Signal-Value Freezing Operator in STL*

• Signal-value freezing operator is an extension of STL that binds dynamic signal values to enable comparisons between current and past (frozen) values.
• It increases expressiveness by allowing explicit formulations of amplitude jumps, local extrema, and adaptive threshold crossings in continuous and hybrid systems.
• The operator utilizes nested freeze variables with efficient monitoring algorithms that balance robustness and computational complexity.

The signal-value freezing operator is a syntactic and semantic extension to Signal Temporal Logic (STL) that enables explicit reference to signal values observed at dynamically determined time points within a trace, via so-called "freeze" or "value-freezing" variables. This extension, formalized as STL*, empowers temporal property specifications to compare current signal values with those captured at earlier or branching moments, permitting the articulation of behavioral properties beyond the expressive capacity of classical STL. In this framework, the value-freezing operator is fundamental for advanced temporal pattern detection in continuous and hybrid systems, allowing, for example, the explicit formulation of amplitude jumps, local extrema, and adaptive threshold crossing properties.

1. Syntax and Semantics of the Value-Freezing Operator

STL* introduces a freeze operator, denoted as $*_i$ in one formulation (Brim et al., 2013) and as $x^*.\varphi$ in another (Ghorbel et al., 2024), that captures the current value (or time) of a signal component at the instant of operator application and binds it to a "freeze slot" or freeze variable. The formal grammar extends STL with:

$$\varphi ::= \mu \mid \top \mid \neg\varphi \mid \varphi_1 \lor \varphi_2 \mid \varphi_1\;\mathbf U_{I}\;\varphi_2 \mid *_i\,\varphi,$$

where $\mu$ is a linear predicate or relational constraint possibly referencing both current and frozen signal values, and $*_i\,\varphi$ freezes the current contextual value into slot $i$ for use in the subformula $\varphi$ (Brim et al., 2013).

The semantics are defined over tuples $(s, t, t^*)$ or, for sampled traces, $(\pi, i, \nu)$, where $t^*$ or $\nu$ tracks the environment of frozen time instants or signal values across the finite set of freeze variables. The freeze operator updates $t^*$ (or $\nu$) with the current time (or value) for slot $i$ (or variable $x^*$) and evaluates the body of the subformula accordingly (Brim et al., 2013, Ghorbel et al., 2024). Atomic predicates can thus compare $s_j(t)$ with $s_j(t^*_i)$ or more generally, functions $f(s)$ with $f(s^*)$.

The key semantic move is: $$(s, t, t^*) \models *_i\,\varphi \iff (s, t, \mathsf{store}_i(t)\,t^*) \models \varphi$$ or, for value freezing,

$$(\pi, i, \nu) \models x^*.\varphi \iff (\pi, i, \nu[x^* := \sigma^k_i]) \models \varphi$$

where $\sigma^k_i$ is the $k$th component of the sampled signal at index $i$.

This mechanism allows recursive and, crucially, nested freezing, where each enclosing freeze operator can bind a distinct variable, resulting in a hierarchical frozen-value environment affecting the evaluation of complex nested subformulas (Ghorbel et al., 2024).

2. Quantitative Robustness Semantics

STL* extends not only the Boolean semantics of STL but also the quantitative robustness semantics. The robustness function $\rho(\varphi, s, t, t^*)$ (or equivalently $\rho(\varphi, \pi, i, \nu)$) generalizes the signed signal-to-specification distance under the frozen-variable context. For predicates involving frozen values, robustness is defined as:

$$\rho(\mu, s, t, t^*) = \frac{\sum_{j}a_{0j}s_j(t) + \sum_{i,j}a_{ij}s_j(t^*_i) + b}{\sqrt{\sum_j a_{0j}^2} + \sum_{i\in I}\sqrt{\sum_j a_{ij}^2}}$$

for linear $\mu$ (Brim et al., 2013), and analogously for general relational predicates involving frozen bindings (Ghorbel et al., 2024).

The robustness of composite formulas propagates following standard min/max conventions, but the freeze operator updates the frozen environment before evaluating the subformula: $$\rho(*_i\,\varphi, s, t, t^*) = \rho(\varphi, s, t, \mathsf{store}_i(t)\,t^*)$$ or, with value freezing,

$$\rho(x^*.\varphi, i, \nu) = \rho(\varphi, i, \nu[x^* := \sigma^k_i])$$

Robustness values provide a lower bound for the signed distance to violation; positive values entail satisfaction, negative values entail falsification, and zero indicates criticality. Freeze operators generally lead to smaller robustness margins than purely temporal operators, as value-based requirements are semantically stricter (Brim et al., 2013).

3. Expressiveness and Illustrative Examples

The value-freezing operator substantially increases the expressive power of temporal logic specifications. It enables reference to signal values at nondeterministically determined past (or present) instants, a capability not realizable with classical STL or temporal logic formulas without freeze.

Typical expressivity patterns include:

• Quantified differences: "Within 5 time units, the signal must increase by at least 8"

$*_{1}\,\mathbf F_{[0,5]}(x \ge x^*_{1} + 8)$

• Local extrema detection: Expressing and enforcing local maxima or minima via freeze-nestings
• Rectangle pulses of unknown amplitude: Detecting unconstrained jump amplitudes and their plateaus using three nested freezes (Ghorbel et al., 2024)
• Peak detection: Augmenting epidemic models to assert that an infection peak occurs by freezing the value at peak and comparing subsequent values (Brim et al., 2013)

Standard STL expresses properties referable only to absolute time windows and lacks the machinery for dynamic value-binding required for such patterns.

4. Monitoring Algorithms and Complexity

Robust and Boolean monitoring under the value-freezing extension requires systematic enumeration of possible freeze environments. Monitoring proceeds by:

• Building the formula tree and identifying all freeze subformulas
• For each combination of freeze variable bindings, recursively evaluating the subformulas under the induced frozen environment
• Maintaining and merging intervals over which subformula truth values remain constant

For a signal sampled at $N$ points and $V$ freeze variables (possibly nested), the naïve algorithmic complexity grows as $O(N^V \cdot N)$ or worse, as each freeze nesting multiplies the state space of possible environments (Brim et al., 2013, Ghorbel et al., 2024). However, an acceleration heuristic that exploits constancy of truth values over contiguous intervals enables practical scaling: by storing intervals of constant valuation and updating only as freeze bindings change, monitoring complexity reduces to $O(N^V \cdot \max(\log N, |\varphi|\cdot M))$ where $M$ is the maximum number of constant intervals per subformula (Ghorbel et al., 2024).

Robustness computation employs a binary-search technique: rewriting the formula to check “robustness $> r$” for candidate $r$ and monitoring the resulting Booleanized formula (Ghorbel et al., 2024).

Empirical results show monitoring traces of length $N = 10^4$ with two nested freeze variables in under 30 seconds (Boolean), and $\approx 900$ seconds for quantitative robustness to $2\%$ accuracy; three freeze variables are feasible for $N \approx 10^3$ (Ghorbel et al., 2024).

5. Practical Applications and Case Studies

The value-freezing operator has been empirically validated in various dynamic systems models:

• Epidemic modeling (SIR): Freeze-augmented temporal properties permit precise specification of peak infection events and decline, exceeding the possibilities of standard STL. Robustness for freeze-augmented properties exhibits narrower "robust tubes" (distance to violation) compared to unaugmented formulas, as the satisfaction sets are smaller (Brim et al., 2013).
• Predator–prey (Lotka–Volterra): Oscillation amplitude constraints can be specified by freezing past extrema and checking future drops, yielding robustness landscapes with respect to system parameters (Brim et al., 2013).
• Signal processing: Detection of rectangular pulses and patterns involving unknown jump amplitudes, as well as local extrema, as illustrated by nested freeze logic over sampled traces (Ghorbel et al., 2024).

A plausible implication is that in practical, engineering-driven temporal property verification, the value-freezing extension enables the direct formulation and monitoring of patterns previously accessible only via unsound or incomplete analytical approximations.

6. Implementation and Tool Support

The Parasim tool (Brim et al., 2013) implements the STL* robustness monitoring algorithm for continuous signals, exploiting memoization and sliding-window optimizations for efficient handling of the min/max computations over time windows. For the discrete-time case, recent implementations realize the interval-acceleration method, supporting two to three nested freeze variables with empirical scaling to moderately large traces (Ghorbel et al., 2024). These implementations are the first to tackle nested value-freezing operators beyond depth one, demonstrating the viability of STL* monitoring in realistic scenarios on moderate hardware.

7. Theoretical and Practical Significance

The signal-value freezing operator fundamentally enhances the temporal logic toolbox for hybrid systems, embedded control, and data-driven verification by allowing expressive reference to signal histories in a logically coherent, compositional manner. Formulas with value freezing typically yield lower robustness, reflecting stricter and more nuanced behavioral requirements. Monitoring algorithms employing freeze-based interval acceleration make the application of STL* feasible for practical systems, at moderate combinatorial cost. The development of STL* and efficient monitoring for nested freeze operators position freeze-augmented STL as a uniquely capable formalism for advanced temporal pattern specification and verification (Brim et al., 2013, Ghorbel et al., 2024).