Draw out a determinism Turing machine that computes a mapping reduction from L1 ≤ m L2 where L1 ＝ {w:w ∈ {a,b}* and |W| is even} and where L2 ＝ {anbn:n ≥ 0}.

Consider the following language L over Σ = {0, 1, 2}:  L = { 0i2 1 2 02i1i | i ≥ 0 } 1. Construct a Turing machine accepting L. Justify your answer, explaining how the TM youconstructed will accept words in the language L. 2. Give an accepting run of your TM for the word 0212001 Hint: think about the TM accepting the language { 0i2 1 2 02i1i | i ≥ 0 } and how can it be changed to accept the language above. Notation: recall that e.g. we write 0i for the word consisting of i-many 0’s.

Please written by computer source Let x be a binary string. The minimal description of x, written d(x), is the shortest string ⟨M, w⟩ where TM M on input w halts with x on its tape. If several such strings exist, select the lexicographically first among them. The descriptive complexity, or Kolmogorov complexity, of x, written K(x), is K(x) = |d(x)|. Show that the function K(x) is not a computable function. HINTS: If K is a computable function, there is some TM which computes it. That TM can be used to find strings of large complexity. Try to design a program which outputs “complex” strings but which contradicts their supposed complexity, and even contradicting the supposed complexity of a single string suffices.

Give the transition diagram of a standard Turing Machine that implements the function. Briefly explain your solution strategies.   DO NOT COPY YOUR ANSWER FROM CHEGG

Exercise 12.9 Given a weighted (connected) undirected graph G = (V, E, w), with m =n+ 0(1) edges (i.e., the number of edges is at most the number of vertices plus some constant), give an O(n)-time algorithm to compute the minimum spanning tree of G. Prove the correctness and the running time of your algorithm. (Hint: Think of how one can eliminate an edge from being in the MST.)

9. Limitation Question. Using the idea of Pumping Lemma, please explain, why we can't build a Finite Automaton to generate all and only strings of the form 1³n0n where n>=0.

Let G be an undirected edge-weighted graph with V vertices and O(V2) edges. Describe an efficient algorithm to compute a minimum spanning tree that runs in O(V2) time. You are only allowed to use data structures such as arrays and priority queues. Do not use complex data structures such as Fibonacci heaps. Give a formal proof of correctness for your algorithm. An answer without any formal proof will be considered incomplete. Analyze the running time of the algorithm.