Multitape Turing machine

Definition

Let $k \ge 1$ be an integer. A $k$-tape Turing machine is a tuple $M = (Q, \Gamma, \sqcup, \Sigma, \delta, q_0, F)$ with the following parts.

1. $Q$ is a finite nonempty set of states, $q_0 \in Q$ is the initial state, and $F \subseteq Q$ is the set of halting states.
2. $\Gamma$ is a finite set of tape symbols, $\sqcup \in \Gamma$ is the blank, and $\Sigma \subseteq \Gamma \setminus \set{\sqcup}$ is the input alphabet.
3. $\delta$ is a partial function from $(Q \setminus F) \times \Gamma^k$ to $Q \times \Gamma^k \times \set{L, R, S}^k$.

A configuration consists of a state and, for each tape $i = 1, \dots, k$, a tape content $t_i \colon \Z \to \Gamma$ and a head position $p_i \in \Z$. Each $t_i$ has only finitely many non-blank cells. Suppose $\delta(q, t_1(p_1), \dots, t_k(p_k)) = (q', s_1, \dots, s_k, D_1, \dots, D_k)$. Then the next configuration has state $q'$. On each tape $i$ it has $s_i$ written at cell $p_i$, and head position $p_i - 1$, $p_i + 1$ or $p_i$ when $D_i$ is $L$, $R$ or $S$. On input $w \in \Sigma^*$, the run starts in state $q_0$ with all heads at cell $0$. Tape $1$ holds $w$ from cell $0$ onward, and all other cells are blank.