Turing Machine accepting{zz | z {a, b} } ∈ ∗Tracing strings aabb & bb

byBrian Morales

let’s trace this TM on the string aabb to show rejection. Begin from the initial configuration:q0∆aabb

∆/∆, R input symbol is ∆ / replace it with ∆ and move the state to the rightq0∆aabb Ⱶ ∆q1aabb

a/A, R input symbol is a / replace it with A and move the state to the rightq0∆aabb Ⱶ ∆q1aabb Ⱶ ∆Aq2abb

a/a, R input symbol is a / replace it with a and move the state to the rightq0∆aabb Ⱶ ∆q1aabb Ⱶ ∆Aq2abb Ⱶ ∆Aaq2bb

b/b, R input symbol is b / replace it with b and move the state to the rightq0∆aabb Ⱶ ∆q1aabb Ⱶ ∆Aq2abb Ⱶ ∆Aaq2bb Ⱶ ∆Aabq2b

b/b, R input symbol is b / replace it with b and move the state to the rightq0∆aabb Ⱶ ∆q1aabb Ⱶ ∆Aq2abb Ⱶ ∆Aaq2bb Ⱶ ∆Aabq2b Ⱶ ∆Aabbq2∆

∆/∆, L input symbol is ∆ / replace it with ∆ and move the state to the leftq0∆aabb Ⱶ ∆q1aabb Ⱶ ∆Aq2abb Ⱶ ∆Aaq2bb Ⱶ ∆Aabq2b Ⱶ ∆Aabbq2∆ Ⱶ ∆Aabq3b

b/B, L input symbol is b / replace it with B and move the state to the leftq0∆aabb Ⱶ ∆q1aabb Ⱶ ∆Aq2abb Ⱶ ∆Aaq2bb Ⱶ ∆Aabq2b Ⱶ ∆Aabbq2∆ Ⱶ ∆Aabq3b Ⱶ ∆Aaq4bB

b/b, L input symbol is b / replace it with b and move the state to the leftq0∆aabb Ⱶ ∆q1aabb Ⱶ ∆Aq2abb Ⱶ ∆Aaq2bb Ⱶ ∆Aabq2b Ⱶ ∆Aabbq2∆ Ⱶ ∆Aabq3b Ⱶ ∆Aaq4bB Ⱶ ∆Aq4abB

a/a, L input symbol is a / replace it with a and move the state to the leftq0∆aabb Ⱶ ∆q1aabb Ⱶ ∆Aq2abb Ⱶ ∆Aaq2bb Ⱶ ∆Aabq2b Ⱶ ∆Aabbq2∆ Ⱶ ∆Aabq3b Ⱶ ∆Aaq4bB Ⱶ ∆Aq4abB Ⱶ ∆q4AabB

A/A, R input symbol is A / replace it with A and move the state to the rightq0∆aabb Ⱶ ∆q1aabb Ⱶ ∆Aq2abb Ⱶ ∆Aaq2bb Ⱶ ∆Aabq2b Ⱶ ∆Aabbq2∆ Ⱶ ∆Aabq3b Ⱶ ∆Aaq4bB Ⱶ ∆Aq4abB Ⱶ ∆q4AabB Ⱶ ∆Aq1abB

a/A, R input symbol is a / replace it with A and move the state to the rightq0∆aabb Ⱶ ∆q1aabb Ⱶ ∆Aq2abb Ⱶ ∆Aaq2bb Ⱶ ∆Aabq2b Ⱶ ∆Aabbq2∆ Ⱶ ∆Aabq3b Ⱶ ∆Aaq4bB Ⱶ ∆Aq4abB Ⱶ ∆q4AabB Ⱶ ∆Aq1abB Ⱶ ∆AAq2bB

b/b, R input symbol is b / replace it with b and move the state to the rightq0∆aabb Ⱶ ∆q1aabb Ⱶ ∆Aq2abb Ⱶ ∆Aaq2bb Ⱶ ∆Aabq2b Ⱶ ∆Aabbq2∆ Ⱶ ∆Aabq3b Ⱶ ∆Aaq4bB Ⱶ ∆Aq4abB Ⱶ ∆q4AabB Ⱶ ∆Aq1abB Ⱶ ∆AAq2bB Ⱶ ∆AAbq2B

B/B, L input symbol is B / replace it with B and move the state to the leftq0∆aabb Ⱶ ∆q1aabb Ⱶ ∆Aq2abb Ⱶ ∆Aaq2bb Ⱶ ∆Aabq2b Ⱶ ∆Aabbq2∆ Ⱶ ∆Aabq3b Ⱶ ∆Aaq4bB Ⱶ ∆Aq4abB Ⱶ ∆q4AabB Ⱶ ∆Aq1abB Ⱶ ∆AAq2bB Ⱶ ∆AAbq2B Ⱶ ∆AAq3bB

b/B, L input symbol is b / replace it with B and move the state to the leftq0∆aabb Ⱶ ∆q1aabb Ⱶ ∆Aq2abb Ⱶ ∆Aaq2bb Ⱶ ∆Aabq2b Ⱶ ∆Aabbq2∆ Ⱶ ∆Aabq3b Ⱶ ∆Aaq4bB Ⱶ ∆Aq4abB Ⱶ ∆q4AabB Ⱶ ∆Aq1abB Ⱶ ∆AAq2bB Ⱶ ∆AAbq2B Ⱶ ∆AAq3bB Ⱶ ∆Aq4ABB

A/A, R input symbol is A / replace it with A and move the state to the rightq0∆aabb Ⱶ ∆q1aabb Ⱶ ∆Aq2abb Ⱶ ∆Aaq2bb Ⱶ ∆Aabq2b Ⱶ ∆Aabbq2∆ Ⱶ ∆Aabq3b Ⱶ ∆Aaq4bB Ⱶ ∆Aq4abB Ⱶ ∆q4AabB Ⱶ ∆Aq1abB Ⱶ ∆AAq2bB Ⱶ ∆AAbq2B Ⱶ ∆AAq3bB Ⱶ ∆Aq4ABB Ⱶ ∆AAq1BB

B/B, L input symbol is B / replace it with B and move the state to the leftq0∆aabb Ⱶ ∆q1aabb Ⱶ ∆Aq2abb Ⱶ ∆Aaq2bb Ⱶ ∆Aabq2b Ⱶ ∆Aabbq2∆ Ⱶ ∆Aabq3b Ⱶ ∆Aaq4bB Ⱶ ∆Aq4abB Ⱶ ∆q4AabB Ⱶ ∆Aq1abB Ⱶ ∆AAq2bB Ⱶ ∆AAbq2B Ⱶ ∆AAq3bB Ⱶ ∆Aq4ABB Ⱶ ∆AAq1BB Ⱶ ∆Aq5ABB

A/a, L input symbol is A / replace it with a and move the state to the leftq0∆aabb Ⱶ ∆q1aabb Ⱶ ∆Aq2abb Ⱶ ∆Aaq2bb Ⱶ ∆Aabq2b Ⱶ ∆Aabbq2∆ Ⱶ ∆Aabq3b Ⱶ ∆Aaq4bB Ⱶ ∆Aq4abB Ⱶ ∆q4AabB Ⱶ ∆Aq1abB Ⱶ ∆AAq2bB Ⱶ ∆AAbq2B Ⱶ ∆AAq3bB Ⱶ ∆Aq4ABB Ⱶ ∆AAq1BB Ⱶ ∆Aq5ABB Ⱶ ∆q5AaBB

A/a, L input symbol is A / replace it with a and move the state to the leftq0∆aabb Ⱶ ∆q1aabb Ⱶ ∆Aq2abb Ⱶ ∆Aaq2bb Ⱶ ∆Aabq2b Ⱶ ∆Aabbq2∆ Ⱶ ∆Aabq3b Ⱶ ∆Aaq4bB Ⱶ ∆Aq4abB Ⱶ ∆q4AabB Ⱶ ∆Aq1abB Ⱶ ∆AAq2bB Ⱶ ∆AAbq2B Ⱶ ∆AAq3bB Ⱶ ∆Aq4ABB Ⱶ ∆AAq1BB Ⱶ ∆Aq5ABB Ⱶ ∆q5AaBB Ⱶ q5∆aaBB

∆/∆, R input symbol is ∆ / replace it with ∆ and move the state to the rightq0∆aabb Ⱶ ∆q1aabb Ⱶ ∆Aq2abb Ⱶ ∆Aaq2bb Ⱶ ∆Aabq2b Ⱶ ∆Aabbq2∆ Ⱶ ∆Aabq3b Ⱶ ∆Aaq4bB Ⱶ ∆Aq4abB Ⱶ ∆q4AabB Ⱶ ∆Aq1abB Ⱶ ∆AAq2bB Ⱶ ∆AAbq2B Ⱶ ∆AAq3bB Ⱶ ∆Aq4ABB Ⱶ ∆AAq1BB Ⱶ ∆Aq5ABB Ⱶ ∆q5AaBB Ⱶ q5∆aaBB Ⱶ ∆q6aaBB

a/A, R input symbol is a / replace it with A and move the state to the rightq0∆aabb Ⱶ ∆q1aabb Ⱶ ∆Aq2abb Ⱶ ∆Aaq2bb Ⱶ ∆Aabq2b Ⱶ ∆Aabbq2∆ Ⱶ ∆Aabq3b Ⱶ ∆Aaq4bB Ⱶ ∆Aq4abB Ⱶ ∆q4AabB Ⱶ ∆Aq1abB Ⱶ ∆AAq2bB Ⱶ ∆AAbq2B Ⱶ ∆AAq3bB Ⱶ ∆Aq4ABB Ⱶ ∆AAq1BB Ⱶ ∆Aq5ABB Ⱶ ∆q5AaBB Ⱶ q5∆aaBB Ⱶ ∆q6aaBB Ⱶ ∆Aq8aBB

a/a, R input symbol is a / replace it with a and move the state to the rightq0∆aabb Ⱶ ∆q1aabb Ⱶ ∆Aq2abb Ⱶ ∆Aaq2bb Ⱶ ∆Aabq2b Ⱶ ∆Aabbq2∆ Ⱶ ∆Aabq3b Ⱶ ∆Aaq4bB Ⱶ ∆Aq4abB Ⱶ ∆q4AabB Ⱶ ∆Aq1abB Ⱶ ∆AAq2bB Ⱶ ∆AAbq2B Ⱶ ∆AAq3bB Ⱶ ∆Aq4ABB Ⱶ ∆AAq1BB Ⱶ ∆Aq5ABB Ⱶ ∆q5AaBB Ⱶ q5∆aaBB Ⱶ ∆q6aaBB Ⱶ ∆Aq8aBB Ⱶ ∆Aaq8BB

There is nowhere to go from q8 with input B, therefore the string aabb is rejected by this TMq0∆aabb Ⱶ ∆q1aabb Ⱶ ∆Aq2abb Ⱶ ∆Aaq2bb Ⱶ ∆Aabq2b Ⱶ ∆Aabbq2∆ Ⱶ ∆Aabq3b Ⱶ ∆Aaq4bB Ⱶ ∆Aq4abB Ⱶ ∆q4AabB Ⱶ ∆Aq1abB Ⱶ ∆AAq2bB Ⱶ ∆AAbq2B Ⱶ ∆AAq3bB Ⱶ ∆Aq4ABB Ⱶ ∆AAq1BB Ⱶ ∆Aq5ABB Ⱶ ∆q5AaBB Ⱶ q5∆aaBB Ⱶ ∆q6aaBB Ⱶ ∆Aq8aBB Ⱶ ∆Aaq8BB Ⱶ ∆AahrBB (reject)

Accept only if state ha is reached!

let’s trace this TM on the string bb to show acceptance. Begin from the initial configuration:q0∆bb

∆/∆, R input symbol is ∆ / replace it with ∆ and move the state to the rightq0∆bb Ⱶ ∆q1bb

b/B, R input symbol is b / replace it with B and move the state to the rightq0∆bb Ⱶ ∆q1bb Ⱶ ∆Bq2b

b/b, R input symbol is b / replace it with b and move the state to the rightq0∆bb Ⱶ ∆q1bb Ⱶ ∆Bq2b Ⱶ ∆Bbq2∆

∆ / ∆, L input symbol is ∆ / replace it with ∆ and move the state to the leftq0∆bb Ⱶ ∆q1bb Ⱶ ∆Bq2b Ⱶ ∆Bbq2∆ Ⱶ ∆Bq3b

b / B, L input symbol is b / replace it with B and move the state to the leftq0∆bb Ⱶ ∆q1bb Ⱶ ∆Bq2b Ⱶ ∆Bbq2∆ Ⱶ ∆Bq3b Ⱶ ∆q4BB

B / B, R input symbol is B / replace it with B and move the state to the rightq0∆bb Ⱶ ∆q1bb Ⱶ ∆Bq2b Ⱶ ∆Bbq2∆ Ⱶ ∆Bq3b Ⱶ ∆q4BB Ⱶ ∆Bq1B

B / B, L input symbol is B / replace it with B and move the state to the leftq0∆bb Ⱶ ∆q1bb Ⱶ ∆Bq2b Ⱶ ∆Bbq2∆ Ⱶ ∆Bq3b Ⱶ ∆q4BB Ⱶ ∆Bq1B Ⱶ ∆q5BB

B / b, L input symbol is B / replace it with b and move the state to the leftq0∆bb Ⱶ ∆q1bb Ⱶ ∆Bq2b Ⱶ ∆Bbq2∆ Ⱶ ∆Bq3b Ⱶ ∆q4BB Ⱶ ∆Bq1B Ⱶ ∆q5BB Ⱶ q5∆bB

∆ / ∆, R input symbol is ∆ / replace it with ∆ and move the state to the rightq0∆bb Ⱶ ∆q1bb Ⱶ ∆Bq2b Ⱶ ∆Bbq2∆ Ⱶ ∆Bq3b Ⱶ ∆q4BB Ⱶ ∆Bq1B Ⱶ ∆q5BB Ⱶ q5∆bB Ⱶ ∆q6bB

b / B, R input symbol is b / replace it with B and move the state to the rightq0∆bb Ⱶ ∆q1bb Ⱶ ∆Bq2b Ⱶ ∆Bbq2∆ Ⱶ ∆Bq3b Ⱶ ∆q4BB Ⱶ ∆Bq1B Ⱶ ∆q5BB Ⱶ q5∆bB Ⱶ ∆q6bB Ⱶ ∆Bq7B

B / ∆, L input symbol is B / replace it with ∆ and move the state to the leftq0∆bb Ⱶ ∆q1bb Ⱶ ∆Bq2b Ⱶ ∆Bbq2∆ Ⱶ ∆Bq3b Ⱶ ∆q4BB Ⱶ ∆Bq1B Ⱶ ∆q5BB Ⱶ q5∆bB Ⱶ ∆q6bB Ⱶ ∆Bq7B Ⱶ ∆q9B

B / B, R input symbol is B / replace it with B and move the state to the rightq0∆bb Ⱶ ∆q1bb Ⱶ ∆Bq2b Ⱶ ∆Bbq2∆ Ⱶ ∆Bq3b Ⱶ ∆q4BB Ⱶ ∆Bq1B Ⱶ ∆q5BB Ⱶ q5∆bB Ⱶ ∆q6bB Ⱶ ∆Bq7B Ⱶ ∆q9B Ⱶ ∆Bq6∆

∆ / ∆, S input symbol is ∆ / replace it with ∆ and halt. String bb is accepted by this TMq0∆bb Ⱶ ∆q1bb Ⱶ ∆Bq2b Ⱶ ∆Bbq2∆ Ⱶ ∆Bq3b Ⱶ ∆q4BB Ⱶ ∆Bq1B Ⱶ ∆q5BB Ⱶ q5∆bB Ⱶ ∆q6bB Ⱶ ∆Bq7B Ⱶ ∆q9B Ⱶ ∆Bq6∆ Ⱶ ∆Bha∆ (accept)