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| Autore principale: | |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| Soggetti: | |
| Accesso online: | https://arxiv.org/abs/2602.16135 |
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| _version_ | 1866911453925081088 |
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| author | Yildiz, Volkan |
| author_facet | Yildiz, Volkan |
| contents | Fully bracketed implication terms on $n$ variables are evaluated in Gödel $m$-valued logic on a finite chain, and we enumerate truth-table rows by output value across all Catalan bracketings. Using the Catalan decomposition, we derive a finite system of generating functions for these value counts and introduce a root-split refinement that records the ordered pair of truth values at the top implication, yielding $m^2$ pair classes. We prove that the associated generating functions share a common dominant square-root singularity, which implies a universal $n^{-3/2}$ asymptotic form with exponential growth rate $(4m)^n$ and a limiting output distribution as $n\to\infty$. The root-split refinement yields matching uniform asymptotics for the pair classes and gives a transparent factorization of the original counts. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_16135 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Godel Implication on Finite Chains: Truth Tables and Catalan-Bracketing Enumerations Yildiz, Volkan Combinatorics Logic 03B50, 05A15, 05A16, 05A19, 05A10 Fully bracketed implication terms on $n$ variables are evaluated in Gödel $m$-valued logic on a finite chain, and we enumerate truth-table rows by output value across all Catalan bracketings. Using the Catalan decomposition, we derive a finite system of generating functions for these value counts and introduce a root-split refinement that records the ordered pair of truth values at the top implication, yielding $m^2$ pair classes. We prove that the associated generating functions share a common dominant square-root singularity, which implies a universal $n^{-3/2}$ asymptotic form with exponential growth rate $(4m)^n$ and a limiting output distribution as $n\to\infty$. The root-split refinement yields matching uniform asymptotics for the pair classes and gives a transparent factorization of the original counts. |
| title | Godel Implication on Finite Chains: Truth Tables and Catalan-Bracketing Enumerations |
| topic | Combinatorics Logic 03B50, 05A15, 05A16, 05A19, 05A10 |
| url | https://arxiv.org/abs/2602.16135 |