# A Classification of Symbolic Transition Systems

## Abstract

We define five increasingly comprehensive classes of infinite-state systems, called STS1–5, whose state spaces have finitary structure. For four of these classes, we provide examples from hybrid systems.

STS1 These are the systems with finite *bisimilarity* quotients. They can be analyzed symbolically by (1) iterating the predecessor and boolean operations starting from a finite set of observable state sets, and (2) terminating when no new state sets are generated. This enables model checking of the *μ*-calculus.

STS2 These are the systems with finite *similarity* quotients. They can be analyzed symbolically by iterating the predecessor and positive boolean operations. This enables model checking of the existential and universal fragments of the *μ*-calculus.

STS3 These are the systems with finite *trace-equivalence* quotients. They can be analyzed symbolically by iterating the predecessor operation and a restricted form of positive boolean operations (intersection is restricted to intersection with observables). This enables model checking of linear temporal logic.

STS4 These are the systems with finite *distance-equivalence* quotients (two states are equivalent if for every distance *d*, the same observables can be reached in *d* transitions). The systems in this class can be analyzed symbolically by iterating the predecessor operation and terminating when no new state sets are generated. This enables model checking of the existential conjunction-free and universal disjunction-free fragments of the *μ*-calculus.

STS5 These are the systems with finite *bounded-reachability* quotients (two states are equivalent if for every distance *d*, the same observables can be reached in *d* or fewer transitions). The systems in this class can be analyzed symbolically by iterating the predecessor operation and terminating when no new states are encountered. This enables model checking of reachability properties.

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