the tape alphabet

The tape alphabet is the full set of symbols the machine is allowed to write on its tape. It is larger than the input alphabet because, beyond the symbols the input is made of, the machine wants a few extra symbols of its own to scribble notes with: tick marks, crossings-out, place-holders, and the blank. Think of writing on a page that arrived with some printed text (the input letters), but you also have a red pen and a highlighter (the extra symbols) to annotate as you work.

Formally the tape alphabet is written Γ (Gamma) and obeys two rules. First, the input alphabet Σ (Sigma) is a subset of Γ, so every symbol that can appear in the input can also sit on the tape. Second, Γ contains at least one symbol not in Σ, namely the blank symbol, used for empty cells. Usually Γ also holds a handful of extra working symbols such as X or a marked version of a letter. The transition function is allowed to write any symbol of Γ into a cell, which is exactly how the machine leaves itself notes (for instance, crossing out an a by overwriting it with X).

Having Γ strictly bigger than Σ is what makes the machine's classic techniques possible. To recognize a^n b^n c^n, a machine cannot count without writing on the tape, so it marks one a, one b, and one c per pass by overwriting them with special symbols from Γ. These extra symbols are pure bookkeeping; they are erased or ignored when reading off any final answer. Honest note: the tape alphabet is always FINITE, however many extra symbols you add; the machine's power comes from the unbounded tape, not from an unlimited alphabet.