Let us consider V = {a, b, c, d, e} and F = (a ∨ b ∨ c ∨ d ∨ e) ∧ (¬a ∨ ¬b ∨ ¬c ∨ ¬d ∨ ¬e) ∧ (a ∨ ¬b ∨ d ∨ ¬e) ∧ (¬b ∨ c ∨ ¬e) ∧ (¬b ∨ c).
a) Is F a 3SAT propositional formula?
b) Find a 3SAT propositional formula F’ equivalent to F.
c) If you design a polynomial algorithm to find the truth assignments for F’, does that mean that you can find a polynomial algorithm to find the truth assignments for F, too?
Justify your answers.
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Q 5. Let F be a field and consider the polynomial ring l (a) State the Division Algorithm for polynomials in Plrl. b) Let a e F. Prove that -a divides f(x) in Fix] if and only if (a)- (c) Prove that z-37 divides 42-1 in F43[z].
Q 5. Let F be a field and consider the polynomial ring l (a) State the Division Algorithm for polynomials in Plrl. b) Let a e F. Prove that -a divides f(x) in...
Problem 4 Let V be the vector space of functions of the form f(x) = e-xp(x), where p(x) is a polynomial of degree (a) Find the matrix of the derivative operator D = d/dx : V → V in the basis ek = e-xXk/k!, k = 0, 1, . .. , n, of V. (b) Find the characteristic polynomial of D. (c) Find the minimal polynomial of D n.
Problem 4 Let V be the vector space of functions of...
Consider the following problem: Section II Con n a truth function f, find a statement S, only intolring the connecti e, ^,V and whose trva function is j. (a) Exhibit an algorithm that solves this problem. (b) Applied the exhibited algorithm to the truth function, 1 given by: TITIT (c) Suppose that the truth function f has n arguments represented by the variables i Consider the first algorithm studied in class to solve the problem of item (a). Let 01,92,.......
File Edit Format View Help Graphs and trees 4. [6 marks] Using the following graph representation (G(V,E,w)): v a,b,c,d,e,f E fa,b), (a,f),fa,d), (b,e), (b,d), (c,f),(c,d),(d,e),d,f)) W(a,b) 4,W(a,f) 9,W(a,d) 10 W(b,e) 12,W(b,d) 7,W(c,d) 3 a) Draw the graph including weights. b) Given the following algorithm for Inding a minimum spanning tree for a graph: Given a graph (G(V,E)) create a new graph (F) with nodes (V) and no edges Add all the edges (E) to a set S and order them...
Problem 4 Let V be the vector space of functions of the form f(x) = e-xp(x), where p(x) is a polynomial of degree (a) Find the matrix of the derivative operator D = d/dx : V → V in the basis ek = e-xXk/k!, k = 0, 1, . .. , n, of V. (b) Find the characteristic polynomial of D. (c) Find the minimal polynomial of D n.
4. The NOT-ALL-EQUAL 3SAT problem is defined as follows: Given a 3-CNF formula F, is there a truth assignment for the variables such that each clause has at least one true literal and at least one false literal? The NOT-ALL-EQUAL 3SAT problem is NP-complete. This question is about trying to reduce the NOT-ALL-EQUAL 3SAT problem to the MAX-CUT problem defined below to show the latter to be NP-complete. A cut in an undirected graph G=(V.E) is a partitioning of the...
Let A={a b c d e f} B={a c e g} C = {b d f}
Find each:
B = {a, c, e,g} C = {b,d,f} A= {a,b,c,d,e,f} Find: (2 points each) (a) AnB (b) AUB (c) Ang (d) COB (e) CUB (f) (An B)UC (g) An(BUC) (h) Ax B (i) C XB G) AB (k) C ( BA) (1) B2
Write down how the expression (A-B) V - (CA D)A (E-F) of wffs (an example is given below): Example: The expression (A A-B) V (CD) 1. A is a wff (propositional variable) 2. B is a wff (propositional variable 3. -B is a wff (obtained by applying to 2) 4. AA-B is a wff (obtained by applying A to 1 and 3) 5. (AA-B) is a wff (obtained by applying parentheses to 4) 6. C is a wff (propositional variable)...
need the answer to b not a. thanks!
2. (40 pts) Let A, B, and C be three strings each n characters long. We want to compute the longest subsequence that is common to all three strings. (a) Let us first consider the following greedy algorithm for this problem. Find the longest common subsequence between any pair of strings, namely, LCS(A, B). LCS(B,C), LCS(A, C). Then, find the longest common subsequence between this LCS and the 3rd string. That is,...
Problem 2 (22+ poits). Consider some unknown vi, ., V/n E Rd with d < n and you are given the corresponding distance matrix Dij = llui-villa (a) Prove that D is not a positive semi-definite matrix unless Dij 0 for all i,j. (b) Show that we cannot recover v\, ...,Vn exactly given D. (c) Design a polynomial time algorithm to recover points x1, ,x,e Rd such that Dij la-x112.
Problem 2 (22+ poits). Consider some unknown vi, ., V/n...