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Introduction to the Theory of Computation

Sipser Michael

Chapter 6: Chapter 6

Problem 14

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Show that for any two languages $A$ and $B$, a language $J$ exists, where $A \leq \mathrm{T} J$ and $B \leq \mathrm{T} J$.

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We have shown the existence of a language $J$ for any two languages $A$ and $B$ such that $A \leq_T J$ and $B \leq_T J$. We defined $J = \{0w : w \in A\} \cup \{1x : x \in B\}$ and showed that both $A$ and $B$ are Turing reducible to $J$ by constructing Turing machines that use an oracle for $J$ to decide if strings are in $A$ or $B$.

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TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Define Turing reducibility

Turing reducibility, also known as Turing-Weihrauch reducibility, is a relation between decision problems. Intuitively, a problem $A$ is Turing reducible to a problem $B$ (denoted $A \leq_T B$), if there exists a Turing machine that can solve problem $A$ using an oracle for problem $B$.

Step 2: Define languages A and B

Let $A$ and $B$ be two arbitrary languages. We want to define a new language $J$ such that both $A$ and $B$ can be Turing reduced to $J$.

Step 3: Define language J

Let's define $J$ as the following: \[ J = \{0w : w \in A\} \cup \{1x : x \in B\} \] Language $J$ is a combination of the languages $A$ and $B$ with an added "tag" at the beginning of the strings. For strings in $A$, a $0$ is added to the beginning, and for strings in $B$, a $1$ is added to the beginning. This allows us to differentiate between strings from $A$ and strings from $B$.

Step 4: Show A is Turing reducible to J

If we have an oracle for $J$, we can decide if any given string $w$ is in $A$ by adding a $0$ to the beginning of $w$ and querying the oracle for $0w$. If the oracle returns that $0w \in J$, then we know $w \in A$, otherwise $w \notin A$. So, we can create a Turing machine that solves $A$ using an oracle for $J$, which by definition means $A \leq_T J$.

Step 5: Show B is Turing reducible to J

Similarly, we can also decide if a given string $x$ is in $B$ by adding a $1$ to the beginning of $x$ and querying the oracle for $1x$. If the oracle returns that $1x \in J$, then we know $x \in B$, otherwise $x \notin B$. This implies that we can create a Turing machine that solves $B$ using an oracle for $J$, which by definition means $B \leq_T J$.

Step 6: Conclusion

In this exercise, we have proven the existence of a language $J$ such that for any two languages $A$ and $B$, both $A$ and $B$ are Turing reducible to $J$.

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Chapter 6: Chapter 6

Show that for any two languages \(A\) and \(B\), a language \(J\) exists, where $A \leq \mathrm{T} J\( and \)B \leq \mathrm{T} J$.

Short Answer

Step by step solution

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Step 1: Define Turing reducibility

Step 2: Define languages A and B

Step 3: Define language J

Step 4: Show A is Turing reducible to J

Step 5: Show B is Turing reducible to J

Step 6: Conclusion

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More chapters from the book ‘Introduction to the Theory of Computation’

Chapter 0

Chapter 1

Chapter 2

Chapter 3

Chapter 4

Chapter 5

Chapter 7

Chapter 8

Chapter 9

Chapter 10

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