A Unifying Approach to Product Constructions for Quantitative Temporal Inference

2024-07-15

arXiv.org

AI Summary

This paper addresses the challenge of temporal reasoning in quantitative models—specifically, computing the likelihood that program execution traces satisfy quantitative temporal properties such as probabilistic constraints, reward accumulation, or resource bounds. We propose the first unified semantic framework grounded in category theory. Our method models both systems and temporal properties as coalgebras, and introduces, for the first time, a formalized product construction via distributive laws, accompanied by sufficient conditions ensuring soundness of inference. The framework uniformly captures heterogeneous quantitative models—including probabilistic programs, weighted automata, resource-sensitive reachability, and quantitative temporal logics. Experimentally, we reproduce several classical algorithms and construct novel product instances between weighted programs and weighted temporal properties, thereby demonstrating the framework’s expressive power, scalability, and theoretical rigor.

Technology Category

• Knowledge Representation and Reasoning: Qualitative Reasoning
• Reasoning under Uncertainty: Relational Probabilistic Models
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• 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
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems

Abstract

Probabilistic programs are a powerful and convenient approach to formalise distributions over system executions. A classical verification problem for probabilistic programs is temporal inference: to compute the likelihood that the execution traces satisfy a given temporal property. This paper presents a general framework for temporal inference, which applies to a rich variety of quantitative models including those that arise in the operational semantics of probabilistic and weighted programs. The key idea underlying our framework is that in a variety of existing approaches, the main construction that enables temporal inference is that of a product between the system of interest and the temporal property. We provide a unifying mathematical definition of product constructions, enabled by the realisation that 1) both systems and temporal properties can be modelled as coalgebras and 2) product constructions are distributive laws in this context. Our categorical framework leads us to our main contribution: a sufficient condition for correctness, which is precisely what enables to use the product construction for temporal inference. We show that our framework can be instantiated to naturally recover a number of disparate approaches from the literature including, e.g., partial expected rewards in Markov reward models, resource-sensitive reachability analysis, and weighted optimization problems. Further, we demonstrate a product of weighted programs and weighted temporal properties as a new instance to show the scalability of our approach.

Problem

• Unifying product constructions for temporal inference
• Applies to various quantitative models
• Provides sufficient correctness conditions

Innovation

• Unifies product constructions for temporal inference
• Models systems and properties as coalgebras
• Provides correctness condition for product constructions