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This question is motivated from the question $\mathrm{SPACE}(o(\log\log n)) = \mathrm{SPACE}(O(1))$. I am wondering, how is it possible to decide a language without reading the entire input. I have seen the similar situation in space complexity ( L and NL ,where we define new model of computation).

My model of computation is Turing machine (for e.g. see lecture note on Turing machine)

My question : Do we need to read the input completely in Turing machine ?

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There is no such requirement. A Turing machine which decides a language can do whatever it wants as long as it always halts; the language it decides is the set of words which it accepts.

There are many languages which don't require reading the entire input:

  • The empty language.
  • The complete language.
  • The language of all words starting with 0.
  • The language of words of length 2.

Here is a more non-trivial example:

  • Given a language $L$, define the language $$ X(L) = \{ 0^\ell 1 w x : |w|=\ell \text{ and } w \in L \}. $$ The language $X(L)$ can be decided by a Turing machine which doesn't always read the entire input. Moreover, generally speaking $L$ and $X(L)$ have roughly the same complexity (though $X(L)$ requires the ability to count).
Yuval Filmus
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