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Draw a Turing machine that recognizes the language $\{w \in \{0,1\}^*|w \text{ contains even number of 1's}\}$

This is where I am at:

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  • $\begingroup$ Can you draw a finite state machine recognizing that language? $\endgroup$ – BrianO Nov 21 '15 at 0:52
  • $\begingroup$ I uploaded a picture to the description $\endgroup$ – Brice Petty Nov 21 '15 at 1:29
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You don't need a full blown Turing machine for this. A finite machine (equivalent to a Turing machine that just reads its tape once in one direction) will do.

The required language is given by the regular expression

$$0^* \left( 10^*10^* \right)^*$$

The minimal finite machine that recognises this has two states:

  • an initial accepting state, say $A$;
  • another non-accepting state, say $B$.

The states behave symmetrically:

  • An input of $0$ leaves the machine in the same state, $A$ or $B$.
  • An input of $1$ takes the machine to the other state: $A$ to $B$ or $B$ to $A$.
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