Godel Implication on Finite Chains: Truth Tables and Catalan-Bracketing Enumerations

Abstract

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.

Theorems & Definitions (47)

• Proposition 1: Recovering the remaining generating functions
• proof
• Proposition 2: $\Phi$--reduction: one equation in one unknown
• proof
• Proposition 3: Explicit inverse and nested radicals
• proof
• Theorem 1: Limiting proportion of $1$'s and asymptotics
• proof
• Lemma 1: Explicit evaluation of the iterates at $x=r$
• proof