CMPSCI 601: Theory of Computation

David Mix Barrington

Spring, 2003

Homework Assignment #1

Due Wednesday 10 March 2004 in class

(Mon 22 March 2:00 p.m. EST for off-campus students)

• Question 1 (20): Define a Garey-Johnson NTM to be a multitape TM that writes a guess string nondeterministicly on one of its tapes, then runs deterministicly, like the example on slide 16 of Lecture 8. Define an ordinary NTM to be a multitape TM whose state table, like that of an NTM, allows it zero, one, or more than one action for each possibility of state and letter seen. Prove that for any language A, A is the language of some GJNTM iff it is the language of some ONTM.

• Question 2 (20): Prove that the definition of NP is the same whether defined in terms of GJNTM's or of ONTM's.

• Question 3 (10): Prove that the reduction relation ≤, as defined in Lecture 10, is reflexive and transitive.

• Question 4 (10): Prove that any two r.e.-complete languages S and T are equivalent in the sense of Lecture 10, i.e., S≤T and T≤S.

• Question 5 (10): Prove that any finite or cofinite language is recursive. (A is cofinite iff A-bar is finite.)

• Question 6 (30): Let P be any Bloop program that takes a single input number n and returns an output number P(n). Prove that there exists a Bloop-computable function f(n) such that the function P(n) is in F(DTIME[f(n)]). Hint: Use induction on the definition of Bloop programs. They are built up from smaller programs by concatenation, inclusion in a for loop, and function calls.

Last modified 9 March 2003