CT-STL: Cumulative-Time Signal Temporal Logic

• CT-STL is a temporal logic that generalizes STL by incorporating cumulative-time operators to specify minimum duration conditions, integral bounds, and rate-of-change constraints.
• It introduces novel grammar constructs for both discrete and continuous signals, enabling qualitative and quantitative reasoning via robust and smooth cumulative semantics.
• CT-STL supports efficient online monitoring and MILP-based control synthesis, yielding significant performance improvements in applications such as microgrid monitoring and autonomous robot missions.

Cumulative-Time Signal Temporal Logic (CT-STL) generalizes Signal Temporal Logic by enabling formal reasoning over cumulative-time objectives and duration-based requirements in cyber-physical systems, both at the semantic and algorithmic levels. CT-STL substantially extends STL's expressiveness, allowing direct specifications over total time spent in prescribed states, integral bounds on state trajectories, and rate-of-change constraints, crucial for control, monitoring, and verification.

1. Formal Syntax of CT-STL

CT-STL extends STL over discrete or continuous time by adding new grammar constructs for cumulative properties:

For discrete time signals $\xi: \mathbb{T} \to \mathbb{R}^n$, the CT-STL grammar includes:

$$\varphi ::= p\ \big|\ \neg\varphi\ \big|\ \varphi_1 \wedge \varphi_2\ \big|\ \varphi_1\,\mathbf{U}_I\,\varphi_2\ \big|\ \mathbf{C}^\tau_I \varphi$$

• $p$: atomic predicates, e.g., $g(\xi(t)) \geq c$.
• $\mathbf{U}_I$: interval until operator.
• $\mathbf{C}^\tau_I$: cumulative-time operator; holds if the nested formula $\varphi$ is satisfied at least $\tau$ time-steps over interval $I$ (Chen et al., 14 Apr 2025).

For continuous time signals $x: \mathbb{R}_{\ge 0} \to \mathbb{R}^n$, CT-STL additionally incorporates bounded-horizon integral and derivative predicates:

$$\varphi ::= \mu\ \big|\ \mu^i_{[a,b]}\ \big|\ \mu^d_+\ \big|\ \mu^d_-\ \big|\ \neg\varphi\ \big|\ \varphi_1\wedge\varphi_2\ \big|\ F_{[t_1,t_2]}\varphi$$

where:

• $$\mu^i_{[a,b]}: \int_{t+a}^{t+b}g(x(\tau))\,d\tau \geq c$$
• $\mu^d_+: \frac{d}{dt^+}g(x(t)) \geq c$, $\mu^d_-: \frac{d}{dt^-}g(x(t)) \geq c$ (Buyukkocak et al., 2021).

Boolean (negation, conjunction) and standard temporal operators (Finally/Globally) are derived as usual.

2. Semantic Framework: Qualitative and Quantitative

Qualitative Semantics

For any signal $\xi$, time $t$, and cumulative-time formula $\mathbf{C}^\tau_I\varphi$:

$$(\xi, t) \models \mathbf{C}^\tau_I\varphi \iff \sum_{t'\in t+I}[(\xi, t') \models \varphi] \geq \tau$$

That is, $\varphi$ holds for at least $\tau$ time-points in $t+I$ (Chen et al., 14 Apr 2025).

For integral predicates: $$(x, t) \models \mu^i_{[a,b]} \iff \int_{t+a}^{t+b}g(x(\tau))\,d\tau \geq c$$ Indicating satisfaction if the cumulative value meets the threshold (Buyukkocak et al., 2021).

Quantitative (Robustness) Semantics

A real robustness measure $\rho(\varphi, \xi, t)$ quantifies satisfaction, notably for cumulative-time constructs:

$$\rho(\mathbf{C}^{\tau}_I \varphi, \xi, t) = \max^{\tau}_{t'\in t+I} \rho(\varphi, \xi, t')$$

where $\max^\tau$ selects the $\lceil\tau\rceil$-th largest value over the window (Chen et al., 14 Apr 2025).

For integral predicates: $$r(x, \mu^i_{[a,b]}, t) = \int_{t+a}^{t+b}g(x(\tau))\,d\tau - c$$

Soundness and completeness are established structurally, ensuring that satisfaction in the qualitative semantics corresponds to positive robustness, and vice versa (Chen et al., 14 Apr 2025).

Smooth Cumulative Robustness

An alternative, differentiable cumulative semantics is proposed via positive/negative cumulative robustness, employing smooth surrogates for min, max, and rectifier functions, yielding quantifiers as in:

$$\widetilde{\max_\beta}(a_1,\ldots,a_m) = \frac{1}{\beta}\ln\left(\sum_{i=1}^m e^{\beta a_i}\right)$$

This enables gradient-based optimization and MPC for control synthesis (Haghighi et al., 2019).

3. Capturing Cumulative-Time and Rate Properties

Classic STL can only enforce state predicates instantaneously or throughout intervals; CT-STL allows explicit cumulative objectives:

• Cumulative-Time: Enforces “$\varphi$ must hold for at least $\tau$ time-units in any interval $I$”. Example: specifying “at least 599,984 ms of normal voltage in every 600,000 ms window” using $\mathbf{C}^{600000-16}_{[0,600000]}(v<1.7)$ (Chen et al., 14 Apr 2025).
• Integral Predicates: Express accumulation, e.g., “must travel at least 5 m in every 10 s window” via $\mu^i_{[-10,0]}$ (Buyukkocak et al., 2021).
• Derivative Predicates: Bound rates of change, such as acceleration limits with $\mu^d_+$ and $\mu^d_-$.

These constructs are pivotal for encoding performance, safety, and resource-usage constraints beyond instantaneous requirements.

4. Algorithmic Monitoring and Control Synthesis

Online Monitoring

Given observed partial traces, CT-STL exploits recursive interval-based satisfaction computations—robust satisfaction intervals (RoSI)—updated bottom-up. For cumulative-time operators, monitoring leverages sliding windows and two-heap data structures:

• Each new sample update: $O(\log \tau + \log(w-\tau))$ time complexity.
• Memory: $O(|\varphi| \cdot \max_n hor(\varphi))$ (Chen et al., 14 Apr 2025).

Early detection is facilitated when violation (or satisfaction) can be conclusively determined before trace completion; substantial simulation speedups are reported (e.g., $100 \times$–$1000 \times$ performance gain over naive approaches).

MILP Encoding and Control Synthesis

For control synthesis under CT-STL, satisfaction predicates are encoded as mixed-integer linear constraints:

• Instantaneous: Big-M constraint encodings.
• Integral: Aggregate over window, e.g., $\sum_{j=k+m}^{k+n-1}g(x_j)\,\delta t - c$ in MILP.
• Derivative: Finite-difference approximations.

Temporal and Boolean operators are imposed by constraints on indicator variables. The top-level CT-STL formula is enforced by $z^\Phi_0=1$. Control synthesis seeks feasible (and optimal) trajectories within such encodings via standard MILP solvers (Buyukkocak et al., 2021).

For smooth cumulative robustness, gradient-based optimization (projected gradient ascent) is applied, and in MPC scenarios, rolling-horizon optimization is performed at each time step (Haghighi et al., 2019).

5. Case Studies and Practical Applications

Microgrid Monitoring

For power-quality constraints (IEC 1547-2018), CT-STL specifies cumulative overvoltage requirements, such as:

• $\mathbf{C}^{600000-16}_{[0,600000]}(v<1.7)$: At least $599,984$ ms in every $600,000$ ms window must see $v<1.7$.
• Event-based offline and online monitoring validates early violation detection, saving $30\%$–$46\%$ runtime (Chen et al., 14 Apr 2025).

Artificial Pancreas

CT-STL encodes regulatory requirements (e.g., keep blood glucose within safe bounds for $>70\%$ of the day):

• Cumulative-time requirements for hypo/hyper/euglycemia are formalized using $\mathbf{C}^\tau_I$ and monitored online.
• Early falsification/satisfaction is observed in simulated patient traces (Chen et al., 14 Apr 2025).

Autonomous Robot Missions

Integral and derivative predicates specify cumulative distance, acceleration limits, and escape speeds in mission planning, rigorously enforcing smooth and energy-efficient trajectories. Comparative analysis demonstrates the impact of cumulative predicates on trajectory smoothness, control effort, and satisfaction of complex temporal tasks (Buyukkocak et al., 2021).

Other Domains

CT-STL is applicable in settings requiring time-in-state guarantees: medical alarms, occupancy detection, noise exposure, and general cyber-physical system requirements (Chen et al., 14 Apr 2025).

6. Limitations and Prospective Extensions

Current CT-STL monitoring is restricted to discrete-time signals and lower-bound cumulative requirements. Dense-time formulations would require measure-theoretic semantics. Monitoring upper bounds necessitates negation. Integration with falsification (e.g., S-TaLiRo) for systematic violation search, specification mining, runtime enforcement, and parameter-mining for cumulative thresholds represent active areas for extension (Chen et al., 14 Apr 2025).

Smooth cumulative semantics approximate the combinatorial aspects with continuous surrogates, trading exactness for tractability in control synthesis, suitable for model predictive control and policy optimization (Haghighi et al., 2019).

7. Connections to Related Temporal Logics

CT-STL unifies and generalizes prior approaches:

• STL’s instantaneously quantified predicates.
• Interval temporal logics expressing dwell-time, cumulative success, or rate constraints.
• Robustness-based synthesis frameworks, incorporating cumulative robustness and smooth optimization (Haghighi et al., 2019).

Cumulative time operators preserve STL’s compositional semantics and fit naturally within established verification and synthesis workflows, providing principled support for specifications involving duration, resource accumulation, and timing tolerances across cyber-physical systems, embedded control, and safety monitoring (Buyukkocak et al., 2021, Chen et al., 14 Apr 2025, Haghighi et al., 2019).