{
  "name": "Term rewriting",
  "version": "1.0.0",
  "area": {
    "name": "Term rewriting",
    "slug": "rewriting",
    "group": "Languages & logic",
    "kind": "rewriting",
    "summary": "Inspect every allowed reduction order in a finite family, and compare all reachable normal forms.",
    "definitions": [
      {
        "term": "Rewrite rule",
        "definition": "A permitted local replacement; here 10 → 01."
      },
      {
        "term": "Normal form",
        "definition": "A word containing no reducible 10 substring."
      },
      {
        "term": "Confluence on a domain",
        "definition": "Every reduction path from each declared input can reach a common result."
      }
    ],
    "methodology": [
      "Enumerate every binary word through the declared length.",
      "Explore all possible one-step adjacent rewrites.",
      "Collect terminal words, memoizing the reduction graph.",
      "Require one normal form per input and replay the supplied concrete trace."
    ],
    "complexity": "Each rewrite decreases the number of inverted 1-before-0 pairs. Exhaustive graph exploration is restricted to the stated length.",
    "common_error": "Two chosen reduction strategies agreeing does not establish that all strategies agree.",
    "next_question": "Publish a general inversion-measure termination argument and a separately checked confluence proof.",
    "references": [
      "https://isa-afp.org/entries/Abstract-Rewriting.html"
    ]
  },
  "records": [
    {
      "id": "KL-FCS-034",
      "version": "1.0.0",
      "domain": "Term rewriting",
      "kind": "rewriting",
      "title": "All reduction orders through length 4",
      "problem": "Establish a unique sorted normal form for every bounded binary word.",
      "specification": {
        "max_length": 4,
        "rule": [
          "10",
          "01"
        ],
        "sample_word": "1100"
      },
      "claim": {
        "unique_normal_forms": true
      },
      "witness": {
        "trace": [
          "1100",
          "1010",
          "0110",
          "0101",
          "0011"
        ]
      },
      "verification_scope": "All reduction paths over 31 bounded inputs",
      "explanation": "The checker explores every enabled rewrite, not just the trace shown. Each result preserves the symbol multiset and reaches a block of zeros followed by ones.",
      "limitations": "The mechanical confluence result is restricted to the declared finite word family.",
      "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://isa-afp.org/entries/Abstract-Rewriting.html"
      ],
      "license_status": "not-yet-selected",
      "dataset": {
        "family": "rewriting",
        "task": "Establish a unique sorted normal form for every bounded binary word.",
        "input_encoding": "Structured JSON; field meanings are stated in the specification.",
        "coverage": "All reduction paths over 31 bounded inputs",
        "acceptance": [
          "Enumerate every binary word through the declared length.",
          "Explore all possible one-step adjacent rewrites.",
          "Collect terminal words, memoizing the reduction graph.",
          "Require one normal form per input and replay the supplied concrete trace."
        ],
        "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": "Inspect every allowed reduction order in a finite family, and compare all reachable normal forms.",
        "definitions": [
          {
            "term": "Rewrite rule",
            "definition": "A permitted local replacement; here 10 → 01."
          },
          {
            "term": "Normal form",
            "definition": "A word containing no reducible 10 substring."
          },
          {
            "term": "Confluence on a domain",
            "definition": "Every reduction path from each declared input can reach a common result."
          }
        ],
        "reasoning": [
          "Enumerate every binary word through the declared length.",
          "Explore all possible one-step adjacent rewrites.",
          "Collect terminal words, memoizing the reduction graph.",
          "Require one normal form per input and replay the supplied concrete trace."
        ],
        "worked_example": "The checker explores every enabled rewrite, not just the trace shown. Each result preserves the symbol multiset and reaches a block of zeros followed by ones.",
        "complexity": "Each rewrite decreases the number of inverted 1-before-0 pairs. Exhaustive graph exploration is restricted to the stated length.",
        "common_error": "Two chosen reduction strategies agreeing does not establish that all strategies agree.",
        "further_work": "Publish a general inversion-measure termination argument and a separately checked confluence proof."
      },
      "related_ids": [
        "KL-FCS-035",
        "KL-FCS-036"
      ]
    },
    {
      "id": "KL-FCS-035",
      "version": "1.0.0",
      "domain": "Term rewriting",
      "kind": "rewriting",
      "title": "All reduction orders through length 6",
      "problem": "Establish a unique sorted normal form for every bounded binary word.",
      "specification": {
        "max_length": 6,
        "rule": [
          "10",
          "01"
        ],
        "sample_word": "101010"
      },
      "claim": {
        "unique_normal_forms": true
      },
      "witness": {
        "trace": [
          "101010",
          "011010",
          "010110",
          "001110",
          "001101",
          "001011",
          "000111"
        ]
      },
      "verification_scope": "All reduction paths over 127 bounded inputs",
      "explanation": "The checker explores every enabled rewrite, not just the trace shown. Each result preserves the symbol multiset and reaches a block of zeros followed by ones.",
      "limitations": "The mechanical confluence result is restricted to the declared finite word family.",
      "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://isa-afp.org/entries/Abstract-Rewriting.html"
      ],
      "license_status": "not-yet-selected",
      "dataset": {
        "family": "rewriting",
        "task": "Establish a unique sorted normal form for every bounded binary word.",
        "input_encoding": "Structured JSON; field meanings are stated in the specification.",
        "coverage": "All reduction paths over 127 bounded inputs",
        "acceptance": [
          "Enumerate every binary word through the declared length.",
          "Explore all possible one-step adjacent rewrites.",
          "Collect terminal words, memoizing the reduction graph.",
          "Require one normal form per input and replay the supplied concrete trace."
        ],
        "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": "Inspect every allowed reduction order in a finite family, and compare all reachable normal forms.",
        "definitions": [
          {
            "term": "Rewrite rule",
            "definition": "A permitted local replacement; here 10 → 01."
          },
          {
            "term": "Normal form",
            "definition": "A word containing no reducible 10 substring."
          },
          {
            "term": "Confluence on a domain",
            "definition": "Every reduction path from each declared input can reach a common result."
          }
        ],
        "reasoning": [
          "Enumerate every binary word through the declared length.",
          "Explore all possible one-step adjacent rewrites.",
          "Collect terminal words, memoizing the reduction graph.",
          "Require one normal form per input and replay the supplied concrete trace."
        ],
        "worked_example": "The checker explores every enabled rewrite, not just the trace shown. Each result preserves the symbol multiset and reaches a block of zeros followed by ones.",
        "complexity": "Each rewrite decreases the number of inverted 1-before-0 pairs. Exhaustive graph exploration is restricted to the stated length.",
        "common_error": "Two chosen reduction strategies agreeing does not establish that all strategies agree.",
        "further_work": "Publish a general inversion-measure termination argument and a separately checked confluence proof."
      },
      "related_ids": [
        "KL-FCS-034",
        "KL-FCS-036"
      ]
    },
    {
      "id": "KL-FCS-036",
      "version": "1.0.0",
      "domain": "Term rewriting",
      "kind": "rewriting",
      "title": "All reduction orders through length 8",
      "problem": "Establish a unique sorted normal form for every bounded binary word.",
      "specification": {
        "max_length": 8,
        "rule": [
          "10",
          "01"
        ],
        "sample_word": "11110000"
      },
      "claim": {
        "unique_normal_forms": true
      },
      "witness": {
        "trace": [
          "11110000",
          "11101000",
          "11011000",
          "10111000",
          "01111000",
          "01110100",
          "01101100",
          "01011100",
          "00111100",
          "00111010",
          "00110110",
          "00101110",
          "00011110",
          "00011101",
          "00011011",
          "00010111",
          "00001111"
        ]
      },
      "verification_scope": "All reduction paths over 511 bounded inputs",
      "explanation": "The checker explores every enabled rewrite, not just the trace shown. Each result preserves the symbol multiset and reaches a block of zeros followed by ones.",
      "limitations": "The mechanical confluence result is restricted to the declared finite word family.",
      "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://isa-afp.org/entries/Abstract-Rewriting.html"
      ],
      "license_status": "not-yet-selected",
      "dataset": {
        "family": "rewriting",
        "task": "Establish a unique sorted normal form for every bounded binary word.",
        "input_encoding": "Structured JSON; field meanings are stated in the specification.",
        "coverage": "All reduction paths over 511 bounded inputs",
        "acceptance": [
          "Enumerate every binary word through the declared length.",
          "Explore all possible one-step adjacent rewrites.",
          "Collect terminal words, memoizing the reduction graph.",
          "Require one normal form per input and replay the supplied concrete trace."
        ],
        "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": "Inspect every allowed reduction order in a finite family, and compare all reachable normal forms.",
        "definitions": [
          {
            "term": "Rewrite rule",
            "definition": "A permitted local replacement; here 10 → 01."
          },
          {
            "term": "Normal form",
            "definition": "A word containing no reducible 10 substring."
          },
          {
            "term": "Confluence on a domain",
            "definition": "Every reduction path from each declared input can reach a common result."
          }
        ],
        "reasoning": [
          "Enumerate every binary word through the declared length.",
          "Explore all possible one-step adjacent rewrites.",
          "Collect terminal words, memoizing the reduction graph.",
          "Require one normal form per input and replay the supplied concrete trace."
        ],
        "worked_example": "The checker explores every enabled rewrite, not just the trace shown. Each result preserves the symbol multiset and reaches a block of zeros followed by ones.",
        "complexity": "Each rewrite decreases the number of inverted 1-before-0 pairs. Exhaustive graph exploration is restricted to the stated length.",
        "common_error": "Two chosen reduction strategies agreeing does not establish that all strategies agree.",
        "further_work": "Publish a general inversion-measure termination argument and a separately checked confluence proof."
      },
      "related_ids": [
        "KL-FCS-034",
        "KL-FCS-035"
      ]
    }
  ],
  "verification": [
    {
      "id": "KL-FCS-034",
      "status": "mechanically-checked",
      "check_units": 31,
      "scope": "All reduction paths over 31 bounded inputs",
      "review_status": "awaiting-independent-review"
    },
    {
      "id": "KL-FCS-035",
      "status": "mechanically-checked",
      "check_units": 127,
      "scope": "All reduction paths over 127 bounded inputs",
      "review_status": "awaiting-independent-review"
    },
    {
      "id": "KL-FCS-036",
      "status": "mechanically-checked",
      "check_units": 511,
      "scope": "All reduction paths over 511 bounded inputs",
      "review_status": "awaiting-independent-review"
    }
  ]
}