Temporal Logic Constraint Formulation

• Temporal Logic Constraint Formulation is the process of rigorously specifying and encoding time-evolving system requirements using logics like STL and LTL.
• It integrates mixed-integer programming and robust constraint techniques to handle uncertainties and ensure constraint satisfaction in control systems.
• This approach enhances control synthesis, formal verification, and policy synthesis across deterministic, stochastic, and adversarial environments.

Temporal logic constraint formulation is the rigorous process of specifying and encoding requirements about the evolution of system states or signals over time, where these requirements are expressed as temporal logic formulas and possibly subject to additional algebraic, logical, or probabilistic constraints. This methodology is central in modern control synthesis, policy synthesis, formal verification, planning, and learning, supporting both discrete and continuous dynamics under deterministic, stochastic, or adversarial environments. The temporal logics involved include syntactically rich languages such as Signal Temporal Logic (STL), Linear Temporal Logic (LTL), and their many-valued or data-constraint extensions.

1. Temporal Logic Syntax and Quantitative Semantics

Signal Temporal Logic (STL) formulas are constructed over secondary signals $y: \mathbb{Z}_{\geq 0} \to \mathbb{R}^p$ via atomic predicates and a composition of Boolean and temporal operators: $$\varphi,\psi ::= \mu_i \mid \neg\varphi \mid \varphi \wedge \psi \mid \varphi \vee \psi \mid \Box_{[a,b]}\varphi \mid \Diamond_{[a,b]}\varphi \mid \varphi\,\mathcal{U}_{[a,b]}\psi$$ with atomic predicates $\mu_i := (y_i[t] \geq 0)$.

Quantitative semantics ("robustness") are recursively assigned by: $$\begin{aligned} \rho_y^{\mu_i}[t] &= y_i[t],\ \rho_y^{\neg\varphi}[t] &= -\rho_y^\varphi[t],\ \rho_y^{\varphi \wedge \psi}[t] &= \min(\rho_y^\varphi[t], \rho_y^\psi[t]),\ \rho_y^{\varphi \vee \psi}[t] &= \max(\rho_y^\varphi[t], \rho_y^\psi[t]),\ \rho_y^{\Box_{[a,b]}\varphi}[t] &= \min_{\tau\in[t+a, t+b]}\rho_y^\varphi[\tau],\ \rho_y^{\Diamond_{[a,b]}\varphi}[t] &= \max_{\tau\in[t+a, t+b]}\rho_y^\varphi[\tau],\ \rho_y^{\varphi\,\mathcal U_{[a,b]}\psi}[t] &= \max_{\tau\in[t+a, t+b]}\Bigl( \min\bigl(\rho_y^\psi[\tau],\min_{k\in[t,\tau]}\rho_y^\varphi[k]\bigr)\Bigr). \end{aligned}$$ A formula is satisfied at time $t$ iff $\rho_y^\varphi[t] \geq 0$.

2. Plant Models and Disturbance Robustification

The typical system is a discrete-time linear plant with bounded additive disturbance: $$x[t+1] = A x[t] + B u[t] + w[t],\qquad w[t] \in \mathcal{W}$$

$y[t] = C x[t] + D u[t] + e$

where $\mathcal{W}$ is a known polytope of admissible disturbances, and admissible controls $u[t] \in \mathcal{U}$ lie in a convex set. The predicted secondary signal over horizon $H$ is: $$y^H[t] = \Phi_0^H x[t] + \Phi_1^H u^H[t] + \Phi_2^H w^H[t] + \mathbf{1} \otimes e$$ Constraints must hold for all possible disturbance realizations, which is conservatively enforced on the "worst-case" corners of the disturbance polytope: $$y^{\omega, H}[t] = \Phi_0^H x[t] + \Phi_1^H u^H[t] + \omega\bigl(\Phi_2^H (\mathbf{1} \otimes W)\bigr) + \mathbf{1}\otimes e$$ where $W$ is the matrix of vertices of $\mathcal{W}$ and $\omega$ applies the row-wise minimum.

3. Mixed-Integer Program Encoding of Temporal Logic Constraints

Temporal and logical formulæ are encoded in mixed-integer linear programs (MILPs) using binary variables $z^\psi[\tau] \in \{0,1\}$ for each subformula at each time.

Atomic predicates: $$y_i^{\omega}[\tau] - M z^{\mu_i}[\tau] \leq 0,\quad y_i^{\omega}[\tau] + M (1 - z^{\mu_i}[\tau]) \geq 0$$

with a suitably large $M$.

Boolean connectives:

For conjunction ($z^\psi = \bigwedge_{i=1}^m z^{\varphi_i}$): $$z^\psi[\tau] \leq z^{\varphi_i}[\tau]\quad \forall i,\qquad z^\psi[\tau] \geq 1 - m + \sum_{i=1}^m z^{\varphi_i}[\tau]$$ For disjunction ($z^\psi = \bigvee_{i=1}^m z^{\varphi_i}$): $$z^\psi[\tau] \geq z^{\varphi_i}[\tau]\quad \forall i,\qquad z^\psi[\tau] \leq \sum_{i=1}^m z^{\varphi_i}[\tau]$$

Temporal operators:

• Eventually ($$z^\psi = \max_{\sigma \in [\tau+a,\tau+b]} z^\varphi[\sigma]$$): encoded as a disjunction.
• Always ($$z^\psi = \min_{\sigma \in [\tau+a,\tau+b]} z^\varphi[\sigma]$$): encoded as a conjunction.
• Until: $$z^\psi[\tau] = \max_{\sigma \in [\tau+a,\tau+b]} \left(z^{\varphi_2}[\sigma] \wedge \min_{k \in [\tau, \sigma]} z^{\varphi_1}[k]\right)$$ constructed via recursive combinations of previously defined connectives.

4. Constraint Formulation in Model Predictive Control

Temporal logic constraints are imposed within a finite-horizon MPC framework. To recover feasibility under robustification, a non-negative slack variable $\zeta \geq 0$ is introduced, softening all secondary-signal entries: $$y^{\text{soft}, \omega, H} = y^{\omega, H} + \mathbf{1}\zeta$$ The MPC optimization problem is then: $$\begin{aligned} \min_{u[t:t+H],\,\zeta \geq 0} \quad& \sum_{\tau = t}^{t+H} J(\hat{x}[\tau], u[\tau]) + M\zeta \ \text{s.t.}\quad & x[\tau+1] = A x[\tau] + B u[\tau] + w[\tau],\quad w[\tau] \in \mathcal{W} \ & y^{\text{soft}, \omega, H} = [\Phi_0 x[t] + \Phi_1 u^H[t] + \omega(\Phi_2(\mathbf{1}\otimes W)) + \mathbf{1}\otimes e] + \mathbf{1}\zeta \ & \rho_{y^{\text{soft}, \omega, H}}^\varphi[\tau] \geq 0\quad \forall\,\tau\in[t-h^\varphi, t+h_p] \ & u[\tau] \in \mathcal{U},\quad z \in \{0,1\} \end{aligned}$$ The big-M coefficient ensures slack is activated only when robust satisfaction is infeasible.

5. MILP Formulation: Variables, Constraints, Objective

Decision variables:

• Continuous: $u[t], \dots, u[t+H] \in \mathbb{R}^m$, $\hat{x}[t], \dots, \hat{x}[t+H] \in \mathbb{R}^n$, $\zeta \in \mathbb{R}_{\ge0}$.
• Binary: $z^\psi[\tau]$ for every subformula at each relevant $\tau$.

Constraints:

• System dynamics (unrolled over horizon).
• STL predicates via big-M.
• Boolean and temporal interrelations.
• Enforce robust STL satisfaction ($z^\varphi[\tau]=1$ throughout horizon).
• Input constraints, non-negativity of slack.

Objective: $$\min \sum_{\tau=t}^{t+H} J(\hat{x}[\tau], u[\tau]) + M\zeta$$ All constraints are linear (or convex quadratic if $J$ quadratic) except for binaries.

6. Computational Complexity and Practical Trade-offs

• Number of binary variables scales as $\#\text{subformulas} \times (H+h^\varphi+1)$.
• MILP (or MIQP) complexity is worst-case exponential in number of binaries.
• Real-time implementation is feasible only for formulas of modest size and horizon, or with formula/horizon-specific heuristics.
• The single-corner robustification is deliberately conservative and reduces the need to enumerate all disturbance vertices, trading complexity for soundness.

7. Implications and Practical Implementation

Robust temporal logic constraint formulation, as instantiated in MILP (or MIQP), enables controller synthesis for linear systems where rich temporal specifications must be respected under bounded disturbances (Sadraddini et al., 2015). The approach is strictly constructive: formulas are transformed into mixed-integer encodings that guarantee robust satisfaction whenever feasible and minimal violation otherwise. This direct translatability from temporal logic to MILP supports rapid prototyping, rigorous verification, and deployment in advanced automation domains, including embedded control, safety-critical monitoring, and intelligent autonomy. The framework is compatible with well-established optimization software, subject to computational complexity determined by formula structure and horizon length, and can be extended with risk-based tightenings and constraint learning.