃): find a #.

• (q₃, q₃): pass all checked symbols.

• (q₃, q₄): replace a 1 before the second # by y.

(4)

• (q₄, q₄): before the second #, only 1 can be found.

• (q₄, q₅): find a #.

• (q₅, q₅): pass all checked symbols.

• (q₅, q₆): replace a 0 by y.

• (q₆, q₆): go back until a x is found.

• (q6, q1): go to the next symbol. It can only be 0 or #.

• (q₁, q₇): In case the next symbol is #, change it to #₁.

• (q₇, q₇): If # is found, change it to #₂. And change all y to z.

• (q₇, q₁): Find the last #₂.

• (q₁, q₁): Pass all z’s.

• (q₁, q_a): If #₁ is found, it means there is exactly two # in the original input string.

Common Mistakes

• # states > 9

• does not accept ##

• accepts ###

Problem 3 (40 pts)

Consider the following definition of nondeterministic TM:

N T M = (Q, Σ, Γ, δ, q₁, q_a, q_r),

where q_a 6= q_r and δ : Q × Γ^k → P (Q × Γ^k× {L, R, S}^k). Give a two-tape NTM to recognize the following language:

{aⁱb^jc^k|i, j, k ≥ 0 and i = j or i = k}.

The # of states ≤ 8 (including q_accept and q_reject). You do not have give a formal definition. (A diagram is enough.) You also need to explain the meaning of each edge.

(6)

Answer

q₁

start q₂

q₆

q₃

q₅ q_a

q₄

q_r a/b/c/t → S

t → R

a → R t → a, R

b → S t → L

b → R t → S

c/t → S t → L

c → R t → L

b → R a → L

c → R t → S

t → S t → S

c → R t → S

b → R t → S

c → R a → L

t → S t → S

All a’s in the first tape are copied to the second tape. Then check if number of b’s or number of c’s is equal to number of a’s.

Meaning of each edge:

• (q₁, q₂): write a t at the beginning of the second tape.

• (q₂, q₂): copy a’s to the second tape.

• (q₂, q₃): in this path, number of b’s should be same as number of a’s.

• (q₂, q₄): in this path, both number of b’s and number of a’s are 0, but number of c is not.

• (q₃, q₃): check if number of b’s is equal to number of a’s.

• (q3, qa): accept if at the end of both tapes.

• (q₃, q₄): a c is found after b’s.

• (q₄, q₄): after a’s and b’s, there should only be c’s.

• (q₄, q_a): accept if at the end of the first tape.

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• (q₂, q₅): in this path, number of c’s should be same as number of a’s, and a b is found after a’s.

• (q₅, q₅): only b∗ can be between a’s and c’s.

• (q5, q6): no more b’s.

• (q₂, q₆): in this path, number of c’s should be same as number of a’s, but no b is found after a’s.

• (q₆, q₆): number of c’s should be same as number of a’s.

• (q₆, q_a): accept if at the end of both tapes.

Another Solution

q₁

start q₂

q3

q₄ q₅

q6 qa

qr

a, b, c, t → S t → $, R

a → R t → R

b, c, t → S t → S

b → R t → S

c, t → S t → L

c → R t → L

t → R

$ → S b, c, t → S

t → L b → R t → L

c, t → S

$ → S

c → R

$ → S

t → R

$ → S

Common Mistakes

• does not accept , aⁿbⁿ, aⁿcⁿ, b⁺(i = k = 0), or c⁺(i = j = 0)

• accepts aⁿbⁿcΣ^∗

• assumes that there is a t at the beginning of tapes (or before the beginning of tapes)