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Problem 14

# 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$.

Expert verified
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 $$B$$.
See the step by step solution

## 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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