CMPSCI 601: Theory of Computation

Offered through the PEEAS distance learning program

Homework Assignment #1

David Mix Barrington

Assignment posted Thu 11 Sept 2003

Due Wed 27 Sept 2003, faxed before 2:00 p.m. EDT or express-mailed to arrive Thu 28 Sept 2003

Make sure you have read the statement on academic honesty on the homework assignment index page.

Question 1 (20): Let A be a language over a one-letter alphabet, that is, A is a subset of a^*. Prove that A is DFA-recognizable (there exists a DFA D with A = L(D)) if and only if A is regular with an expression of the form: the expression is a union of finitely many terms, and each term is of the form uv^*, where u and v are words in a^*. (Statement corrected 12 September.)
Question 2 (20): Again let A be a language over a one-letter alphabet. Prove that A is a CFL iff it is regular. (Hint: This is an old well-known result so you might find a reference. It might help to use the CFL Pumping Lemma, which simplifies somewhat if there is only one letter in the alphabet.)
Question 3 (10): Let M be a pushdown automaton with the property that its stack never contains more than 17 letters (on any input, on any possible sequence of nondeterministic choices). Prove that L(M) is regular.
Question 4 (10): Let M be a Turing machine with the property that it never, on any input, looks at any input letter beyond the first 17 letters. Prove that L(M) is regular.
Question 5 (20): Let M be a deterministic Turing machine. We say that M hangs on input w if when started in the standard position on w, M goes off the left end of its tape before accepting or rejecting. Define the language H(M) to be the set of strings w such that M hangs on w. Prove that for any language A, A is r.e. iff there exists a machine M such that A = H(M).
Question 6 (20): Let A be an r.e. language over {0,1} (so that A is a subset of {0,1}^*) with the property that if u and v are any two strings of the same length that differ in exactly one position, then exactly one of u and v is in A.
- (a,15) Prove that there exists a Turing machine M such that for any nonempty string w, M halts on w and M accepts w if and only if w is in A.
- (b,5) Prove that A is recursive. (Note that the machine in part (a) does not necessarily decide A because it need not halt if its input is the empty string. But remember that either the empty string is in A or it isn't -- break into two cases and design a machine to decide A in each case.)
Last modified 12 September 2003