Top/RedoHwks

If you got less than a 23/30 on homework 2, 3, or 4 and want to redo it (and you haven't already) you can. Turn any hwks you want to redo in by wednesday. Your new grade, minus a 7 point penalty, will replace your current grade on that hwk. Don't redo the same problems, but do the following substitutes:

hwk 2

1) problem 5.1

2) Let L = { <D> : D is a DFA and there are w,w₁,w₂,...,w₉ ∈ L(D) such that w = w₁w₂w₃w₄...w₉ } . Prove or disprove that L is decidable.

3) Let ONE_TM = { <M> : L(M) = { 1 } }. Prove or disprove that ONE_TM is decidable.

hwk 3

1) problem 5.19

2) Describe a poly-time algorithm that decides whether a given graph G has a 2-coloring. (An assignment of a color RED or BLUE to each vertex such that for every edge (u,w), the endpoints have different colors.) Argue that your algorithm is correct.

3) Describe a polytime algorithm for 2-SAT. Argue that your algorithm is correct. (2-SAT is the following problem: "given a Boolean formula F in 2-cnf form, is the formula satisfiable." 2-cnf form means an "and" of clauses, where each clause is an "or" of two literals.")

hwk 4

1) Let U = { <M, x, 1^t> : (∃ y ∈ Σ^*) M accepts input <x, y> within t steps }. Show that U is NP-complete.

2) problem 7.26

3) Show that the RULER problem is NP-complete. (Recall the RULER problem described in class. You are given n non-negative numbers x₁,x₂,...,x_n and a width W. You must decide whether a carpenter's ruler with segments of lengths x₁,x₂,...,x_n can be folded so that the result has width < W.

To explain more carefully, a folding is a sequence b₁,b₂,...,b_n where each b_i ∈ {-1,1}. Given a folding, define the partial sums W₀,W₁,W₂,...,W_n as follows:

W₀ = 0. W_i+1 = W_i + b_i x_i.

Then the width of the folding is

max_i,i' |W_i - W_i'|.)