A Topological Framework for Finite Behavioural Observations and Verification

2026-06-22

AI Summary

This work investigates formal verification and runtime monitoring of properties in concurrent systems based on limited behavioral observations, such as traces or simulations. By introducing point-set topological methods, it establishes a precise correspondence between the topology induced by different observation mechanisms on the space of processes and verifiability: precisely those properties that are open sets in the respective topology are verifiable. The main contributions include a general verification theorem unifying monitorability under trace, simulation, and finite-depth bisimulation semantics; a rigorous proof that the topologies τ_O and τ_sim, induced respectively by observational closure and simulation relations, satisfy a strict inclusion τ_sim ⊂ τ_O; and the insight that stronger behavioral equivalences yield fundamentally distinct topologies, thereby deepening the understanding of the relationship between behavioral semantics and verification capabilities.

Abstract

Formal verification and monitorability are based on finite observations, which allow properties to be verified from finite information about system behaviour. We study such observations through the topologies they generate on spaces of processes. We first consider trace-based topologies and show that finite trace observations on $Σ^ω$ induce the Cantor topology, while the topology corresponding to full trace inclusion is the discrete one. We then move to arbitrary process spaces, where finite trace observations define the topology $τ_O$, and show that simulation observations generate a strictly finer topology $τ_{\mathrm{sim}}$. Next, we prove a general verification theorem showing that, for any topology generated by finite observations, open sets are exactly the properties verifiable by those observations. We instantiate this result for $τ_O$ and $τ_{\mathrm{sim}}$, obtaining multi-trace and simulation monitorability as concrete cases. Finally, we examine the effect of replacing simulation with stronger relations, showing that finite-depth bisimulation yields a genuinely different topology.

Problem

• finite observations
• topological framework
• formal verification
• monitorability
• process spaces

Innovation

• topological framework
• finite observations
• simulation
• monitorability
• bisimulation