Initial, accepting, and rejecting configurations

Fix a deterministic one-tape Turing machine $M=(Q,\mathrm{\Sigma },\mathrm{\Gamma },\bigsqcup ,q_0,q_{\mathrm{a}\mathrm{c}\mathrm{c}},q_{\mathrm{r}\mathrm{e}\mathrm{j}},\delta )$ and an input word $w\in {\mathrm{\Sigma }}^{\ast }$.

The initial configuration of $M$ on input $w$ is $I_M(w):=(q_0,0,t_w),$ where $t_w$ is the initial tape from Right-infinite tapes as finite-support functions.

A configuration $C=(q,h,t)$ of $M$ is:

• accepting if $q=q_{\mathrm{a}\mathrm{c}\mathrm{c}}$,
• rejecting if $q=q_{\mathrm{r}\mathrm{e}\mathrm{j}}$,
• halting if it is accepting or rejecting.

Thus the machine starts in state $q_0$, scans the leftmost input cell, and halts exactly when it enters one of the two designated halting states.

• If $w=\epsilon$, then the initial tape is blank everywhere and the head is still at cell $0$.

• Accepting and rejecting are properties of configurations, not of words. The same input may lead different machines to different outcomes.