CMPSCI 601: Theory of Computation

Offered through the PEEAS distance learning program

Homework Assignment #3

David Mix Barrington

Assignment posted Wed 8 Oct 2003

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

• Question 1 (10): Let C be a boolean circuit of depth d. Prove that there exists a boolean expression of length O(2^d) that computes the same function as C. (Problem corrected 15 September.)

• Question 2 (20): Consider boolean straight-line programs as in the example on slide 16 of Lecture #10.

  • (a,10) A straight-line program is said to be read-once if each gate is used at most once on the right-hand side of an instruction. What kind of circuit computes the same function as a read-once straight-line program? Prove your answer, that is, prove that a function has a circuit of size s with your condition if and only if it has a read-once SLP of length s.

  • (b,10) Consider straight-line programs where we can also have instructions of the form "gate[i] = gate[j] xor gate[k]" where j and k are less than i. Prove that such a program can be simulated by a program with only "and", "or", and "not" instructions. If the program with "xor"'s has length s, what bound can you prove on the length of your simulation? (correction made 17 Oct)

• Question 3 (20): A groupoid is a set with a multiplication operation, which need not be commutative or associative. The groupoid multiplication problem GMP is to input a table for a groupoid G, a string w of elements of G, and a target element t. The output is a boolean saying whether it is possible, by giving w the right parenthesization, to have w multiply to t. Show (by following the model of slide 6 of Lecture #9) that GMP is in the class P.

• Question 4 (30): A queue machine (QM) is much like a PDA but has a queue instead of a stack. Instead of pushing and popping from its stack as it reads the input, it enqueues and dequeues characters from its queue. Explain how a QM can decide the four languages on slide 11 of Lecture #9. Use this fact to prove that the problem Σ^*-QM, of determining whether a given QM accepts all strings, is co-r.e. complete.

• Question 5 (20): Consider the first-order vocabulary for strings described on slide 11 of Lecture #11. For each of the two sentences on that slide, determine exactly for which strings the sentence is true. (Give a regular expression for the language of strings making each sentence true.) Write a first-order sentence over this vocabulary that is true for a string iff it is in the language 10^*1. Do the same for the language of strings that contain the consecutive substring 101, that is Σ^*101Σ^*.

Last modified 17 October 2003