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Chapter 6: Q16P (page 271)

Prove that there exist two languages that are Turing-incomparable鈥攖hat is,A 'TB where B'T A.

Short Answer

Expert verified

The given statement is proved.

Step by step solution

01

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鈥檛 belong to language L.

02

Proof of the statement.

Two languages A are B said to be Turing incompatible if language A is not Turing Compatible with language B and vice versa.

In this question, we don鈥檛 have any Turing Machine that can work as Oracle Turing Machine to detect decidability and Turing reducibility.

So there is no language from A and B which could be replaced by any intermediate language.

Also, Turing Machine cannot be constructed until decidability is proved.

Thus, no solution can be defined in our case to prove decidability of two languages so that language A and B are not Turing Reducible.

This proves that A and B is Turing Incomparable, A ' TB and B'TA.

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Most popular questions from this chapter

Describe two different Turing machines, M and N, where M outputs <N> and N outputs when started on any input.

Let be two disjoint languages. Say that language separates if A鈯�CandB鈯�C'. Describe two disjoint Turing-recognizable languages that aren鈥檛 separable by any decidable language.

Give a model of the sentence

蠁It=鈭�鈭赌X,Y[R1(x,x)]鈭�鈭赌x.y[R1(x,y)鈫�(R1(y,x)]鈭�鈭赌x,y,z[(R1(x,y)鈭�R1(y,z))鈫�R1(x,z)]

Let R鈯�螡 kbe a -kary relation. Say that Ris definable in Th(螡,+)if we can give a formula蠁with kfree variablesx1,...,xksuch that for all,a1,...,ak鈭�螡,蠁(a1,...,ak)is true exactly when .a1,...,ak鈭�RShow that each of the following relations is definable in.Th(螡,+)

a.R0={0}

b.R1={1}

c.R=={(a,a)|a鈭�螡}

d.R<={(a,b)|a,b鈭�螡anda<b}

Let (N,<)be the model with universe N and the 鈥渓ess than鈥� relation. Show that Th(N,<) is decidable.
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