{
  "name": "Formal languages",
  "version": "1.0.0",
  "area": {
    "name": "Formal languages",
    "slug": "formal-languages",
    "group": "Languages & logic",
    "kind": "languages",
    "summary": "Characterize bounded membership in a recursively structured language while keeping the length bound visible.",
    "definitions": [
      {
        "term": "Dyck word",
        "definition": "A balanced parenthesis string whose prefix balance never goes negative."
      },
      {
        "term": "Prefix balance",
        "definition": "Opening parentheses minus closing parentheses in an input prefix."
      },
      {
        "term": "Membership",
        "definition": "Whether a particular word belongs to the specified language."
      }
    ],
    "methodology": [
      "Generate every parenthesis string up to the stated length.",
      "Scan each prefix and reject any negative balance.",
      "Accept only strings with terminal balance zero.",
      "Compare the complete accepted-word list and its cardinality."
    ],
    "complexity": "A length bound n produces 2ⁿ⁺¹−1 candidate words; each membership scan takes O(n).",
    "common_error": "Equal numbers of opening and closing symbols do not guarantee proper nesting.",
    "next_question": "Add context-free grammar membership, CYK charts, and parse-tree certificates.",
    "references": [
      "https://ocw.mit.edu/courses/18-404j-theory-of-computation-fall-2020/download/"
    ]
  },
  "records": [
    {
      "id": "KL-FCS-031",
      "version": "1.0.0",
      "domain": "Formal languages",
      "kind": "languages",
      "title": "Balanced words through length 4",
      "problem": "List every balanced parenthesis word of length at most 4, including the empty word.",
      "specification": {
        "max_length": 4,
        "alphabet": [
          "(",
          ")"
        ],
        "language": "Every prefix has nonnegative balance and the terminal balance is zero."
      },
      "claim": {
        "accepted_count": 4
      },
      "witness": {
        "words": [
          "",
          "()",
          "(())",
          "()()"
        ]
      },
      "verification_scope": "Complete bounded membership · 31 candidates",
      "explanation": "A prefix can invalidate a word before its final symbol. The accepted list includes different nesting structures and concatenations, not only fully nested strings.",
      "limitations": "This enumerates a bounded language slice; it is not a grammar-equivalence theorem.",
      "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://ocw.mit.edu/courses/18-404j-theory-of-computation-fall-2020/download/"
      ],
      "license_status": "not-yet-selected",
      "dataset": {
        "family": "formal-languages",
        "task": "List every balanced parenthesis word of length at most 4, including the empty word.",
        "input_encoding": "Structured JSON; field meanings are stated in the specification.",
        "coverage": "Complete bounded membership · 31 candidates",
        "acceptance": [
          "Generate every parenthesis string up to the stated length.",
          "Scan each prefix and reject any negative balance.",
          "Accept only strings with terminal balance zero.",
          "Compare the complete accepted-word list and its cardinality."
        ],
        "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": "Characterize bounded membership in a recursively structured language while keeping the length bound visible.",
        "definitions": [
          {
            "term": "Dyck word",
            "definition": "A balanced parenthesis string whose prefix balance never goes negative."
          },
          {
            "term": "Prefix balance",
            "definition": "Opening parentheses minus closing parentheses in an input prefix."
          },
          {
            "term": "Membership",
            "definition": "Whether a particular word belongs to the specified language."
          }
        ],
        "reasoning": [
          "Generate every parenthesis string up to the stated length.",
          "Scan each prefix and reject any negative balance.",
          "Accept only strings with terminal balance zero.",
          "Compare the complete accepted-word list and its cardinality."
        ],
        "worked_example": "A prefix can invalidate a word before its final symbol. The accepted list includes different nesting structures and concatenations, not only fully nested strings.",
        "complexity": "A length bound n produces 2ⁿ⁺¹−1 candidate words; each membership scan takes O(n).",
        "common_error": "Equal numbers of opening and closing symbols do not guarantee proper nesting.",
        "further_work": "Add context-free grammar membership, CYK charts, and parse-tree certificates."
      },
      "related_ids": [
        "KL-FCS-032",
        "KL-FCS-033"
      ]
    },
    {
      "id": "KL-FCS-032",
      "version": "1.0.0",
      "domain": "Formal languages",
      "kind": "languages",
      "title": "Balanced words through length 6",
      "problem": "List every balanced parenthesis word of length at most 6, including the empty word.",
      "specification": {
        "max_length": 6,
        "alphabet": [
          "(",
          ")"
        ],
        "language": "Every prefix has nonnegative balance and the terminal balance is zero."
      },
      "claim": {
        "accepted_count": 9
      },
      "witness": {
        "words": [
          "",
          "()",
          "(())",
          "()()",
          "((()))",
          "(()())",
          "(())()",
          "()(())",
          "()()()"
        ]
      },
      "verification_scope": "Complete bounded membership · 127 candidates",
      "explanation": "A prefix can invalidate a word before its final symbol. The accepted list includes different nesting structures and concatenations, not only fully nested strings.",
      "limitations": "This enumerates a bounded language slice; it is not a grammar-equivalence theorem.",
      "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://ocw.mit.edu/courses/18-404j-theory-of-computation-fall-2020/download/"
      ],
      "license_status": "not-yet-selected",
      "dataset": {
        "family": "formal-languages",
        "task": "List every balanced parenthesis word of length at most 6, including the empty word.",
        "input_encoding": "Structured JSON; field meanings are stated in the specification.",
        "coverage": "Complete bounded membership · 127 candidates",
        "acceptance": [
          "Generate every parenthesis string up to the stated length.",
          "Scan each prefix and reject any negative balance.",
          "Accept only strings with terminal balance zero.",
          "Compare the complete accepted-word list and its cardinality."
        ],
        "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": "Characterize bounded membership in a recursively structured language while keeping the length bound visible.",
        "definitions": [
          {
            "term": "Dyck word",
            "definition": "A balanced parenthesis string whose prefix balance never goes negative."
          },
          {
            "term": "Prefix balance",
            "definition": "Opening parentheses minus closing parentheses in an input prefix."
          },
          {
            "term": "Membership",
            "definition": "Whether a particular word belongs to the specified language."
          }
        ],
        "reasoning": [
          "Generate every parenthesis string up to the stated length.",
          "Scan each prefix and reject any negative balance.",
          "Accept only strings with terminal balance zero.",
          "Compare the complete accepted-word list and its cardinality."
        ],
        "worked_example": "A prefix can invalidate a word before its final symbol. The accepted list includes different nesting structures and concatenations, not only fully nested strings.",
        "complexity": "A length bound n produces 2ⁿ⁺¹−1 candidate words; each membership scan takes O(n).",
        "common_error": "Equal numbers of opening and closing symbols do not guarantee proper nesting.",
        "further_work": "Add context-free grammar membership, CYK charts, and parse-tree certificates."
      },
      "related_ids": [
        "KL-FCS-031",
        "KL-FCS-033"
      ]
    },
    {
      "id": "KL-FCS-033",
      "version": "1.0.0",
      "domain": "Formal languages",
      "kind": "languages",
      "title": "Balanced words through length 8",
      "problem": "List every balanced parenthesis word of length at most 8, including the empty word.",
      "specification": {
        "max_length": 8,
        "alphabet": [
          "(",
          ")"
        ],
        "language": "Every prefix has nonnegative balance and the terminal balance is zero."
      },
      "claim": {
        "accepted_count": 23
      },
      "witness": {
        "words": [
          "",
          "()",
          "(())",
          "()()",
          "((()))",
          "(()())",
          "(())()",
          "()(())",
          "()()()",
          "(((())))",
          "((()()))",
          "((())())",
          "((()))()",
          "(()(()))",
          "(()()())",
          "(()())()",
          "(())(())",
          "(())()()",
          "()((()))",
          "()(()())",
          "()(())()",
          "()()(())",
          "()()()()"
        ]
      },
      "verification_scope": "Complete bounded membership · 511 candidates",
      "explanation": "A prefix can invalidate a word before its final symbol. The accepted list includes different nesting structures and concatenations, not only fully nested strings.",
      "limitations": "This enumerates a bounded language slice; it is not a grammar-equivalence theorem.",
      "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://ocw.mit.edu/courses/18-404j-theory-of-computation-fall-2020/download/"
      ],
      "license_status": "not-yet-selected",
      "dataset": {
        "family": "formal-languages",
        "task": "List every balanced parenthesis word of length at most 8, including the empty word.",
        "input_encoding": "Structured JSON; field meanings are stated in the specification.",
        "coverage": "Complete bounded membership · 511 candidates",
        "acceptance": [
          "Generate every parenthesis string up to the stated length.",
          "Scan each prefix and reject any negative balance.",
          "Accept only strings with terminal balance zero.",
          "Compare the complete accepted-word list and its cardinality."
        ],
        "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": "Characterize bounded membership in a recursively structured language while keeping the length bound visible.",
        "definitions": [
          {
            "term": "Dyck word",
            "definition": "A balanced parenthesis string whose prefix balance never goes negative."
          },
          {
            "term": "Prefix balance",
            "definition": "Opening parentheses minus closing parentheses in an input prefix."
          },
          {
            "term": "Membership",
            "definition": "Whether a particular word belongs to the specified language."
          }
        ],
        "reasoning": [
          "Generate every parenthesis string up to the stated length.",
          "Scan each prefix and reject any negative balance.",
          "Accept only strings with terminal balance zero.",
          "Compare the complete accepted-word list and its cardinality."
        ],
        "worked_example": "A prefix can invalidate a word before its final symbol. The accepted list includes different nesting structures and concatenations, not only fully nested strings.",
        "complexity": "A length bound n produces 2ⁿ⁺¹−1 candidate words; each membership scan takes O(n).",
        "common_error": "Equal numbers of opening and closing symbols do not guarantee proper nesting.",
        "further_work": "Add context-free grammar membership, CYK charts, and parse-tree certificates."
      },
      "related_ids": [
        "KL-FCS-031",
        "KL-FCS-032"
      ]
    }
  ],
  "verification": [
    {
      "id": "KL-FCS-031",
      "status": "mechanically-checked",
      "check_units": 31,
      "scope": "Complete bounded membership · 31 candidates",
      "review_status": "awaiting-independent-review"
    },
    {
      "id": "KL-FCS-032",
      "status": "mechanically-checked",
      "check_units": 127,
      "scope": "Complete bounded membership · 127 candidates",
      "review_status": "awaiting-independent-review"
    },
    {
      "id": "KL-FCS-033",
      "status": "mechanically-checked",
      "check_units": 511,
      "scope": "Complete bounded membership · 511 candidates",
      "review_status": "awaiting-independent-review"
    }
  ]
}