AGM Robustness Interval Semantics

• AGM Robustness Interval Semantics is a quantitative framework assigning interval bounds to partial trajectories, enabling robust reasoning in STL.
• It replaces classical min/max operations with arithmetic–geometric mean aggregation to smooth and distribute the influence of all subformulae.
• The incremental monitoring algorithm updates interval estimates in real time, supporting efficient tree construction and decision-making in RRTη planning.

Searching arXiv for papers on AGM robustness interval semantics and adjacent AGM semantics. I’m checking arXiv for directly relevant papers and adjacent semantic frameworks. AGM robustness interval semantics is an interval-based quantitative semantics for partial trajectories introduced in the sampling-based planning framework RRT$^\eta$, where AGM denotes Arithmetic-Geometric Mean robustness for Signal Temporal Logic (STL). Rather than evaluating only complete trajectories, it assigns each prefix $\mathbf{s}_{t_0,t'}$ and formula $\phi$ an interval

$$[\eta]_{\mathbf{s}_{t'},\phi}=[\underline{\eta},\overline{\eta}]$$

that soundly contains the true AGM robustness of every possible completion of that prefix. The construction extends the normalized, signed AGM robustness semantics for STL, originally proposed as an alternative to classical min-max robustness, to incremental monitoring, steering, pruning, and rewiring during sampling-based tree construction (Ahmad et al., 18 Feb 2026, Mehdipour et al., 2019).

1. Foundations in Arithmetic-Geometric Mean robustness

The underlying robustness notion is the AGM robustness semantics for STL. In the original AGM formulation, the robustness score $\eta(\varphi,S,t)$ is normalized to $[-1,1]$, with positive values denoting satisfaction, negative values denoting violation, and magnitude measuring how strongly the specification is satisfied or violated (Mehdipour et al., 2019). For predicates, the base cases include

$\eta(\top,S,t)=1,\qquad \eta(\bot,S,t)=-1,$

and for a predicate of the form $\varphi:s_i-\pi\ge 0$,

$\eta(\varphi,S,t)=\frac{1}{2}(s_i[t]-\pi).$

A central departure from classical STL robustness is the replacement of min/max aggregation by arithmetic and geometric means. The positive/negative decomposition is defined by

$$[F]_+ := \begin{cases} F, & F>0\ 0, & \text{otherwise} \end{cases} \qquad [F]_- := -[-F]_+,$$

so that $\mathbf{s}_{t_0,t'}$0 (Ahmad et al., 18 Feb 2026). For robustness values $\mathbf{s}_{t_0,t'}$1, the AGM operators are

$\mathbf{s}_{t_0,t'}$2

and

$\mathbf{s}_{t_0,t'}$3

In recursive STL form, AGM robustness is defined over signal segments $\mathbf{s}_{t_0,t'}$4. The Boolean cases use $\mathbf{s}_{t_0,t'}$5 and $\mathbf{s}_{t_0,t'}$6, while the temporal operators $\mathbf{s}_{t_0,t'}$7 and $\mathbf{s}_{t_0,t'}$8 aggregate over the relevant time window (Ahmad et al., 18 Feb 2026). This yields a semantics in which all relevant subformulae and time points influence the robustness value, rather than a single extremal witness.

The contrast with standard robustness is explicit. Classical robustness uses $\mathbf{s}_{t_0,t'}$9 for $\phi$0 and $\phi$1, and $\phi$2 for $\phi$3 and $\phi$4, so a single “critical” time point or subformula dominates the result (Mehdipour et al., 2019, Ahmad et al., 18 Feb 2026). AGM robustness instead spreads influence across all relevant subformulae and time points. The original STL paper further states that $\phi$5 is smooth in $\phi$6 almost everywhere, and that its gradient with respect to relevant signal components is non-zero wherever smooth (Mehdipour et al., 2019).

2. Interval semantics for partial trajectories

AGM robustness interval semantics addresses the fact that sampling-based planners construct trajectories incrementally. At time $\phi$7, only a prefix $\phi$8 is available, so the exact robustness of the eventual trajectory is unknown. The interval semantics therefore works over

$\phi$9

and associates to each prefix and formula an interval that bounds all robustness values achievable by completions (Ahmad et al., 18 Feb 2026).

If $$[\eta]_{\mathbf{s}_{t'},\phi}=[\underline{\eta},\overline{\eta}]$$0 denotes the set of completions of the prefix, the soundness requirement is

$$[\eta]_{\mathbf{s}_{t'},\phi}=[\underline{\eta},\overline{\eta}]$$1

This is the formal guarantee that makes the interval semantics safe for planning: no feasible completion is assigned a robustness value outside the maintained interval (Ahmad et al., 18 Feb 2026).

The interval arithmetic mirrors AGM aggregation. For interval inputs $$[\eta]_{\mathbf{s}_{t'},\phi}=[\underline{\eta},\overline{\eta}]$$2,

$$[\eta]_{\mathbf{s}_{t'},\phi}=[\underline{\eta},\overline{\eta}]$$3

and

$$[\eta]_{\mathbf{s}_{t'},\phi}=[\underline{\eta},\overline{\eta}]$$4

These rules are then used recursively by the interval monitor (Ahmad et al., 18 Feb 2026).

Temporal operators require special handling because only part of the temporal window may have been observed. If the current time has not yet reached the temporal window, the interval is empty. If the prefix already covers the entire temporal interval, the interval collapses to a singleton exact value. Otherwise the observed segment contributes the currently known lower and upper information, while the unobserved future is handled conservatively so that no feasible completion is excluded (Ahmad et al., 18 Feb 2026). For the eventual operator, the appendix gives

$$[\eta]_{\mathbf{s}_{t'},\phi}=[\underline{\eta},\overline{\eta}]$$5

A common misconception is to treat the interval merely as a heuristic uncertainty band. In the RRT$$[\eta]_{\mathbf{s}_{t'},\phi}=[\underline{\eta},\overline{\eta}]$$6 construction, the interval is defined semantically by completion containment and not just by numerical approximation (Ahmad et al., 18 Feb 2026).

3. Incremental monitoring and exact interval updates

The monitoring procedure is designed to update intervals incrementally as new observations arrive. The main monitor is

$$[\eta]_{\mathbf{s}_{t'},\phi}=[\underline{\eta},\overline{\eta}]$$7

taking the formula $$[\eta]_{\mathbf{s}_{t'},\phi}=[\underline{\eta},\overline{\eta}]$$8, the current robustness interval, an auxiliary interval, the new observation $$[\eta]_{\mathbf{s}_{t'},\phi}=[\underline{\eta},\overline{\eta}]$$9, the current time $\eta(\varphi,S,t)$0, and the start time $\eta(\varphi,S,t)$1, and returning the updated interval (Ahmad et al., 18 Feb 2026).

To avoid recomputing robustness over the entire prefix, the framework introduces incremental modification functions. For disjunction,

$\eta(\varphi,S,t)$2

and for conjunction,

$\eta(\varphi,S,t)$3

The correctness lemma states that these updates exactly match full recomputation: $\eta(\varphi,S,t)$4 This exactness is significant because the planner obtains an incremental monitor without changing the underlying AGM aggregation logic (Ahmad et al., 18 Feb 2026).

The temporal subroutine $\eta(\varphi,S,t)$5 handles $\eta(\varphi,S,t)$6 and $\eta(\varphi,S,t)$7 by returning $\eta(\varphi,S,t)$8 before the temporal window starts, a singleton exact robustness value when the temporal window is fully observed, and otherwise an interval updated using the incremental AGM modification functions (Ahmad et al., 18 Feb 2026). The paper emphasizes that this yields complexity $\eta(\varphi,S,t)$9 per new observation rather than $[-1,1]$0 for recomputation from scratch.

4. Use in RRT$[-1,1]$1 planning

Within RRT$[-1,1]$2, the interval semantics is not an isolated monitor; it directly informs search decisions. The framework’s stated contributions are: AGM robustness interval semantics for reasoning about partial trajectories during tree construction, an efficient incremental monitoring algorithm computing these intervals, and enhanced Direction of Increasing Satisfaction vectors leveraging Fulfillment Priority Logic for principled objective composition (Ahmad et al., 18 Feb 2026).

During tree expansion, the planner computes a Direction of Increasing AGM Satisfaction, $[-1,1]$3. For Boolean compositions $[-1,1]$4, the composition rule uses interval estimates of the subformulae: $[-1,1]$5 The paper describes both a stochastic composition scheme using interval dominance with Bernoulli tie-breaking and an FPL-based composition using fulfillment values derived from interval bounds (Ahmad et al., 18 Feb 2026).

The interval bounds also control admissibility and rewiring. A node is only added if the upper bound is nonnegative,

$[-1,1]$6

meaning that satisfaction remains feasible for some completion. Rewiring seeks parent connections that improve the lower bound $[-1,1]$7 while preserving formula consistency (Ahmad et al., 18 Feb 2026). This implements a conservative planning strategy: feasibility is certified by the upper bound, whereas robustness quality is driven by the lower bound.

The framework is presented as synthesizing dynamically feasible control sequences satisfying STL specifications with high robustness while maintaining the probabilistic completeness and asymptotic optimality of RRT$[-1,1]$8 (Ahmad et al., 18 Feb 2026). Validation is reported on three robotic systems: a double integrator point robot, a unicycle mobile robot, and a 7-DOF robot arm.

5. Theoretical properties and semantic interpretation

Three properties organize the semantics. The first is soundness: $[-1,1]$9 for every completion $\eta(\top,S,t)=1,\qquad \eta(\bot,S,t)=-1,$0. The second is chain inclusion or monotonicity: $\eta(\top,S,t)=1,\qquad \eta(\bot,S,t)=-1,$1 so intervals shrink or remain unchanged as the prefix grows. The third is convergence: $\eta(\top,S,t)=1,\qquad \eta(\bot,S,t)=-1,$2 meaning that the interval becomes exact once the trajectory reaches the formula horizon (Ahmad et al., 18 Feb 2026).

These properties explain why the semantics is operationally useful. Soundness prevents exclusion of feasible completions. Monotonic shrinking provides progressively tighter guidance during tree growth. Convergence ensures that the interval semantics is conservative only while the trajectory is incomplete and reduces to ordinary AGM robustness on full trajectories (Ahmad et al., 18 Feb 2026).

The smoother optimization behavior relative to min-max robustness comes from the AGM aggregation itself. The RRT$\eta(\top,S,t)=1,\qquad \eta(\bot,S,t)=-1,$3 paper states that AGM robustness is smoother because it incorporates all relevant subformulae and time points rather than only an extremal witness, uses geometric mean in homogeneous-sign regimes to reflect gradual compound improvement, and uses arithmetic mean in mixed-sign regimes to produce a more continuous tradeoff (Ahmad et al., 18 Feb 2026). The earlier STL paper states that AGM robustness rewards satisfaction frequency and duration, for example interpreting $\eta(\top,S,t)=1,\qquad \eta(\bot,S,t)=-1,$4 as “eventually satisfy $\eta(\top,S,t)=1,\qquad \eta(\bot,S,t)=-1,$5 with the maximum possible satisfaction as early as possible and for as long as possible” (Mehdipour et al., 2019). This suggests that the interval semantics inherits not only a bound structure for partial trajectories but also the broader AGM preference for distributed, non-critical-point satisfaction.

6. Terminological scope and adjacent AGM semantics

The acronym “AGM” is overloaded across research areas. In RRT$\eta(\top,S,t)=1,\qquad \eta(\bot,S,t)=-1,$6 and the STL robustness lineage, AGM means Arithmetic-Geometric Mean robustness (Mehdipour et al., 2019, Ahmad et al., 18 Feb 2026). In belief change theory, AGM refers to the paradigm of minimal change associated with Alchourrón, Gärdenfors, and Makinson (Falakh et al., 2021). The two uses are distinct, though several papers in the second tradition are semantically adjacent because they also organize reasoning around model selection, closeness, or path dependence.

In generalized AGM-style belief revision over Tarskian logics, revision is characterized by a base-indexed preference relation over interpretations such that

$\eta(\top,S,t)=1,\qquad \eta(\bot,S,t)=-1,$7

with totality required but transitivity not always available (Falakh et al., 2021). A Kripke-Lewis semantics for contraction and revision uses belief relations together with selection functions, yielding characterizations such as

$\eta(\top,S,t)=1,\qquad \eta(\bot,S,t)=-1,$8

for contraction, and

$\eta(\top,S,t)=1,\qquad \eta(\bot,S,t)=-1,$9

for revision (Bonanno, 2023, Bonanno, 2023). These frameworks do not provide AGM robustness interval semantics in the RRT$\varphi:s_i-\pi\ge 0$0 sense, but they are adjacent semantic characterizations based on closest-state selection and minimal change.

The distinction becomes sharper in trace-indexed work on constructive AGM belief revision. There, belief states are indexed by traces $\varphi:s_i-\pi\ge 0$1, revision is defined by Levi revision over a constructive partial meet contraction algorithm, and the paper proves that strict sheaf-like composition fails for sequential AGM revision (Chen, 2 Jun 2026). This result shows that path-dependent belief revision is not functorial over trace concatenation. A plausible implication is that “interval semantics” in belief-change settings often concerns trace-local coherence and restriction maps rather than numerical robustness bounds.

Broader robustness semantics provide further contrast. “Omega-Regular Robustness” derives a semantic robustness preference relation from an $\varphi:s_i-\pi\ge 0$2-regular language and refines Tabuada–Neider’s five-valued semantics into an infinite domain (Fisman et al., 16 Mar 2025). This suggests that AGM robustness interval semantics belongs to a wider family of semantics that replace Boolean satisfaction by structured quantitative or ordered information, but its distinctive mechanism is the interval bounding of Arithmetic-Geometric Mean robustness over partial trajectories rather than a language-induced preorder or a belief-revision operator.