# Reduction from HALT to MPCP via TM Configuration Transitions

I have book with a very simple reduction from HALT to MPCP and I suspect that it is wrong. Here is the reduction:

Add the following dominos $$(top, bottom$$):

• Start: $$(\langle \varepsilon \rangle ,\langle \varepsilon\rangle \langle\kappa_0\rangle)$$ for initial configuration $$\kappa_0$$.

• Step: $$(\langle\kappa_i\rangle, \langle\kappa_j\rangle)$$ for all configuration transitions $$\kappa_i \rightarrow \kappa_j$$

• Finish: $$(\langle\kappa_e\rangle \langle\varepsilon\rangle, \langle\varepsilon\rangle)$$ for all final configurations $$\kappa_e$$

This is much easier than what I see e.g. in Sipser. Is there an error in this proof? I think it is impossible to add a domino for all possible configuration transitions.

You may need the definition of configuration transitions:

Let $$T = (S, \Sigma, \Pi, \delta, s_0, \_, E)$$ be a TM. Every triple $$(\nu,s,\omega)$$ with $$\nu, \omega \in \Pi^{+}$$ and $$s \in S$$ is called configuration of $$T$$. The configuration transitions $$\rightarrow_{T}$$ is defined as

movement to the right: $$\delta(s,\sigma) = (s’, \sigma’, \rightarrow), \varrho, \sigma \in \Pi$$

\begin{align*} (\nu \varrho, s, \sigma \omega) \rightarrow_{T} \left\{\begin{array}{ll}% (\nu \varrho \sigma’ ,s’, \omega) & \mbox{if \omega \neq \epsilon } \ (\nu \varrho \sigma’ ,s’, \_) & \mbox{if \omega = \epsilon } \end{array}\right. \end{align*} movement to the left: $$\delta(s, \sigma) = (s’, \sigma’, \leftarrow), \varrho, \sigma \in \Pi$$ \begin{align*} (\nu \varrho, s, \sigma \omega) \rightarrow_{T} \left\{\begin{array}{ll}% (\nu, {s’}, \varrho {\sigma’} \omega) & \mbox{if \nu \neq \epsilon } \ (\_, {s’}, \varrho {\sigma’} \omega) & \mbox{if \nu = \epsilon } \end{array}\right. \end{align*}