Induction and Recursion Principles in a Higher-Order Quantitative Logic

2025-01-30

AI Summary

Probabilistic program verification lacks high-order quantitative logics to formally reason about probabilistic behaviors. Method: We propose a novel induction principle over probability measures and natural numbers, and develop a controlled recursive reasoning mechanism grounded in the Banach fixed-point theorem. Our approach integrates affine type systems, 1-bounded complete metric spaces, the probability monad, and the Kantorovich distance, enabling, for the first time in a higher-order logic, a unified formal treatment of couplings, process distances, and convergence. Contributions: (1) We provide rigorous, machine-checkable proofs of upper bounds on distances between probabilistic processes; (2) we embed coupling techniques directly into the logical system, supporting consistent verification of both equivalence and convergence; and (3) we overcome expressive limitations of conventional logics regarding probabilistic structure and inductive reasoning, thereby establishing a new foundation for highly trustworthy probabilistic program verification.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Data transparency and provenance
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

Quantitative logic reasons about the degree to which formulas are satisfied. This paper studies the fundamental reasoning principles of higher-order quantitative logic and their application to reasoning about probabilistic programs and processes. We construct an affine calculus for 1-bounded complete metric spaces and the monad for probability measures equipped with the Kantorovic distance. The calculus includes a form of guarded recursion interpreted via Banach's fixed point theorem, useful, e.g., for recursive programming with processes. We then define an affine higher-order quantitative logic for reasoning about terms of our calculus. The logic includes novel principles for guarded recursion, and induction over probability measures and natural numbers. Examples of reasoning in the logic include proofs of upper bounds on distances of processes. We also show how our logic can express coupling proofs - a powerful technique for comparing probabilistic processes.

Problem

• Probabilistic Programs
• Higher-Order Quantified Logic
• Inductive Reasoning

Innovation

• Restricted Metric Spaces
• Probabilistic Mathematical Objects
• Recursive Logic Rules