Turing Machines

Definition: A Turing machine is a 7-tuple, $\left(Q,\Sigma ,\Gamma ,\delta ,q_0,q_{\text{accept }},q_{\text{reject }}\right)$, where $Q$, $\Sigma$, $\Gamma$ are all finite sets and

1. $Q$ is the set of states
2. $\Sigma$ is the input alphabet not containing the blank symbol $\_$
3. $\Gamma$ is the tape alphabet, where $\_\in \Gamma$ and $\Sigma \subseteq \Gamma$,
4. $\delta :Q\times \Gamma ⟶Q\times \Gamma \times \{\mathrm{L},\mathrm{R}\}$ is the transition function,
5. $q_0\in Q$ is the start state,
6. $q_{\text{accept }}\in Q$ is the accept state, and
7. $q_{\text{reject }}\in Q$ is the reject state, where $q_{\text{reject }}\ne q_{\text{accept }}$.

As a Turing machine computes, changes occur in the current state, the current tape contents, and the current head location. A setting of these three items is called a configuration of the Turing machine.
We write $uqv$ for the configuration where the current state is $q$, the current tape contents is $uv$, and the current head location is the first symbol of $v$