An STREL-based Formulation of Spatial Resilience in Cyber-Physical Systems

2025-12-13

AI Summary

Existing approaches lack formal modeling and quantitative methods for spatial resilience—defined as the dual properties of recoverability (rapid restoration after violations) and persistence (long-term avoidance of recurrent violations)—in Cyber-Physical Systems (CPS). Method: This paper introduces SpaRS, the first formal framework for spatial resilience. It innovatively defines recoverability (rec) and persistence (per) as orthogonal, distance-based metrics; develops a compositional SpaRS specification language and its quantitative semantics SpaRV; proposes S-atom atomic units and a non-dominated (rec, per) optimization mechanism; unifies syntax and semantics via the STREL sublogic SREL; and designs an automated verification algorithm with a prototype tool. Results: SpaRS establishes the first sound, complete, and computable semantic model for spatial resilience. Evaluated on two representative CPS case studies, it enables multi-objective resilient path analysis and comparative evaluation.

Technology Category

• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Knowledge Representation and Reasoning: Geometric, Spatial, and Temporal Reasoning

Application Category

• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Security and Privacy: Large-scale security measurements
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

Abstract

Resiliency is the ability of a system to quickly recover from a violation (recoverability) and avoid future violations for as long as possible (durability). In the spatial setting, recoverability and durability (now known as persistency) are measured in units of distance. Like its temporal counterpart, spatial resiliency is of fundamental importance for Cyber-Physical Systems (CPS) and yet, to date, there is no widely agreed-upon formal treatment of spatial resiliency. We present a formal framework for reasoning about spatial resiliency in CPS. Our framework is based on the spatial fragment of STREL, which we refer to as SREL. In this framework, spatial resiliency is given a syntactic characterization in the form of a Spatial Resiliency Specification (SpaRS). An atomic predicate of SpaRS is called an S-atom. Given an arbitrary SREL formula $varphi$, distance bounds $d_1, d_2$, the S-atom of $varphi$, $S_{d_1, d_2} (varphi)$, is the SREL formula $ egvarphi R_{[0,d_1]} (varphi R_{[d_2, +infty)}varphi)$, specifying that recovery from a violation of $varphi$ occurs within distance $d_1$ (recoverability), and subsequently that $varphi$ be maintained along a route for a distance greater than $d_2$ (persistency). S-atoms can be combined using spatial STREL operators, allowing one to express composite resiliency specifications. We define a quantitative semantics for SpaRS in the form of a Spatial Resilience Value (SpaRV) function $σ$ and prove its soundness and completeness w.r.t. SREL's Boolean semantics. The $σ$-value for $S_{d_1,d_2}(varphi)$ is a set of non-dominated (rec, per) pairs, quantifying recoverability and persistency, given that some routes may offer better recoverability while others better persistency. In addition, we design algorithms to evaluate SpaRV for SpaRS formulas. Finally, two case studies demonstrate the practical utility of our approach.

Problem

• Formalizes spatial resilience for Cyber-Physical Systems using STREL.
• Quantifies recoverability and persistency via distance-based metrics.
• Provides algorithms to evaluate spatial resilience specifications.

Innovation

• Uses STREL spatial logic for formal spatial resilience framework
• Introduces S-atoms with distance bounds for recoverability and persistency
• Defines quantitative semantics with Spatial Resilience Value function