Year: 2026
Type: preprint
Abstract: We study the distribution of output values produced by all fully bracketed implication terms on $n$ distinct variables in Gödel $m$-valued logic on the finite chain $V_m$. The Catalan decomposition of bracketings yields a stratified system of recurrences for the output counts. Passing to tail sums reduces this system to a finite iteration of a single map $Φ_x$ and its inverse branch $Ψ_x$. For each fixed $m$, this reduction gives a common dominant square-root singularity at $r=1/(4m)$, coefficient asymptotics of order $(4m)^n n^{-3/2}$, and limiting proportions for every truth level. In the four-valued case these proportions are obtained explicitly in nested radicals. The main new phenomenon appears when the number of truth values also tends to infinity. If $K_m$ denotes the output level under the limiting $n\to\infty$ distribution and $X_m=K_m/(m-1)$, then $X_m\Rightarrow X$, where $X$ has density $(1+2t)^{-3/2}$ on $[0,1)$ together with an atom of mass $1/\sqrt{3}$ at $1$. In particular, the top mass converges to $1/\sqrt{3}$, while the bottom mass is asymptotic to $1/m$. We further show that the same probability law arises intrinsically when Gödel implication is defined directly on $[0,1]$ and valuations are sampled from Lebesgue product measure: the continuum tail generating function satisfies an exact first-order differential equation whose critical specialization is the ODE appearing in the finite-chain Euler limit. We also record a root-split refinement whose pair-count generating functions factor as products of the level generating functions and inherit the same square-root scale.
Source: arXiv (Cornell University)
Author: Yildiz, Volkan
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7130537316
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