Tropical Mathematics and the Lambda-Calculus II: Tropical Geometry of Probabilistic Programming Languages

2025-01-26

AI Summary

This work addresses the challenge of modeling statistical behaviors and quantifying differential privacy in higher-order probabilistic programs—particularly probabilistic PCF. Methodologically, it introduces tropical geometry into probabilistic programming semantics for the first time, employing the min-plus semiring algebra to encode the most probable program behaviors as tropical polynomials. It establishes a semantic bridge between weighted relational semantics and linear logic, and designs the first intersection type system capable of characterizing tropical polynomials. The core contributions are: (i) a tropical algebraic framework for representing probabilistic program behaviors; (ii) an algebraic, verifiable method for estimating privacy guarantees in security protocols; and (iii) a breakthrough over traditional statistical inference approaches by enabling rigorous modeling of higher-order functions and recursive structures. The framework thus provides a novel semantic tool for probabilistic programming languages that combines mathematical rigor with computational tractability.

Technology Category

• Reasoning under Uncertainty: Probabilistic Programming
• Machine Learning: Probabilistic Circuits and Graphical Models
• Knowledge Representation and Reasoning: Logic Programming

Application Category

• Security and Privacy: Data transparency and provenance
• Semantics and Knowledge: Provenance, trust, security and privacy, and ethical issues in managing semantic data
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems

Abstract

In the last few years there has been a growing interest towards methods for statistical inference and learning based on ideas from computational algebraic geometry, and notably from tropical geometry, that is, the study of algebraic varieties over the min-plus semiring. At the same time, recent work has demonstrated the possibility of interpreting a higher-order probabilistic programming language in the framework of tropical mathematics, by exploiting the weighted relational semantics from linear logic. In this paper we try to put these two worlds in contact, by showing that actual methods from tropical geometry can indeed be exploited to perform statistical inference on higher order programs. For example, we show that the problem of describing the most-likely behavior of a probabilistic PCF program reduces to studying a tropical polynomial function associated with the program. We also design an intersection type system that captures such polynomials. As an application of our approach, we finally show that the tropical polynomial associated with a probabilistic protocol expressed in our language can be used to estimate its differential privacy.

Problem

• Tropical Geometry
• Probabilistic Programming Languages
• Statistical Analysis

Innovation

• Tropical Geometry
• Probabilistic Programming
• Privacy Analysis