General Decidability Results for Systems with Continuous Counters

2025-11-26

AI Summary

To address the high computational complexity of modeling concurrent resources using natural-number counters, this paper proposes relaxing discrete counters to continuous counters over nonnegative rationals, enabling efficient approximate modeling and verification of dynamic resource behavior. The key contribution is a proof that, although continuous counter systems are infinite-state, the language of operation sequences leading to reachable target configurations remains regular—and a finite automaton recognizing this language can be effectively constructed. This result significantly extends the decidability frontier for reachability to broader classes of systems, including higher-order recursion schemes and well-structured transition systems. Furthermore, the paper establishes a non-elementary lower bound on the size of the constructed automaton, revealing the inherent theoretical cost of preserving decidability through continuous abstraction.

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Abstract

Counters that hold natural numbers are ubiquitous in modeling and verifying software systems; for example, they model dynamic creation and use of resources in concurrent programs. Unfortunately, such discrete counters often lead to extremely high complexity. Continuous counters are an efficient over-approximation of discrete counters. They are obtained by relaxing the original counters to hold values over the non-negative rational numbers. This work shows that continuous counters are extraordinarily well-behaved in terms of decidability. Our main result is that, despite continuous counters being infinite-state, the language of sequences of counter instructions that can arrive in a given target configuration, is regular. Moreover, a finite automaton for this language can be computed effectively. This implies that a wide variety of transition systems can be equipped with continuous counters, while maintaining decidability of reachability properties. Examples include higher-order recursion schemes, well-structured transition systems, and decidable extensions of discrete counter systems. We also prove a non-elementary lower bound for the size of the resulting finite automaton.

Problem

• Continuous counters approximate discrete counters efficiently
• Main result proves reachability language is regular
• Enables decidability in various infinite-state transition systems

Innovation

• Continuous counters approximate discrete counters efficiently
• Regular language describes reachable counter instruction sequences
• Finite automaton effectively computed for reachability analysis