Forward Analysis for WSTS, Part III: Karp-Miller Trees

Abstract

This paper is a sequel of Forward Analysis for WSTS, Part I: Completions [STACS 2009, LZI Intl. Proc. in Informatics 3, 433–444] and Forward Analysis for WSTS, Part II: Complete WSTS [Logical Methods in Computer Science 8(3), 2012]. In these two papers, we provided a framework to conduct forward reachability analyses of WSTS, using finite representations of downwards-closed sets. We further develop this framework to obtain a generic Karp-Miller algorithm for the new class of very-WSTS. This allows us to show that coverability sets of very-WSTS can be computed as their finite ideal decompositions. Under natural assumptions on positive sequences, we also show that LTL model checking for very-WSTS is decidable. The termination of our procedure rests on a new notion of acceleration levels, which we study. We characterize those domains that allow for only finitely many accelerations, based on ordinal ranks.