2. (a) Prove the transitive property for polynomial-time mapping reductions (b) Using the transitivity, show that...
Prove that polynomial-time reducibility is transitive: that is, if L1is polynomial-time-reducible to L2, and L2 is polynomial-time-reducible to L3, then L1 is polynomial-time-reducible to L3.
Prove that polynomial-time reducibility is transitive: that is, if L1is polynomial-time-reducible to L2, and L2 is polynomial-time-reducible to L3, then L1 is polynomial-time-reducible to L3.
(complexity) prove: if P=NP, then there's an algorithm with a polynomial running time for the following problem: input: a boolean formula φ output: a satisfying assignment of φ if φ satisfiable. if φ not satisfiable, a "no" will be returned. explanation: the algorithm accepts φ as an input (boolean formula). if φ doesn't have a satisfiable assignment, a "no" is returned. if φ does have a satisfiable assignment, one of the satisfying assignment is returned,. so we assign 0 or...
4. a) Define the concept of NP-Completeness B) Show that there is a polynomial time algorithm that finds a longest path in a directed graph, under the condition that A is NP-complete and A has a polynomial time algorithm.
2. Fix m, n E N. Define a mapping f:Z/nZ+Z/mZ by f([a]n) = [a]m. a. Prove that if m | n then f is a well-defined function. That is, prove that if (a)n = [b]n then f([a]n) = f([b]n). b. Let n = 12 and m = 3. Write PreImp({[1]3, [2]3}) in roster notation. c. Suppose mfn. Show that f is ill-defined. That is, show there exist a, b E Z such that (a)n = [b]n but f([a]n) + f([b]n).
2. Prove that {a"6"c" |m,n0}is not a regular language. Answer: 3. Let L = { M M is a Turing machine and L(M) is empty), where L(M) is the language accepted by M. Prove L is undecidable by finding a reduction from Aty to it, where Arm {<M.w>M is a Turing machine and M accepts Answer: 4. a) Define the concept of NP-completeness b) If A is NP-complete, and A has a polynomial time algorithm, then a polynomial time algorithm...
EXP is the class of languages decidable in exponential time (i.e. in 2" steps for some k) Much like the relationship between P and time can be decided in exponential time (i.e., NP EXP), but it is an open question if all problems decidable in exponential time are verifiable in polynomial time (i.e., EXP NP), though this is not expected to be true. Formally, the EXP class can be defined similarly to how we define P: NP, all languages that...
PLEASE ANSWER ALL! SHOWS STEPS 2. (a) Prove by using the definition of convergence only, without using limit theo- (b) Prove by using the definition of continuity, or by using the є_ó property, that 3. Let f be a twice differentiable function defined on the closed interval [0, 1]. Suppose rems, that if (S) is a sequence converging to s, then lim, 10 2 f (x) is a continuous function on R r,s,t e [0,1] are defined so that r...
2 seperate questions multiple choice Solve the system of linear equations using row reductions or show that it is inconsistent. x1 - x2 +3xz =-8 2xy + x3 = 0 X; +5x2 + x3 = 40 No solution (-8,0,0) (8, 8, 0) (0,8,0) Solve the system of linear equations using row reductions or show that it is inconsistent. 2x; – 5x2 + 3x3 = -1 - 2x + 6x2 - 5x₂=6 --4x; + 7x2 =-13 X1 12 17127 x2 =...
Here you are asked to prove the Fundamental Theorem of Algebra a different way by using Rouché's Theorem. Where n E N, consider the polynomial n-1 Pn (z)z" k-0 Using the circular contour C-[z : zR with R appropriately chosen, (a) prove that pn(2) has (counting multiplicity) precisely n zeros in the open disc D(0, R); (b) also show that Pn(z) has no zeros in C \ D(0, R) Here you are asked to prove the Fundamental Theorem of Algebra...