← 研踪 PaperTrailarXiv:2611.01825 · v2
The Scope Grammar of Natural-Language Rules: Counting and Enumerating Readings modulo Logical Equivalence
首次提交 2026-11-01 19:45
版本更新 2026-11-04 09:32
雷达观测 2026-11-05 10:08
摘要
A rule written for people fixes its atomic conditions and connectives but leaves open how the connectives are bracketed, and each bracketing is a different logical reading that an autoformalizer must resolve or expose. We study the scope grammar, a deliberately ambiguous context-free grammar whose derivation trees are the readings of a connective skeleton and whose quotient by logical equivalence is what matters downstream. A skeleton with $k$ binary connectives has $C_k$ derivation trees, the $k$-th Catalan number, but far fewer logically distinct readings. For the positive, repetition-free fragment we make the gap exact: through the uniqueness of the read-once normal form we characterize the realizable normal forms by lowest common ancestors and count them with a two-label recurrence over the operator word, verified against exhaustive enumeration up to eleven atoms. The count is invariant under reversal and $\wedge/\vee$ duality; a skeleton is ambiguous if and only if it uses both $\wedge$ and $\vee$, a one-pass test; and the number of classes is at least $2^{\lfloor n/3\rfloor}$. Up to fifteen atoms the alternating pattern is the worst case; its count has a cubic generating function, is the 3-Schröder sequence (OEIS A137842), and grows with base $\varphi^{5/2}\approx3.330$, below the Catalan base 4. We reduce maximality to an inequality on increments, verified to sixteen atoms, and explain why local exchange arguments cannot prove it. We also treat negation, exceptions, and background axioms, where normal-form uniqueness still decides equivalence but counting is open.
作者备注：13 pages, 1 figure, 1 table
观测到的版本
相邻记录可能跳过中间版本；首次发现 v2 及以上也会标为版本更新。
v2 · 2026-11-04 09:32 · 最新观测版本
The Scope Grammar of Natural-Language Rules: Counting and Enumerating Readings modulo Logical Equivalence
A rule written for people fixes its atomic conditions and connectives but leaves open how the connectives are bracketed, and each bracketing is a different logical reading that an autoformalizer must resolve or expose. We study the scope grammar, a deliberately ambiguous context-free grammar whose derivation trees are the readings of a connective skeleton and whose quotient by logical equivalence is what matters downstream. A skeleton with $k$ binary connectives has $C_k$ derivation trees, the $k$-th Catalan number, but far fewer logically distinct readings. For the positive, repetition-free fragment we make the gap exact: through the uniqueness of the read-once normal form we characterize the realizable normal forms by lowest common ancestors and count them with a two-label recurrence over the operator word, verified against exhaustive enumeration up to eleven atoms. The count is invariant under reversal and $\wedge/\vee$ duality; a skeleton is ambiguous if and only if it uses both $\wedge$ and $\vee$, a one-pass test; and the number of classes is at least $2^{\lfloor n/3\rfloor}$. Up to fifteen atoms the alternating pattern is the worst case; its count has a cubic generating function, is the 3-Schröder sequence (OEIS A137842), and grows with base $\varphi^{5/2}\approx3.330$, below the Catalan base 4. We reduce maximality to an inequality on increments, verified to sixteen atoms, and explain why local exchange arguments cannot prove it. We also treat negation, exceptions, and background axioms, where normal-form uniqueness still decides equivalence but counting is open.
查看此版本解读 →查看此版本原文 ↗v1 · 2026-11-01 19:45
The Scope Grammar of Natural-Language Rules: Counting and Enumerating Readings modulo Logical Equivalence
A rule written for people fixes its atomic conditions and connectives but leaves open how the connectives are bracketed, and each bracketing is a different logical reading that an autoformalizer must resolve or expose. We study the scope grammar, a deliberately ambiguous context-free grammar whose derivation trees are the readings of a connective skeleton and whose quotient by logical equivalence is what matters downstream. A skeleton with $k$ binary connectives has $C_k$ derivation trees, the $k$-th Catalan number, but far fewer logically distinct readings. For the positive, repetition-free fragment we make the gap exact: through the uniqueness of the read-once normal form we characterize the realizable normal forms by lowest common ancestors and count them with a two-label recurrence over the operator word, verified against exhaustive enumeration up to eleven atoms. The count is invariant under reversal and $\wedge/\vee$ duality; a skeleton is ambiguous if and only if it uses both $\wedge$ and $\vee$, a one-pass test; and the number of classes is at least $2^{\lfloor n/3\rfloor}$. Up to fifteen atoms the alternating pattern is the worst case; its count has a cubic generating function, is the 3-Schröder sequence (OEIS A137842), and grows with base $\varphi^{5/2}\approx3.330$, below the Catalan base 4. We reduce maximality to an inequality on increments, verified to sixteen atoms, and explain why local exchange arguments cannot prove it. We also treat negation, exceptions, and background axioms, where normal-form uniqueness still decides equivalence but counting is open.
查看此版本解读 →查看此版本原文 ↗