{
  "machine": "2DFT",
  "sigma": [
    "a",
    "b"
  ],
  "outputAlpha": [
    "a",
    "b"
  ],
  "states": [
    {
      "id": "s1",
      "x": 140,
      "y": 180,
      "name": "start"
    },
    {
      "id": "s2",
      "x": 430,
      "y": 180,
      "name": "copy 1"
    },
    {
      "id": "s3",
      "x": 700,
      "y": 180,
      "name": "rewind"
    },
    {
      "id": "s4",
      "x": 430,
      "y": 420,
      "name": "copy 2"
    },
    {
      "id": "s5",
      "x": 760,
      "y": 420,
      "name": "done"
    }
  ],
  "startId": "s1",
  "accepts": [
    "s5"
  ],
  "transitions": [
    {
      "id": "t1",
      "from": "s1",
      "to": "s2",
      "symbol": "⊢",
      "dir": "R",
      "output": ""
    },
    {
      "id": "t2",
      "from": "s2",
      "to": "s2",
      "symbol": "a",
      "dir": "R",
      "output": "a"
    },
    {
      "id": "t3",
      "from": "s2",
      "to": "s2",
      "symbol": "b",
      "dir": "R",
      "output": "b"
    },
    {
      "id": "t4",
      "from": "s2",
      "to": "s3",
      "symbol": "⊣",
      "dir": "L",
      "output": ""
    },
    {
      "id": "t5",
      "from": "s3",
      "to": "s3",
      "symbol": "a",
      "dir": "L",
      "output": ""
    },
    {
      "id": "t6",
      "from": "s3",
      "to": "s3",
      "symbol": "b",
      "dir": "L",
      "output": ""
    },
    {
      "id": "t7",
      "from": "s3",
      "to": "s4",
      "symbol": "⊢",
      "dir": "R",
      "output": ""
    },
    {
      "id": "t8",
      "from": "s4",
      "to": "s4",
      "symbol": "a",
      "dir": "R",
      "output": "a"
    },
    {
      "id": "t9",
      "from": "s4",
      "to": "s4",
      "symbol": "b",
      "dir": "R",
      "output": "b"
    },
    {
      "id": "t10",
      "from": "s4",
      "to": "s5",
      "symbol": "⊣",
      "dir": "S",
      "output": ""
    }
  ],
  "notes": [
    {
      "id": "n1",
      "x": 120,
      "y": 60,
      "color": "blue",
      "anchorStates": [],
      "anchorTransitions": [],
      "text": "w ↦ ww, by reading the input twice. Pass 1 sweeps right printing every symbol, bounces off ⊣ and rewinds left in silence, then pass 2 sweeps right printing everything again. The two-way head is the whole trick — the input is re-read, not remembered."
    },
    {
      "id": "n2",
      "x": 120,
      "y": 520,
      "color": "green",
      "anchorStates": [],
      "anchorTransitions": [],
      "text": "No one-way transducer can do this. A Mealy machine or FST prints a bounded amount per input symbol and never goes back, so its output grows at a fixed rate; doubling requires visiting each cell twice. Two-way transducers compute exactly the MSO-definable (regular) transductions — strictly more than one-way ones, but still far short of a pushdown transducer."
    }
  ],
  "config": {
    "transducerAccepts": true
  },
  "meta": {
    "title": "Copy twice: w ↦ ww",
    "blurb": "Sweeps the input printing it, rewinds, then sweeps and prints again — doubling the word. The canonical transduction that separates two-way transducers from one-way ones.",
    "inputs": [
      {
        "w": "ab",
        "expect": "accept",
        "out": "abab"
      },
      {
        "w": "a",
        "expect": "accept",
        "out": "aa"
      },
      {
        "w": "abb",
        "expect": "accept",
        "out": "abbabb"
      },
      {
        "w": "ba",
        "expect": "accept",
        "out": "baba"
      },
      {
        "w": "aabb",
        "expect": "accept",
        "out": "aabbaabb"
      }
    ],
    "library": {
      "author": {
        "login": "thethinkmachine"
      },
      "license": "CC-BY-4.0",
      "tags": [
        "showcase",
        "2dft"
      ],
      "difficulty": "intermediate"
    }
  },
  "format": "automata-studio/workspace",
  "schema": 1,
  "app": "dev"
}