Introduction to Algorithms
3rd Edition
ISBN: 9780262033848
Author: Thomas H. Cormen, Ronald L. Rivest, Charles E. Leiserson, Clifford Stein
Publisher: MIT Press
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Chapter 34.3, Problem 8E
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To describe the fault in professor’s reasoning.
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We say that a Turing machine is verbose on sif, at the completion of its computation on s, it leaves at least as many non-blank characters on the left of the tape as shastotal characters.
Let VERBOSETM= { <M, w> | M is a TM and is verbose on s}.
Show that VERBOSETM is undecidable by reduction from ATM. Do not use Rice’s theorem.
Consider L={(TM) | TM stands for the Turing machine, which halts on all input, and L(TM)= L' for some undecidable language L'}. The encoding of a Turing machine as a string over the alphabet 0-1 is (TM), and L is?
decidable and recursively enumerable
decidable and recursive
decidable and non-recursive
undecidable and recursively enumerable
We say that a Turing machine is verbose on sif, at the completion of its computation on s, it leaves at least as many non-blank characters on the left of the tape as shastotal characters.
Let VERBOSETM= { <M, s> | M is a TM and is verbose on s}.
Show that VERBOSETMis undecidable by reduction from ATM. Do not use Rice’s theorem.
Chapter 34 Solutions
Introduction to Algorithms
Ch. 34.1 - Prob. 1ECh. 34.1 - Prob. 2ECh. 34.1 - Prob. 3ECh. 34.1 - Prob. 4ECh. 34.1 - Prob. 5ECh. 34.1 - Prob. 6ECh. 34.2 - Prob. 1ECh. 34.2 - Prob. 2ECh. 34.2 - Prob. 3ECh. 34.2 - Prob. 4E
Ch. 34.2 - Prob. 5ECh. 34.2 - Prob. 6ECh. 34.2 - Prob. 7ECh. 34.2 - Prob. 8ECh. 34.2 - Prob. 9ECh. 34.2 - Prob. 10ECh. 34.2 - Prob. 11ECh. 34.3 - Prob. 1ECh. 34.3 - Prob. 2ECh. 34.3 - Prob. 3ECh. 34.3 - Prob. 4ECh. 34.3 - Prob. 5ECh. 34.3 - Prob. 6ECh. 34.3 - Prob. 7ECh. 34.3 - Prob. 8ECh. 34.4 - Prob. 1ECh. 34.4 - Prob. 2ECh. 34.4 - Prob. 3ECh. 34.4 - Prob. 4ECh. 34.4 - Prob. 5ECh. 34.4 - Prob. 6ECh. 34.4 - Prob. 7ECh. 34.5 - Prob. 1ECh. 34.5 - Prob. 2ECh. 34.5 - Prob. 3ECh. 34.5 - Prob. 4ECh. 34.5 - Prob. 5ECh. 34.5 - Prob. 6ECh. 34.5 - Prob. 7ECh. 34.5 - Prob. 8ECh. 34 - Prob. 1PCh. 34 - Prob. 2PCh. 34 - Prob. 3PCh. 34 - Prob. 4P
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- With Automata Theory , describe an algorithm that given two regular expressions R1 and R2 outputs YES if andonly if every string generated by R1 is also generated by R2. (I.e., output YES exactlywhen the set of strings generated by R1 is a subset of the set of strings generated by R2.)arrow_forwardLet L be a language: L = {: M be TM which runs in polyno time}, and prove L is undecidable for the following scenario: A TM runs in polyno time if there exists a constant n > 0 such that, for all inputs y ∈ {0, 1} *, M(y) will halt at most |y| steps.arrow_forwardOnly Typing answer use pumping lemma to show that the language L = {a^m b^m+1 c^m} is not context freearrow_forward
- A common problem that arises in software maintenance is identifying (and then removing) dead code, code that will never be executed no matter what input the program is given. The analogous problem for TMs would be to determine if a state is never entered, no matter what input the TM is given. Prove by reduction that Ldead, the set of pairs (T,s) where T is a Turing machine and s is a dead state, is not recursive.arrow_forwardThink about the concept of extended transition functions δ* and ∆* of Deterministic Finite Automata and Nondeterministic Finite Automata respectively. Prove the following facts about those functions using mathematical induction. 1. Given a DFA M = (Q, Σ, δ, s0, F) and strings x, y ∈ Σ*, we have δ*(q, xy) = δ*(δ*(q, x), y). 2. Given a NFA N = (Q, Σ, ∆, S0, F), subsets A ⊆ Q, B ⊆ Q and a string x ∈ Σ∗, we have ∆* (A ∪ B, x) = ∆*(A, x) ∪ ∆*(B, x).arrow_forwardLet Σ = {a, b}. Give a DFA/RE, CFG/PDA, a Turing machine, if it exists, for the language L = {w = wR|w ∈ Σ ∗ , l(w) is odd}, where wR denotes the reverse of w and l(w) denotes the length of w. If it does not exist, prove why it does not exist.arrow_forward
- Please provide step by step explanation and construct the TM to show it is undecidable. Please use prove by reduction. Prove that the language {<M>| M is a Turing machine and M accepts finitely many strings} is undecidable by using only the fact that HALT TM is undecidable.arrow_forwardFor each language below, indicate whether the language is: 1) in P, 2) in NP but probably not in P, or 3) probably not in NP. Give a brief informal justification for your answer. Note that all languages are encoded over the alphabet {0, 1}. L = { (M, w) | M is a Turing machine, w is a string and M should accept the input w}. A: L = {[w,x] | w is list of cities with distance between them, x is a visiting order and x must be the shortest path that travels between all the cities visiting each one only once}. A: L = {[M, w] | M is a context free language in Chomsky normal form, w is a string and w is accepted by the language M}. A:arrow_forwardINPUT: A graph G, a non-negative integer k ≥ 0, and a boolean formula F in CNF. OUTPUT: “Yes” if and only if either G has an independent set of size k or if the boolean formula F is satisfiable. Prove that this problem is NP-Complete.arrow_forward
- Draw out a 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}.arrow_forwardThere are two key steps in using Pumping lemma to prove a language to be nonregular: pick w and i such that we can make proof reasoning easier and correct to lead to a contradiction. When we try to prove the following language L to be nonregular using Pumping lemma,L = {w∈Σ*, na(w) < nb(w)}, we pick w = ambm+1, then we should pick i = ______. Group of answer choices A)2 B)0 C)1arrow_forwardNeed help in theoretical computer science Given the following language over the alphabet Σ = {a, b, c, d}. • L = {a ^ r b ^ r c ^ r d ^ m | r, m∈IN} Answer the following questions. (i) Are infinitely many words in this language 1-inflatable? If so, include such words. (ii) Are infinitely many words in this language not 1-inflatable? If so, include such words. (iii) Can one use the pumping lemma for context-free languages to show that the language L is not a context-free language? If so, briefly outline how one can then argue. Subtask (b) Now let the following language be given over the alphabet Σ = {a, b, c, d} • L = {a^r b^ r c^r d ^ m | r, m∈IN} ∪ {a^i b ^ j c^k | i, j, k∈IN} (i) Are infinitely many words in this language 1-inflatable? If so, include such words. (ii) Are infinitely many words in this language not 1-inflatable? If so, include such words. (iii) Can one use the pumping lemma for context-free languages to show that the language L is not a context-free…arrow_forward
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