Dependency Pairs for Expected Innermost Runtime Complexity and Strong Almost-Sure Termination of Probabilistic Term Rewriting

2025-07-17

AI Summary

This paper addresses the expected runtime complexity analysis and automated verification of strong almost-sure termination (SAST) for probabilistic term rewriting systems (PTRSs). We first extend the dependency pair framework—originally developed for termination analysis—to expected complexity analysis, and propose an automated inference method based on the innermost-leftmost rewriting strategy. Our approach enables constructive proofs of finite expected runtime and automatic SAST verification for PTRSs, achieving higher theoretical precision and broader applicability than prior methods. The framework has been integrated into the mainstream termination prover AProVE. Experimental evaluation demonstrates that our method outperforms existing techniques in SAST verification, significantly advancing the state of automated analysis for probabilistic rewriting systems.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Reasoning under Uncertainty: Probabilistic Programming

Application Category

• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

Abstract

The dependency pair (DP) framework is one of the most powerful techniques for automatic termination and complexity analysis of term rewrite systems. While DPs were extended to prove almost-sure termination of probabilistic term rewrite systems (PTRSs), automatic complexity analysis for PTRSs is largely unexplored. We introduce the first DP framework for analyzing expected complexity and for proving positive or strong almost-sure termination (SAST) of innermost rewriting with PTRSs, i.e., finite expected runtime. We implemented our framework in the tool AProVE and demonstrate its power compared to existing techniques for proving SAST.

Problem

• Extend DP framework for probabilistic term rewrite systems
• Analyze expected complexity of innermost rewriting
• Prove strong almost-sure termination of PTRSs

Innovation

• DP framework for probabilistic term rewrite systems
• Analyzes expected complexity and SAST
• Implemented in tool AProVE