Spatial Resiliency Specification (SpaRS)

• Spatial Resiliency Specification (SpaRS) is a formal, algorithmic framework that quantifies and verifies a system’s capacity to recover from spatial violations and maintain resilience over extended domains.
• It integrates spatial logic via SREL, quantitative semantics, and fault-tolerance combinatorics to rigorously evaluate recovery and persistency in both abstract models and applied designs.
• By computing resiliency margins and identifying hidden essential sensor pairs, SpaRS provides actionable insights for improving the robustness of cyber-physical systems and sparse sensor arrays.

A Spatial Resiliency Specification (SpaRS) provides a formal, algorithmic, and quantitative framework for reasoning about and verifying spatial resilience properties in spatially distributed systems, notably in cyber-physical systems (CPS) and sparse sensor arrays. SpaRS enables the rigorous specification, evaluation, and archival of a system’s capacity to recover from spatial violations and maintain desired properties over extended spatial domains. The framework synthesizes spatial logic (via the Spatial Reach and Escape Logic, SREL), quantitative semantics, and fault-tolerance combinatorics, supporting both abstract models (routes over weighted graphs) and applied design scenarios (e.g., sensor placement and redundancy).

1. Formal Foundations and Definitions

Spatial resiliency is the combined capability of a system to (a) recover from violation of a property within a bounded distance (recoverability) and (b) maintain the property for a further minimal distance (persistency/durability) (Zhang et al., 14 Dec 2025). In SpaRS, these requirements are encoded syntactically via SREL—a modal spatial logic for expressing neighborhood and reachability properties over spatial models.

A spatial model consists of:

• $M = (L, W)$, where $L$ is a finite set of spatial locations and $W \subseteq L \times A \times L$ is an undirected, weighted adjacency relation.
• A distance function $f: A \to D$ with $D = \mathbb{R}_{\ge 0} \cup \{\infty\}$ quantifies distances on edges.
• Atomic spatial signals $\xi: L \to \mathbb{R}$ define which locations satisfy specified predicates.

The key semantic unit is the S-atom: given an SREL formula $\varphi$ and distance parameters $d_1$, $d_2$, the S-atom is defined

$$S_{d_1,d_2}(\varphi) \equiv \neg\,\varphi\, R_{[0, d_1]}^f\,\left(\varphi\, R_{[d_2,\infty)}^f\,\varphi\right)\,,$$

stipulating that violations of $\varphi$ must be recoverable within $d_1$, and, post-recovery, $\varphi$ is sustained for more than $d_2$.

For sparse linear arrays (SLAs), spatial resiliency is represented combinatorially in terms of sensor position sets $S = \{n_1, ..., n_N\}$ (in units of $\lambda/2$), their difference coarray (DCA), and a weight function $w(m)$ giving the multiplicity of each spatial lag. An $\eta$-fold redundant SLA requires

$$w(m) \geq \eta \quad \forall |m| \leq L - (\eta-1),$$

with precisely defined decrements for boundary lags. The central resiliency criterion is: each lag must persist with weight at least 1 in the face of up to $\eta-1$ arbitrary sensor failures (Patwari et al., 9 Sep 2025).

2. Syntactic and Semantic Structure of SpaRS

The syntax of SpaRS formulas extends SREL by allowing:

• S-atoms (basic spatial recovery-persistence requirements)
• Boolean connectives ($\neg, \land, \lor$)
• Reachability and escape modalities ($R^f_{[d_1,d_2]}, E^f_{[d_1,d_2]}$)
• “Somewhere” and “everywhere” quantification

The semantics depart from Boolean truth: each SpaRS formula at a location $\ell$ returns a set $\sigma$ of non-dominated integer-valued pairs $(x_r, x_p)$, where $x_r$ (recoverability margin) and $x_p$ (persistency margin) quantify how much faster recovery is possible and how much longer persistence is maintained relative to the bounds $d_1, d_2$. The dominance relation $\succ_{re}$ promotes solutions with more positive margins, and all mutually non-dominated pairs are retained to capture route-specific trade-offs.

Soundness and completeness are guaranteed: a SpaRS formula is satisfied at $\ell$ if and only if there is at least one $(x_r, x_p)\succ_{re}(0,0)$. Failure is evidenced by a strictly negative pair.

3. Algorithmic Evaluation of Spatial Resilience

For S-atoms $S_{d_1, d_2}(\varphi)$, the resilience value $\sigma$ is computed by:

1. Partitioning $L$ into satisfying and non-satisfying loci for $\varphi$.
2. For every recovery candidate $v$ where $\varphi$ holds, computing (a) recoverability via shortest paths (Dijkstra), and (b) maximal persistency via longest edge-simple paths (DFS/bitmask).
3. Aggregating $(d_1 - \mathrm{rec}[s], \mathrm{per}(v) - d_2)$ over sources and recoveries, returning the $\succ_{re}$ maxima.

Composite formulas are evaluated recursively using modified flooding algorithms for the modal reach/escape operators, employing $\maxre$ and $\minre$ set operations in place of scalar $\max/\min$.

In combinatorial sensor array resiliency, a systematic framework evaluates all unordered pairs of sensor failures (excluding known essential pairs) and recalculates DCA weights to reveal hidden essential pairs (HESPs). Pseudocode is provided for exhaustive testing (Patwari et al., 9 Sep 2025).

4. Applications and Case Studies

SpaRS has been applied in both abstract graph-based CPS and structured sensor array design.

In cyber-physical networks (Zhang et al., 14 Dec 2025):

• A solar-powered rover traversing a weighted graph of sites: SpaRS quantifies alternative routes’ trade-offs between prompt recovery (minimizing distance to a “power-valid” location) and maximum persistence (distance along which the power condition holds).
• Networked microgrids: S-atoms express minimum-distanced recovery of “supply ≥ demand,” and SpaRV values directly identify spatial vulnerabilities or resilience deficits at nodes.
• Urban bike-sharing: SpaRS efficiently pinpoints docking stations that do not meet resilience requirements (i.e., cannot recover from bike outage within prescribed spatial bounds).

In sparse array design (Patwari et al., 9 Sep 2025):

• For 15-element three-fold redundant arrays, the SpaRS process identifies both non-vulnerable sensor pairs and hidden essential pairs whose removal would catastrophically reduce spatial DOF by introducing DCA holes. Tabulated results clearly distinguish healthy configurations from vulnerable ones, quantifying robustness.

Scenario	Removed Sensors	min $w(m)$ after removal	DOFs (post-removal)	DCA Hole Location
Healthy	–	$\ge$ 1 $\forall m$	59	none
Non-vulnerable failure	{6,8}	$\ge$ 1	59	none
Hidden essential failure	{2,15}	$w'(14)=0$	27	$m = \pm14$

5. Specification, Verification, and Archival

A rigorous SpaRS workflow comprises:

1. Specification: Documenting the spatial domain, spatial signals, target atomic formulas, aperture (in arrays), weight function, and redundancy parameter $\eta$.
2. Verification: Applying exhaustive or algorithmic evaluation to enumerate all permissible failure scenarios (sensor failures or spatial violations), computing the minimal residual resilience in all cases.
3. Archival: Cataloguing hidden essential pairs and performance losses for reproducibility and future design reference. Visualizations of resilience margins pre- and post-failure form part of the archival record.
4. Design Guidance: New designs must enforce the formal multiplicity (in arrays) or SpaRS recoverability/persistency (in graphs), with certification via the framework to avoid unanticipated resilience gaps.

Spatial resiliency that is not verified using SpaRS may suffer catastrophic structural loss or violation persistence during realistic failure scenarios.

6. Broader Impact and Extensions

Beyond the modalities summarized above, SpaRS enables formal, multi-valued spatial resilience characterizations scalable to a wide variety of spatially structured systems. By offering both compositional specification and quantitative, algorithmically checkable semantics, SpaRS bridges the gap between logical verification and practical design for resilience.

A plausible implication is that SpaRS will facilitate standardized, transparent evaluation of spatial resilience in emerging CPS domains, including robotics, critical infrastructure, and sensing systems, enabling both theoretical insight and certified robustness in applied contexts (Patwari et al., 9 Sep 2025, Zhang et al., 14 Dec 2025).