# Prove the equivalence of the modified Turing Machines and the standard Turing Machines

We have a Turing Machine that cannot write the same symbol it has read in a transition, meaning it should always alter the symbol when passing it. How can we prove that such machines have equal processing power compared with the standard Turing Machines?

Now suppose $$S$$ is a stand Turing machine with $$\Gamma$$ as its alphabet on tape.
Let $$\Gamma'$$ be a copy of $$\Gamma$$, i.e., for all $$\gamma\in\Gamma$$, there is a corresponding $$\gamma'\in\Gamma'$$.
Let Turing machine $$M$$ be the same as $$S$$ except that
• $$M$$'s alphabet on tape is $$\Gamma\sqcup\Gamma'$$.
• Suppose $$M$$ read $$\alpha\in\Gamma$$ or $$\alpha'\in \Gamma'$$. Then $$M$$ will behave like $$S$$ as if $$S$$ had read $$\alpha$$, except that $$M$$ will write $$\beta'$$ instead if $$S$$ would write $$\beta\in\Gamma$$ when $$\alpha$$ was read.
We see that $$M$$ is a modified Turing machine that simulates $$S$$. Hence the power of modified Turing machines is at least that of standard Turing machines.