CMPSCI 501: Theory of Computation

David Mix Barrington

Spring, 2016

Homework Assignment #4

Posted Sunday 6 March 2016

Due on paper in class, Wednesday 23 March 2016

There are twelve questions for 100 total points plus 10 extra credit. All but one are from the textbook, Introduction to the Theory of Computation by Michael Sipser (third edition, with second edition numbers given where different). The number in parentheses following each problem is its individual point value.

Students are responsible for understanding and following the academic honesty policies indicated on this page.

• Problem 4.14 (second)/4.15 (third) (10XC) (Look at the solution in the text for Problem 4.13/4.14)

• Problem 4.26 (second)/4.28 (third) (10)

• Problem D-1 (20): A language A is said to be in the class Π₂ if there is a Turing decidable language R such that for any string w, w ∈ A ↔ ∀x:∃y: R(w, x, y).
  • (a,10) Show that the language EQ_TM is in the class Π₂. (Hint: Use the TD predicate ACH(<M>, x, c) meaning "c is an accepting computation history of M on x". Then you can translate "w ∈ L(M)" as "∃c:ACH(<M>, w, c)". Apply this translation to the statement "L(M₁) = L(M₂) ↔ ∀x: (x ∈ L(M₁) → x ∈ L(M₂)) ∧ (x ∈ L(M₂) → x ∈ L(M₁)).)
  • (b, 10) Prove that if A is any language in the class Π₂, then A ≤_m EQ_TM. (Hint: For any string w, build a TM depending on w that accepts all inputs x if and only if w is in A.)

• Exercise 5.1 (both editions) (5)

• Exercise 5.2 (both) (5)

• Problem 5.12 (both) (10).

• Problem 5.16 (both) (10).

• Problem 5.22 (both) (5)

• Problem 5.29 (both) (10)

• Problem 5.32 (both) (10)

• Exercise 6.2 (both) (5)

• Problem 6.6 (both) (10)

Last modified 7 March 2016