{
  "id": "KL-FCS-021",
  "version": "1.0.0",
  "domain": "Model checking",
  "kind": "model-checking",
  "title": "Branching safety closure",
  "problem": "Explore the complete transition graph from state 0.",
  "specification": {
    "initial": [
      0
    ],
    "transitions": {
      "0": [
        1,
        2
      ],
      "1": [
        3
      ],
      "2": [
        3
      ],
      "3": [
        3
      ]
    },
    "safe": [
      0,
      1,
      2,
      3
    ]
  },
  "claim": {
    "invariant_holds": true
  },
  "witness": {
    "reachable": [
      0,
      1,
      2,
      3
    ]
  },
  "verification_scope": "Complete four-state reachability",
  "explanation": "Every successor is included in the closure. The safe-set comparison identifies whether the property holds across all reachable executions.",
  "limitations": "This is a finite safety check; no liveness or fairness statement is included.",
  "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": "model-checking",
    "task": "Explore the complete transition graph from state 0.",
    "input_encoding": "Structured JSON; field meanings are stated in the specification.",
    "coverage": "Complete four-state reachability",
    "acceptance": [
      "Initialize the frontier with every initial state.",
      "Follow all transitions, deduplicating visited states.",
      "Compare the supplied reachable-state certificate with the computed closure.",
      "Check whether every reachable state lies in the declared safe set."
    ],
    "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": "Explore the states actually reachable from an initial condition, and evaluate safety on that closure.",
    "definitions": [
      {
        "term": "Reachability",
        "definition": "The least set containing all initial states and closed under transitions."
      },
      {
        "term": "Safety invariant",
        "definition": "A predicate true at every reachable state."
      },
      {
        "term": "Unreachable state",
        "definition": "A declared state that no allowed execution from an initial state reaches."
      }
    ],
    "reasoning": [
      "Initialize the frontier with every initial state.",
      "Follow all transitions, deduplicating visited states.",
      "Compare the supplied reachable-state certificate with the computed closure.",
      "Check whether every reachable state lies in the declared safe set."
    ],
    "worked_example": "Every successor is included in the closure. The safe-set comparison identifies whether the property holds across all reachable executions.",
    "complexity": "Breadth-first exploration is O(V+E) for an explicit finite graph; implicit system state spaces can grow exponentially.",
    "common_error": "Ignoring an enabled transition can make an unsafe system appear safe.",
    "further_work": "Add counterexample paths, temporal properties, and fairness-aware liveness checks."
  },
  "related_ids": [
    "KL-FCS-008",
    "KL-FCS-020"
  ]
}