When the tape head of a TM is scanning a blank symbol, what will happen?

Question

I am reading Introduction to automata theory, languages, and computation 3ed, by John E. Hopcroft, et al. The wikipedia article (https://en.wikipedia.org/wiki/Turing_machine#Formal_definition) for Turing machine also cites the TM definition in the 1ed of the book. (I assume the 3ed still use the same TM definition as 1ed)

When the tape head of a TM is scanning a blank symbol, what will happen?

• Is the transition function undefined on a blank tape symbol?

• Does the TM halt?

Thanks.

Answer 1

If you look at Wikipedia's definition more carefully, you will see that $b$ (the blank symbol) is an element of $\Gamma$. So the transition function, with domain $(Q\setminus F)\times\Gamma$, is defined.

Answer 2

First, there are lots of slightly different formalizations for Turing machines, so I would look closely at that particular book if you want the answer relevant to that particular book.

Some formalizations see a blank symbol still as a 'symbol', i.e. a 'blank', and are happy define transitions for a 'blank'. Other formalizations, however, may treat a blank square as containing no symbol at all, and thus presumably will say that no transition will apply.

For most formalizatons, if no transitions apply, then the machine will halt. However, for some formalizations, the machine is said to halt only when the machines reaches an explicitly defined halting state.