{
  "id": "KL-FCS-058",
  "version": "1.0.0",
  "domain": "Finite probability",
  "kind": "markov",
  "title": "Success and failure after four steps",
  "problem": "Compute the exact distribution at each step of a finite Markov chain.",
  "specification": {
    "transition": [
      [
        "1/2",
        "1/4",
        "1/4"
      ],
      [
        "0",
        "1",
        "0"
      ],
      [
        "0",
        "0",
        "1"
      ]
    ],
    "initial": [
      "1",
      "0",
      "0"
    ],
    "steps": 4,
    "semantics": "Discrete time, row-stochastic transition matrix, no nondeterministic scheduler."
  },
  "claim": {
    "distribution": [
      "1/16",
      "15/32",
      "15/32"
    ]
  },
  "witness": {
    "trajectory": [
      [
        "1",
        "0",
        "0"
      ],
      [
        "1/2",
        "1/4",
        "1/4"
      ],
      [
        "1/4",
        "3/8",
        "3/8"
      ],
      [
        "1/8",
        "7/16",
        "7/16"
      ],
      [
        "1/16",
        "15/32",
        "15/32"
      ]
    ]
  },
  "verification_scope": "Exact rational trajectory · 4 transitions",
  "explanation": "Each step distributes the current mass across outgoing transitions. Absorbing rows retain their mass, and every row of the artifact preserves total probability one.",
  "limitations": "The result is for the declared horizon. The browser chart uses floating-point display; Python fractions establish the exact certificate.",
  "verification_status": "mechanically-checked",
  "review_status": "awaiting-independent-review",
  "provenance": {
    "origin": "Original Kenton Labs reference instance, authored with Codex assistance on 2026-10-11.",
    "external_dataset": null,
    "model_run": null
  },
  "references": [
    "https://www.prismmodelchecker.org/doc/whatsinprism.php"
  ],
  "license_status": "not-yet-selected",
  "dataset": {
    "family": "probability",
    "task": "Compute the exact distribution at each step of a finite Markov chain.",
    "input_encoding": "Structured JSON; field meanings are stated in the specification.",
    "coverage": "Exact rational trajectory · 4 transitions",
    "acceptance": [
      "Validate the initial distribution and each matrix row.",
      "Multiply the row distribution by the transition matrix.",
      "Repeat for the exact declared horizon.",
      "Compare every trajectory row and the final rational distribution."
    ],
    "generation": "Deterministic finite fixture; full enumeration or witness replay as stated.",
    "split_policy": "Reference corpus for exposition and reproduction; no train/test evaluation split is claimed."
  },
  "lesson": {
    "motivation": "Propagate a probability distribution through an explicitly finite stochastic model with exact fractions.",
    "definitions": [
      {
        "term": "DTMC",
        "definition": "A discrete-time Markov chain whose row probabilities determine the next-state distribution."
      },
      {
        "term": "Stochastic matrix",
        "definition": "A nonnegative matrix with every row summing to one."
      },
      {
        "term": "Absorbing state",
        "definition": "A state that transitions to itself with probability one."
      }
    ],
    "reasoning": [
      "Validate the initial distribution and each matrix row.",
      "Multiply the row distribution by the transition matrix.",
      "Repeat for the exact declared horizon.",
      "Compare every trajectory row and the final rational distribution."
    ],
    "worked_example": "Each step distributes the current mass across outgoing transitions. Absorbing rows retain their mass, and every row of the artifact preserves total probability one.",
    "complexity": "A dense n-state chain over h steps costs O(hn²) arithmetic operations, with fraction sizes growing over time.",
    "common_error": "A finite-horizon probability is not automatically an eventual-reachability answer.",
    "further_work": "Add absorbing-state equations, expected hitting time, and nondeterministic MDP choices."
  },
  "related_ids": [
    "KL-FCS-059",
    "KL-FCS-060"
  ]
}