Local Softmax and Global Weights in Non-Boolean Event Structures

TL;DR

This paper investigates how local softmax normalization in non-Boolean event structures relates to global probability weights, revealing conditions under which local rules can be globally consistent and characterizing exotic weights.

Contribution

It demonstrates that imposing no-disturbance constraints simplifies softmax rules to coordinate parametrizations of admissible weights, clarifying the structure of weights in non-Boolean event models.

Findings

Local normalization does not automatically produce global weights in non-Boolean structures.

Imposing no-disturbance collapses softmax rules to coordinate parametrizations of the positive weight polytope.

Exotic weights beyond classical or quantum bounds are properties of the structure and chosen weights, not the normalization link.

Abstract

Softmax and related normalized response functions are widely used in choice theory, machine learning, and cognitive science. In non-Boolean event structures with overlapping contexts, however, local normalization does not automatically yield a global probability weight. We show that imposing single-valuedness on shared atoms -- equivalently, no-disturbance or consistent connectedness -- collapses generalized softmax rules to coordinate parametrizations of the strictly positive part of the admissible-weight polytope. Any strictly positive admissible weight can be represented in this way, while boundary weights arise as limits. Exotic weights that exceed classical or quantum bounds are therefore properties of the event structure and the chosen weight, not of the normalizing link. The resulting hierarchy separates local normalization, cross-context gluing, Cauchy--Gleason linearity, and physical or cognitive realizability.