{
  "machine": "PFA",
  "sigma": [
    "a",
    "b"
  ],
  "states": [
    {
      "id": "s1",
      "x": 240,
      "y": 280,
      "name": "saw b"
    },
    {
      "id": "s2",
      "x": 640,
      "y": 280,
      "name": "saw a"
    }
  ],
  "startId": "s1",
  "accepts": [
    "s2"
  ],
  "transitions": [
    {
      "id": "t1",
      "from": "s1",
      "to": "s2",
      "symbol": "a",
      "weight": 0.9
    },
    {
      "id": "t2",
      "from": "s1",
      "to": "s1",
      "symbol": "a",
      "weight": 0.1
    },
    {
      "id": "t3",
      "from": "s1",
      "to": "s1",
      "symbol": "b",
      "weight": 0.9
    },
    {
      "id": "t4",
      "from": "s1",
      "to": "s2",
      "symbol": "b",
      "weight": 0.1
    },
    {
      "id": "t5",
      "from": "s2",
      "to": "s2",
      "symbol": "a",
      "weight": 0.9
    },
    {
      "id": "t6",
      "from": "s2",
      "to": "s1",
      "symbol": "a",
      "weight": 0.1
    },
    {
      "id": "t7",
      "from": "s2",
      "to": "s1",
      "symbol": "b",
      "weight": 0.9
    },
    {
      "id": "t8",
      "from": "s2",
      "to": "s2",
      "symbol": "b",
      "weight": 0.1
    }
  ],
  "notes": [
    {
      "id": "n1",
      "x": 150,
      "y": 70,
      "color": "blue",
      "anchorStates": [],
      "anchorTransitions": [],
      "text": "A noisy last-symbol detector. Each read moves to the state matching the symbol 90% of the time and the wrong way 10% of the time — a channel that garbles one bit in ten. The language is still exactly \"ends in a\", because the accepting mass lands on 0.9 or 0.1 and the cut-point λ = 0.5 sits between them."
    },
    {
      "id": "n2",
      "x": 150,
      "y": 430,
      "color": "green",
      "anchorStates": [],
      "anchorTransitions": [],
      "text": "Watch the Pr row: unlike an NFA's set of states, a PFA carries a probability per state, and every (state, symbol) row sums to 1. The verdict compares the accepting mass against λ — strictly greater accepts. Because 0.9 and 0.1 stay well clear of 0.5, the cut-point is isolated and the language stays regular."
    }
  ],
  "config": {
    "pfaCutPoint": 0.5
  },
  "meta": {
    "title": "Noisy channel: does it end in a?",
    "blurb": "A two-state probabilistic automaton over a 10% noisy channel. The accepting mass is 0.9 after an a and 0.1 after a b, so cut-point 0.5 recovers the language \"ends in a\" despite the noise.",
    "inputs": [
      {
        "w": "a",
        "expect": "accept"
      },
      {
        "w": "b",
        "expect": "reject"
      },
      {
        "w": "ba",
        "expect": "accept"
      },
      {
        "w": "ab",
        "expect": "reject"
      },
      {
        "w": "bbba",
        "expect": "accept"
      },
      {
        "w": "aaab",
        "expect": "reject"
      }
    ],
    "library": {
      "author": {
        "login": "thethinkmachine"
      },
      "license": "CC-BY-4.0",
      "tags": [
        "showcase",
        "pfa"
      ],
      "difficulty": "intermediate"
    }
  },
  "format": "automata-studio/workspace",
  "schema": 1,
  "app": "dev"
}