Spatio-Temporal Logic (SpaTiaL)

• SpaTiaL is a formal framework that integrates learned spatial classifiers with temporal logic to specify and monitor dynamic, pattern-rich systems.
• It employs SVM-STL to construct spatial predicates via machine learning and uses robust parameter synthesis techniques like Particle Swarm Optimization.
• The framework has practical applications in reaction-diffusion systems, swarm robotics, and cyber-physical systems, enhancing interpretability and automation.

Spatio-Temporal Logic (SpaTiaL)

Spatio-Temporal Logic (SpaTiaL) constitutes a formal framework for specifying and learning properties of dynamical systems that exhibit both spatial and temporal structure. The framework extends the expressiveness of traditional Signal Temporal Logic (STL) by enabling the specification and learning of properties involving time-varying spatial patterns, incorporating spatial classification through machine learning, and supporting robust parameter synthesis and interpretability for complex spatio-temporal dynamics. The SVM-STL formulation provides a unified technical foundation for learning, monitoring, and synthesizing spatio-temporal logic specifications in dynamical systems and serves as a prototype for second-generation SpaTiaL methodologies (Alsalehi et al., 2021).

1. Syntax and Semantics of SVM-STL

Let $S:\mathbb{N}\to\mathbb{R}^{L\times W\times C}$ denote a spatio-temporal trajectory over a discrete time index (e.g., system simulations producing image-like grid data over time). Formulas in (parametric) SVM-STL are generated as follows: $$\varphi ::= \top \;\mid\; \mu_j \;\mid\; \neg\varphi \;\mid\; \varphi_1\wedge\varphi_2 \;\mid\; \varphi_1\vee\varphi_2 \;\mid\; \mathcal{G}_{[a,b]}\varphi \;\mid\; \mathcal{F}_{[a,b]}\varphi$$

• $\mu_j$ is a spatial predicate of the form either $h_j(S[k]) > r$ or $h_j(S[k]) \le r$, where $h_j:\mathbb{R}^{L\times W\times C}\to\mathbb{R}$ is a learned spatial classifier and $r\in\mathbb{R}$ is a threshold.
• The temporal interval operators $\mathcal{G}_{[a,b]}$ (“always in $[a,b]$”) and $\mathcal{F}_{[a,b]}$ (“eventually in $[a,b]$”) quantify over time indices.

Qualitative semantics are inductively defined:

$$\begin{aligned} & (S, k)\models \mu_j \iff h_j(S[k]) \succ r \ & (S, k)\models \neg\varphi \iff \neg\big((S, k)\models\varphi\big) \ & (S, k)\models \varphi_1\wedge\varphi_2 \iff (S, k)\models \varphi_1\text{ and }(S, k)\models \varphi_2 \ & (S, k)\models \mathcal{F}_{[a,b]} \varphi \iff \exists k'\in[k+a,k+b]: (S, k')\models\varphi \ & (S, k)\models \mathcal{G}_{[a,b]} \varphi \iff \forall k'\in[k+a,k+b], (S, k')\models\varphi \end{aligned}$$

Quantitative (robustness) semantics return a real-valued signal $\rho(S, \varphi, k)$: $$\begin{aligned} &\rho(S, \mu_j, k) = \begin{cases} h_j(S[k]) - r &\text{if } h_j(S[k]) > r\ r - h_j(S[k]) &\text{if } h_j(S[k]) \le r \end{cases}\ &\rho(S, \neg\varphi, k) = -\rho(S,\varphi,k)\ &\rho(S, \varphi_1\wedge\varphi_2, k) = \min\big(\rho(S, \varphi_1, k),\, \rho(S, \varphi_2, k)\big)\ &\rho(S, \mathcal{F}_{[a,b]}\varphi, k) = \max_{k'\in[k+a, k+b]} \rho(S, \varphi, k')\ &\rho(S, \mathcal{G}_{[a,b]}\varphi, k) = \min_{k'\in[k+a, k+b]} \rho(S, \varphi, k')\ \end{aligned}$$ This semantic lifting supports both binary satisfaction and margin-based robustness.

2. Spatial Predicates and Classifier Construction

Classical STL predicates operate as $f(s[k])>0$ for some real-valued function $f$. In SVM-STL, these are generalized to $h_j$, which are learned spatial classifiers operating on each spatio-temporal frame (e.g., an image, grid of concentrations).

Construction of $h_j$:

• Extract features using a pre-trained CNN feature extractor $$f_{cnn}:\mathbb{R}^{L\times W \times C} \to \mathbb{R}^m$$.
• Form a one-vs-rest binary classification problem for each spatial class $j$.
• Solve the convex SVM: $$\min_{\omega_j, b_j} \|\omega_j\| \quad \text{s.t. } \ell^{(i)} \big(\omega_j^T f_{cnn}(I^{(i)}) + b_j\big) \ge 1, \quad i=1,\dots,N$$
• Define $$h_j(I) = \frac{\omega_j^T f_{cnn}(I) + b_j}{\|\omega_j\|}$$, which provides a normalized margin indicating the presence of spatial pattern $j$.

These spatial classifiers can thus detect complex visual patterns, acting as the bridge from geometric structure to Boolean/spatial logic predicates.

3. Learning SVM-STL Specifications from Data

The framework for learning SVM-STL formulas from data consists of several pipeline stages:

3.1 Clustering unlabeled spatial data:

• Treat spatial frames $I^{(i)}$ as points in the CNN feature space.
• Run $k$-means clustering (optimized via Particle Swarm Optimization, PSO) on feature vectors, yielding $n_I$ spatial classes.

3.2 Learning predicates (SVMs):

• For each cluster $j$, perform one-vs-rest classification to train $h_j$.

3.3 Clustering spatio-temporal trajectories:

• Map each trajectory $S$ to a time series $s[k] = (h_1(S[k]), \dots, h_{n_I}(S[k]))$.
• Cluster these multivariate time series using dynamic time warping (DTW) distance and PSO-tuned $k$-means.

3.4 Temporal logical structure:

• Using the above-labeled trajectories and atomic primitives $$\mathcal{P} = \{\mathcal{F}_{[a,b]}(h_j>r), \mathcal{G}_{[a,b]}(h_j\le r)\}$$, learn temporal formulas:
  • Fit small depth-2 decision trees (impurity = misclassification gain).
  • Apply AdaBoost to aggregate $K$ such trees into a final SVM-STL classifier.

The result is a compositional spatio-temporal logic formula in which spatial structure is captured via learned image classifiers and temporal evolution is encoded via STL-like operators (Alsalehi et al., 2021).

4. Parameter Synthesis for Spatio-Temporal Specifications

Given a parametric SVM-STL formula $\varphi_\theta$ and a parametric dynamical system $S_\pi$, the problem is to identify system parameters $\pi^*$ that maximize the robustness of $\varphi_\theta$ on the induced trajectory $S_\pi$: $$\pi^* = \arg\max_{\pi\in\Pi} \rho(S_\pi, \varphi_\theta, 0)$$ Since the robustness function is non-differentiable, parameter optimization employs Particle Swarm Optimization (PSO): $$\begin{aligned} &v_i \leftarrow W v_i + c_1 \operatorname{rand}() (pbest_i-\pi_i) + c_2 \operatorname{rand}() (gbest-\pi_i) \ &\pi_i \leftarrow \pi_i + v_i \end{aligned}$$ This approach permits synthesis of, for example, reaction-diffusion system parameters that enforce the emergence of specified spatial patterns with temporal constraints.

5. Illustrative Case: Reaction-Diffusion Patterns

In the reaction-diffusion testbed, the system evolves on a $32\times 32$ grid involving species $x^1_{i,j}, x^2_{i,j}$ with spatio-temporal couplings; system trajectories $S$ are image sequences representing species concentrations.

Learned SVM-STL Specification Example:

$$\varphi_3 = \varphi_{31}^{2.8} \wedge \varphi_{32}^{0.6} \wedge \varphi_{33}^{0.1}$$

Each $\varphi_{3k}$ is a shallow decision tree formula combining temporal windows and spatial predicates, such as: $$\varphi_{31} = \left(\mathcal{G}_{[19,49]} h_1 \le -4.0 \wedge \mathcal{G}_{[36,47]} h_2 > 3.1\right) \vee \left(\neg \mathcal{G}_{[19,49]} h_1 \le -4.0 \wedge \mathcal{F}_{[38,40]} h_2 > 4.2\right)$$ Here $h_1$ and $h_2$ distinguish “no spots” vs. “spots” and the windowing captures phase transitions in pattern formation. Parameter synthesis identifies diffusion coefficients $(D_1^*, D_2^*)$ favoring desired spatial patterns such as “large spots.” Robustness $\rho \approx 0.11$ indicates margin of pattern satisfaction under optimized parameters (Alsalehi et al., 2021).

6. Expressiveness, Complexity, and Applications

SVM-STL achieves high expressiveness by coupling per-frame spatial classification with full temporal logical structure:

• Each frame is embedded as a vector of $h_j$-values, yielding data-driven detection of complex spatial types.
• Temporal formulas over spatial classes enable compositional, phase-structured modeling.

Complexity:

• Training: $n_I$ convex SVMs ($\text{poly}(N_I)$) and boosting over decision trees ($\text{poly}(N_S, K)$).
• Monitoring: $\mathcal{O}(|\varphi|\cdot T)$ for trajectory of length $T$.
• Parameter synthesis: heuristic, scaling linearly with the product of PSO particles, iterations, and system simulation cost.

Applications:

• Formal verification and automatic parameter tuning of spatio-temporal phenomena in reaction-diffusion systems, swarm robotics, smart-city monitoring, and anomaly detection in cyber-physical systems—specifically where interpretable, time-varying spatial patterns are key targets (Alsalehi et al., 2021).

7. Broader Connections and Research Context

SpaTiaL, exemplified by the SVM-STL framework, emerges as a second-generation approach for expressing learned, data-driven specifications in high-dimensional spatio-temporal systems. This approach marks a shift from hand-tuned logical templates toward logics whose atomic predicates are induced via machine learning from raw system data, enabling automated discovery and operationalization of domain-relevant spatial concepts. As research progresses, extensions to spatial logics with richer geometric primitives, finer-grained temporal operators, and robust online monitoring infrastructures are plausible developments. The SVM-STL methodology stands as a current prototype, distinct from pure hand-crafted logics and from older, limited spatial operators, in supporting practical, interpretable spatio-temporal logic specification and learning (Alsalehi et al., 2021).