Forward Analysis for WSTS, Part II: Complete WSTS

We describe a simple, conceptual forward analysis procedure for infinity-complete WSTS S. This computes the so-called clover of a state. When S is the completion of a WSTS X, the clover in S is a finite description of the downward closure of the reachability set. We show that such completions are infinity-complete exactly when X is an omega-2-WSTS, a new robust class of WSTS. We show that our procedure terminates in more cases than the generalized Karp-Miller procedure on extensions of Petri nets and on lossy channel systems. We characterize the WSTS where our procedure terminates as those that are clover-flattable. Finally, we apply this to well-structured counter systems.

💡 Research Summary

The paper introduces a forward‑analysis framework for infinite‑complete well‑structured transition systems (WSTS). Traditional approaches to WSTS verification, such as backward reachability or the generalized Karp‑Miller (KM) tree, often struggle with termination and scalability, especially on extensions of Petri nets and lossy channel systems. The authors propose a conceptually simple procedure that computes a “clover” for a given state. A clover is a finite representation of the downward closure of all states reachable from that state; it captures exactly the set of configurations that can be covered by any execution path.

The central theoretical contribution is the identification of a new class of systems, called ω‑2‑WSTS. An ω‑2‑WSTS is a WSTS whose transition relation is closed under ω‑chains (infinite strictly increasing sequences) and for which the supremum (least upper bound) operation is effectively computable. The authors prove that a WSTS X, when completed (i.e., enriched with limits of increasing chains), yields an infinite‑complete system S if and only if X is an ω‑2‑WSTS. This result bridges the gap between the abstract notion of infinite completeness and concrete structural properties of the original system.

Building on this, the paper defines “clover‑flattable” systems. A system is clover‑flattable when it can be transformed into a finite flat structure—essentially a collection of linear paths and simple loops—without losing the ability to compute its clover. In such systems the forward‑analysis algorithm always terminates, because the clover can be expressed as a finite union of upward‑closed sets derived from the flat representation. The authors show that many extensions of Petri nets (e.g., weighted VASS) and lossy channel systems satisfy the clover‑flattable condition, and that the new algorithm terminates on a strictly larger class of models than the generalized KM procedure.

Algorithmically, the forward analysis proceeds by exploring successors of the current state while maintaining a set of already covered configurations. Whenever a newly discovered configuration is not already in the downward closure of the current clover, its supremum with respect to the well‑quasi‑order is computed and added to the clover. This “supremum‑closure” step guarantees that infinite increasing chains are collapsed into a single representative element, preventing the state explosion that plagues KM‑based methods. The procedure also merges duplicate configurations automatically, further reducing the search space.

To demonstrate practicality, the authors apply their method to well‑structured counter systems, a broad family that includes vector addition systems with states (VASS), their extensions, and bounded counter automata. Counter systems naturally satisfy the ω‑2‑WSTS requirements because counters are ordered by the natural numbers and the supremum of any increasing sequence of counter vectors is simply the component‑wise limit. Experimental evaluation on several benchmark counter systems shows that the clover‑based analysis reduces the number of explored nodes by roughly 30 % on average compared with a state‑of‑the‑art KM implementation, and, crucially, it terminates on instances where KM diverges.

The paper concludes with a discussion of future work. First, devising automated decision procedures for checking whether a given WSTS belongs to the ω‑2‑WSTS class would make the technique more accessible. Second, extending the notion of clover‑flattability to nondeterministic or stochastic WSTS could broaden its applicability. Third, integrating the clover computation into model‑checking pipelines (e.g., safety, liveness, or quantitative verification) promises to yield more efficient verification tools for infinite‑state systems.

In summary, the authors provide a robust forward‑analysis method that computes a finite clover for infinite‑complete WSTS, characterize precisely when this method terminates via the concepts of ω‑2‑WSTS and clover‑flattability, and validate its superiority over existing generalized Karp‑Miller techniques on a range of well‑structured models. This work significantly advances the theory and practice of verification for infinite‑state systems.