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| --- | ||
| title: "CS135-lecture-20210423" | ||
| # date: 2021-04-23T13:59:40-07:00 | ||
| draft: false | ||
| bookToc: true | ||
| tags: ["turing machine"] | ||
| --- | ||
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| ## Introduction to Turing machines | ||
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| {{< hint info >}} | ||
| Note: See text book readings for formal history of the Turing machine. | ||
| {{< /hint >}} | ||
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| A Turing machine has a tape that is infinitely long in both directions. | ||
| The machine has a tape head that can read and write to the tape. | ||
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| The control unit of the Turing machine is an automata. | ||
| Each transition has a pair {{<k>}} x,y {{</k>}}, follow the arrow if | ||
| the tape head is over {{<k>}} x {{</k>}}, and do operation {{<k>}} y {{</k>}}. | ||
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| {{<k>}} y {{</k>}} can be: | ||
| - {{<k>}} R {{</k>}}, move right | ||
| - {{<k>}} L {{</k>}}, move left | ||
| - {{<k>}} y {{</k>}}, write {{<k>}} y {{</k>}}, can be a blank (indicated by {{<k>}} B {{</k>}}). | ||
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| Turing machines are deterministic, and have no {{<k>}} \lambda {{</k>}} transitions. | ||
| The machine halts when no transition matches, so Turing machines don't require a transition for every alphabet character. | ||
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| There are two types: | ||
| - Recognizer, which accepts/rejects input. | ||
| - Transducer, which takes input and produces output. | ||
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| By convention, the tape head always starts at the far left of the input, and ends at the far left of the output. | ||
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| ### First example | ||
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| With a tape starting with {{<k>}} n {{</k>}} 1s, we want the output to be {{<k>}} n {{</k>}} 0s. | ||
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| Some simple pseudo code might be | ||
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| ``` | ||
| while head is over 1 | ||
| write 0 | ||
| move R | ||
| move L | ||
| while head is over 0 | ||
| move L | ||
| move R | ||
| ``` | ||
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| Make sure to check edge cases, i.e. only a single 1 being on the tape. | ||
| Also check if the tape is empty initially. | ||
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| This starts out as a self loop on the initial state that has the pair {{<k>}} 1,0 {{</k>}}. | ||
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| The `move R` can be added as another pair {{<k>}} 0,R {{</k>}}. | ||
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| {{< hint info >}} | ||
| Note: This would be a problem if we had any 0s in the input to begin with, we are assuming the input is good. | ||
| To handle this we could have a state like | ||
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| Which would halt on a 0 in the initial input. | ||
| {{< /hint >}} | ||
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| When this loop finishes, it means we are no longer over a 1. | ||
| The head should be over a blank. | ||
| So to transition to the next state we have the pair {{<k>}} B,L {{</k>}}. | ||
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| The next loop can be handled by {{<k>}} 0,L {{</k>}}. | ||
| And when we reach the far right blank we can move right. | ||
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| The last state has no arrows, so the program will halt. | ||
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| ### Second example | ||
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| The pair {{<k>}} B,B {{</k>}} is kind of like "no operation". | ||
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| ### Testing | ||
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