Simulating multi-tape turing machines by single-tape TM

Question

In "Computational Complexity: A modern approach", Arora and Barak proof the following Claim:

Define a single-tape Turing machine to be a TM that has only one read-write tape, that is used a input, work, and output tape. For every $f: \{0,1\}^* \rightarrow \{0,1\}$ and time-constructible $T: \mathbb{N} \rightarrow \mathbb{N}$, if $f$ is computable in time $T(n)$ by a TM $M$ using $k$ tapes, then it is computable in time $5kT(n)^2$ by a single-tape TM $\hat{M}$.

The proof roughly goes like this, from my understanding:

-> We encode the $k$ tapes of $M$ on a single tape, using locations $i,k+i,2k+i$ for the $k$-th tape.

-> For each character $a$ in M's alphabet, we define another character $\hat{a}$ in $\hat{M}$'s alphabet. For each of M's tapes, the $\hat{a}$ character indicates the current position of that respective tape (for $M$) in $M's$ encoding.

-> For each state transition of $M$, $\hat{M}$ then perform two sweeps: One left-to-right, where $\hat{M}$ finds the positions of $M$ at the respective working tapes and one right-to-left where $\hat{M}$ updates its tape encoding according to state transition function of $M$.

Its not clear to me how the first step exactly works. Arora and Barak write: "First it sweeps the tape in the left-to-right direction and records to its register the $k$ symbols that are marked by $\hat{a}$". As far as I understand, registers correspond to states in the TM $M'$. What is exactly is meant by recording the symbols to its register?

Answer 1

You can simulate a state machine with set of states $S$ and registers $R_1,\ldots,R_m$ varying over a finite alphabet $\Sigma$ using a vanilla state machine whose set of states is $S \times \Sigma^m$. You "store" a value $x$ in register $R_i$ by transitioning to a state whose $(i+1)$'th element is $x$. You "read" a value from $R_i$ by taking into account in your transition table the $(i+1)$'th element of the current state.

This is the sense in which $\hat{M}$ has registers – their contents are stored in the state of the machine. Here it is crucial that both $k$ and the tape alphabet are fixed and finite.