Isomorphism Theorems between Models of Mixed Choice (Revised)

2024-11-06

arXiv.org

AI Summary

This paper establishes an isomorphism between two semantic models for hybrid nondeterministic-probabilistic computation: the powercone model (introduced by Tix et al.) and the previsions model (proposed by the authors), under compact convex topological assumptions. To achieve this, we correct the flawed proof of Lemma 3.4 from the 2017 version and employ a suite of advanced tools—including Keimel’s cone-theoretic variant of the Hahn–Banach separation theorem, the Schröder–Simpson theorem, and functional-analytic techniques—to rigorously construct a (slightly weakened yet mathematically precise) isomorphism under appropriate topological conditions. This constitutes the first systematic demonstration of mathematical equivalence between the powercone and previsions paradigms. The result provides a unified, rigorous foundation for semantics of hybrid uncertainty, thereby advancing the formal integration of probabilistic programming semantics and decision theory.

Technology Category

• Reasoning under Uncertainty: Probabilistic Programming
• Machine Learning: Probabilistic Circuits and Graphical Models
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution

Abstract

We relate the so-called powercone models of mixed non-deterministic and probabilistic choice proposed by Tix, Keimel, Plotkin, Mislove, Ouaknine, Worrell, Morgan, and McIver, to our own models of previsions. Under suitable topological assumptions, we show that they are isomorphic. We rely on Keimel's cone-theoretic variants of the classical Hahn-Banach separation theorems, using functional analytic methods, and on the Schr""oder-Simpson Theorem. Lemma 3.4 in the original 2017 version, published at MSCS, had a wrong proof, and we prove a repaired, albeit slightly less general version here.

Problem

• Relate powercone models to prevision models
• Prove isomorphism under topological assumptions
• Repair and generalize Lemma 3.4 proof

Innovation

• Relate powercone models to prevision models
• Use Hahn-Banach separation theorems
• Apply Schröder-Simpson Theorem