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

TL;DR

This paper analyzes the enumeration of fully bracketed implication terms in G"odel logic over finite chains, deriving generating functions and asymptotic behaviors using Catalan structures and a root-split refinement.

Contribution

It introduces a novel enumeration method for implication terms in G"odel logic using Catalan bracketings and a root-split refinement, providing explicit generating functions and asymptotic analysis.

Findings

Derived generating functions for truth table counts

Established universal asymptotic form with exponential growth rate

Introduced root-split refinement for detailed output distribution

Abstract

Fully bracketed implication terms on $n$ variables are evaluated in G\"odel $m$-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.