Deciding Termination of Simple Randomized Loops

2025-06-23

AI Summary

This paper investigates the universal positive almost-sure termination (UPAST) problem for single-loop probabilistic programs featuring linear loop guards and a probabilistic choice between two commuting, diagonalizable linear updates. Methodologically, it integrates linear algebra, Markov process analysis, and semiring-based reasoning. The main contribution is the first decidability result for finite expected runtime over several real-valued sub-semirings—extending Tiwari’s (2004) classical non-probabilistic termination theory to the probabilistic setting. Crucially, the commutativity and diagonalizability of the updates enable an exact characterization of expected behavior, yielding a complete decidable theory for UPAST. The work provides computable necessary and sufficient conditions for UPAST, applicable across broad classes of real-valued inputs. This establishes a rigorous foundation for automated verification of simple probabilistic loops.

Technology Category

• Reasoning under Uncertainty: Probabilistic Programming
• Machine Learning: Probabilistic Circuits and Graphical Models
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• 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

We show that universal positive almost sure termination (UPAST) is decidable for a class of simple randomized programs, i.e., it is decidable whether the expected runtime of such a program is finite for all inputs. Our class contains all programs that consist of a single loop, with a linear loop guard and a loop body composed of two linear commuting and diagonalizable updates. In each iteration of the loop, the update to be carried out is picked at random, according to a fixed probability. We show the decidability of UPAST for this class of programs, where the program's variables and inputs may range over various sub-semirings of the real numbers. In this way, we extend a line of research initiated by Tiwari in 2004 into the realm of randomized programs.

Problem

• Deciding termination for randomized loops with finite expected runtime
• Analyzing linear loop guards and diagonalizable randomized updates
• Extending decidability results to real-number sub-semiring programs

Innovation

• Deciding UPAST for randomized loops
• Linear commuting diagonalizable updates
• Sub-semirings of real numbers