What finite-state machine captures binary strings with equal numbers of '01' and '10'?

Question

I need help designing a finite state machine that accepts binary strings containing as many occurrences of the pattern 01 as occurrences of the pattern 10

Answer 1

What is the language in question?

[...] binary strings containing as many occurrences of the pattern 01 as occurrences of the pattern 10. I kinda have a hard time understanding exactly which strings should be accepted and which should be rejected.

The language defined by your specs is in fact none other than the set composed of

• the empty string,
• all strings that start and end with the same character.

The empty string is accepted because it contains zero occurrences of either pattern; easy. To understand why all non-empty accepted strings must start and end with the same character, instead of coming up with a formal proof, let's take a look at a couple of examples. I'll use

• -- to highlight occurrences of the 01 pattern, and
• ** to highlight occurrences of the 10 pattern.

String 10001010

This string contains

• 2 occurrences of 01, and
• 3 occurrences of 10,

as shown below:

10001010
**  ****
   ----

Therefore, it is not accepted. Note that it doesn't start and end with the same character.

String 11001111

This string contains

• 1 occurrence of 01, and
• 1 occurrence of 10,

as shown below:

11001111
 **--

Therefore, it is accepted. Note that it starts and ends with the same character (1).

You get the idea...

A finite-state machine that describes the language in question

I need help designing a finite state machine [...]

Here is an FSM that describes the language in question:

To convince yourself that it does indeed describe the language of interest here, you can think of

• s0 as the state that accepts only the empty string;
• s1 as the state that accepts only strings that start and end with a 0;
• s2 as the state in which the next character needs to be a 0 for the input string so far to get accepted;
• s3 as the state that accepts only strings that start and end with a 1;
• s4 as the state in which the next character needs to be a 1 for the input string so far to get accepted.

Bonus!

Here the LaTeX code I wrote for drawing the FSM above.

\documentclass{standalone}

\usepackage{tikz}
\usetikzlibrary{
  automata,
  positioning,
}

\begin{document}
  \begin{tikzpicture}[
    node distance=2cm,
    on grid,
    auto,
    scale=.8,
    transform shape,
  ]
    \node[state, initial, accepting] (s0)                     {$s_0$};
    \node[state,          accepting] (s1) [above right=of s0] {$s_1$};
    \node[state                    ] (s2) [right      =of s1] {$s_2$};
    \node[state,          accepting] (s3) [below right=of s0] {$s_3$};
    \node[state                    ] (s4) [right      =of s3] {$s_4$};

    \path[->] (s0) edge              node        {0}    (s1)
              (s1) edge [bend left]  node        {1}    (s2)
                   edge [loop above] node        {0}    ()
              (s2) edge [loop right] node        {1}    ()
                   edge [bend left]  node        {0}    (s1);
    \path[->] (s0) edge              node [swap] {1}    (s3)
              (s3) edge [bend right] node [swap] {0}    (s4)
                   edge [loop below] node        {1}    ()
              (s4) edge [loop right] node        {0}    ()
                   edge [bend right] node [swap] {1}    (s3);
  \end{tikzpicture}
\end{document}