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Chapter 0: Q22P (page 1)

Show that A is Turing-recognizable Am≤ATM

Short Answer

Expert verified

It is proved that A is Turing Recognizable.

Step by step solution

01

Introduction of Turing Recognizable

A language L is said to be Turing Recognizable if and only if there exist any Turing Machine (TM) which recognize it i.e., TM halt and accept strings belong to language L and will reject or not halt on the input strings that doesn’t belong to language L .

02

To show A is Turing recognizable

Before we proceed, we know that ATM is Turing Recognizable.

Let us assume that A is Turing Recognizable, this means we can construct a Turing Machine M for A , such that LM=A.

Assume f is Mapping Reduction Function which is defined as:
fw=M,w

So,w∈Aif and only ifM,w∈ATM.

So, if, Am≤ATMthen A must be Turing Recognizable as ATM is Turing Recognizable

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Most popular questions from this chapter

Use the procedure described in Lemma 1.55 to convert the following regular expressions to nondeterministic finite automata.


a.(0∪1)*000(0∪1)*b.((00*11)∪01)*c.∅*

Let ∑=0,1. Let B be the collection of strings that contain at least one 1 in their second half. In other words,

a. Give a PDA that recognizes B

b. Give a CFG that generates B .

Show that the single-tape TMs that cannot write on the portion of the tape containing the input string recognize only regular languages.

A Turing machine with left reset is similar to an ordinary Turing machine, but the transition function has the form

δ : Q × Γ−→Q × Γ × {R, RESET}.

If δ(q, a) = (r, b, RESET), when the machine is in state q reading an a, the machine’s head jumps to the left-hand end of the tape after it writes b on the tape and enters state r. Note that these machines do not have the usual ability to move the head one symbol left. Show that Turing machines with left reset recognize the class of Turing-recognizable languages.

Let AMBIGCFG=<G>|GisanambiguousCFG. Show that AMBIGCFG is undecidable. (Hint: Use a reduction from PCP. Given an instance

P=t1b1,t2b2,…,tkbk

of the Post Correspondence Problem, construct a CFG Gwith the rules

S→T|B

T→t1Ta1|.....|tkTak|t1a1|....|tkak

B→b1B|...|b1Bak|...|bkak

where a1,...,ak are new terminal symbols. Prove that this reduction works.)
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