Robust Interval Semantics

• Robust interval semantics is a framework that converts exact point values into interval representations, ensuring that every operation encloses the true result under uncertainty.
• It employs directed rounding and interval arithmetic to redefine previously undefined operations, enabling error-free and certified numerical verification.
• The approach finds applications in numerical computing, probabilistic verification, robust control, and neural network and quantum computation through techniques like IMDPs and temporal logics.

Robust interval semantics is a class of mathematical frameworks and formal methods in which the semantics of numerical, probabilistic, logical, or optimization objects is generalized from point-valued (exact) to interval-valued (set-based) representations. These frameworks enforce that all relevant operations, computations, or deductions yield intervals that provably (often minimally) enclose the range of all possible true outcomes under specified uncertainty, rounding, or non-determinism. Robust interval semantics is central to verification, error estimation, robust control, optimization, and formal reasoning under uncertainty across numerical, symbolic, and hybrid systems.

1. Foundational Principles and Formalization

Robust interval semantics extends traditional concrete semantics over real numbers, floating-point values, distributions, or functions to interval-valued abstractions. The main principles include:

• Interval Representation: Each base value $x$ is represented as an interval $[a, b]$, sometimes degenerate (with $a = b$) or proper ($a < b$), encoding uncertainty due to rounding, finite precision, epistemic uncertainty, or model perturbation.
• Soundness: For every admissible concrete object within the uncertainty (e.g., true real number, trajectory, probability, quantum state), its corresponding value is guaranteed to lie within the computed interval.
• Closure Under Operations: All algebraic, logical, or computational operations (arithmetic, min/max, predicates, system dynamics, etc.) are extended to the interval domain such that the result interval is a provable enclosure of all possible outcomes for all variations of the input within the input intervals.

A canonical example is robust interval semantics for floating-point numbers, where every finite nonzero float $x$ is interpreted as the point-interval $[x, x]$; signed zeroes as $[0, m]$ or $[-m, 0]$ ($m$ denoting the least positive normal/subnormal number); and infinities as $[M, +\infty)$, $(-\infty, -M]$ respectively. Arithmetic operations are reinterpreted as operations over intervals, with roundings realized as bound selectors under appropriate rounding modes, ensuring all results are intervals with no undefined or exceptional values (0810.4196).

General interval semantics can be succinctly summarized with the following operational form for two intervals $X = [a, b], Y = [c, d]$:

Operation	Definition
$X \oplus Y$	$[a + c,\, b + d]$
$X \ominus Y$	$[a - d,\, b - c]$
$X \otimes Y$	$[\min\{ac, ad, bc, bd\},\, \max\{ac, ad, bc, bd\}]$
$X \oslash Y$	$$[\min\{a/c, a/d, b/c, b/d\},\, \max\{a/c, a/d, b/c, b/d\}]$$

2. Robust Interval Semantics in Numerical Computing

In standard floating-point arithmetic, interval semantics clarify and robustify numerical computation by:

• Defining all previously undefined operations: E.g., $0 \ast \infty, \infty - \infty, \infty/\infty, 0/0$ yield well-defined intervals such as $[0,+\infty)$ or $\mathbb{R}$, rather than NaN-exceptions. No operation results in an error; every algebraic expression produces an interval result (0810.4196).
• Rounding as Bound Selection: Directed rounding modes pick out interval endpoints. For example, roundToward$+\infty$ gives the upper bound of the result interval, roundToward$-\infty$ the lower bound; roundToZero and roundToNearest select corner endpoints or the closest endpoint to the real value, respectively.
• Verification via Enclosures: Every floating point program, under interval semantics, computes an interval such that the result of every possible real-valued execution is guaranteed to be inside the reported interval—yielding a foundation for certified numerical verification.
• Hardware Realizability: Point-interval operands, and more generally interval semantics, are implementable with only minor FPU changes; bounds may be computed using parallel FPUs operating in directed rounding modes and require no NaN-logic (0810.4196).

For error analysis and robust numerical estimation, further refinements such as error intervals, movability flags, and input search are introduced. Here, error intervals track the possible versus guaranteed occurrence of domain errors (e.g., taking $\sqrt{x}$ for $x < 0$), and movability flags indicate whether refining the computation precision could reduce the output interval. Input search algorithms partition the input space to focus validation and sampling efficiently on the valid, samplable points. These extensions have demonstrated practical success at refining error diagnosis, eliminating futile computation, and improving robustness over state-of-the-art technical computing environments (Flatt et al., 2021).

3. Robust Interval Semantics in Probability, Optimization, and Control

Robust interval semantics underpin several rigorous frameworks in probabilistic verification, optimization, and robust synthesis:

• Interval Markov Decision Processes (IMDPs): Here, transitions are specified by intervals $$I(s,a,s') = [\underline{p}_{s,a,s'}, \overline{p}_{s,a,s'}]$$, representing epistemic or model uncertainty (Hahn et al., 2017, Huynh et al., 3 Oct 2025). Strategies are called robust if, for every admissible realization of transition probabilities (within these intervals), key objectives (e.g., reachability, long-run reward) are met in the worst case. Multi-objective robust synthesis is generally PSPACE-hard, but practical Pareto-approximation algorithms based on robust value iteration and mixed-integer linear programming are available (Hahn et al., 2017, Huynh et al., 3 Oct 2025).
• Permissive Controller Synthesis Under Interval Uncertainty: Recent advances provide fully robust, maximally permissive controllers for IMDPs, ensuring that all compliant strategies satisfy temporal/reward constraints under all admissible uncertainties, with scalable MILP encodings (Huynh et al., 3 Oct 2025).
• Interval Linear Programming with Forall-Exists Robustness: In robust optimization, solutions are sought that remain optimal for all realizations of some coefficients within intervals, provided the right-hand sides can adapt within their intervals. Execution relies on checking feasibility through interval linear algebra and verifying optimality via enumeration or dual-based LP tests. The robust-optimal set is always a finite union of convex polyhedra, but may be nonconvex or disconnected (Hladík, 2014).

4. Robust Interval Semantics in Temporal Logics and Formal Methods

Extended to temporal logics, robust interval semantics encode satisfaction of formulas over uncertain or partially observed signals:

• Interval Signal Temporal Logic (I-STL): Given interval-valued signals and interval inclusion predicates, the semantics of each STL operator (min, max, etc.) is lifted to its minimal inclusion function. The resulting robustness interval is provably a natural inclusion (enclosure) for the true robustness over all signals in the interval tube. This enables monitoring and controller synthesis over uncertain cyberphysical signals with negligible overhead and soundness guarantees (Baird et al., 2023).
• Robust Interval Semantics for Online Monitoring: In online scenarios, for partial traces (prefixes), robust interval semantics compute the tightest possible interval for satisfaction values over all completions. This supports early detection of inevitable property satisfaction or violation, reducing simulation time and computational burden in high-value cases (Deshmukh et al., 2015).

Domain	Robust Interval Object	Main Application
Floating Point	$[x, x]$, $[0, m]$, $[M, +\infty)$ etc.	Error-free, total arithmetic, verification
IMDPs	$[\underline{p}, \overline{p}]$ transition	Robust control and synthesis
Linear Program	$[A], [c], [b]$ (forall-exists quantif.)	Optimization under data uncertainty
Temporal Logic	$[\rho]^{\mathbb{I}}(\varphi, [x], t)$	Sound monitoring, synthesis

5. Robust Interval Semantics in Machine Learning and Quantum Computation

• Neural Network Verification: Abstract interpretation using the interval domain provides sound over-approximation of the output range of neural networks for input intervals ("boxes"). The interval universal approximation theorem (IUA) establishes that for any continuous function, neural networks with (suitable) activation can, in principle, approximate the set semantics up to arbitrary precision—but the problem of efficiently constructing such networks or tight approximations is computationally $\Delta_2^P$-intermediate, fundamentally limiting practicality for large models (Wang et al., 2020).
• Quantization-aware Robustness: For quantized neural networks, quantization-aware interval bound propagation (QA-IBP) extends interval semantics into discrete domains, enabling both robust training and complete verification procedures over low-bit QNNs, with soundness and completeness guarantees under the $L_\infty$ norm (Lechner et al., 2022).
• Quantum Robustness Certification: In near-term quantum algorithms, robust interval semantics certify expectation values (e.g., ground state energies) under noise and imperfect state preparation. Using fidelity certificates and moments of observables, robustness intervals—often produced via semi-definite programming or Gram-matrix constraints—provide explicit, guaranteed error bars for all ideal states within a given fidelity (Weber et al., 2021).

6. Theoretical Properties and Practical Impact

Robust interval semantics yield desirable theoretical properties:

• Closure and Totality: All basic operations are closed, with no undefined or exceptional cases.
• Soundness: True values (of computations, properties, optimal values) are always enclosed, permitting sound verification and worst-case analysis.
• Compositionality: Interval semantics are generally compositional, supporting modular reasoning.
• Algorithmic Complexity: For linear problems, evaluation is often tractable; for nonlinear problems (e.g., range of neural nets), fundamental intractability results hold.

Practically, robust interval semantics underpin several advances:

• Sound, early-aborting runtime verification of CPS models (Deshmukh et al., 2015).
• Maximally permissive robust controllers and shields in robotics (Huynh et al., 3 Oct 2025).
• Reliable error estimation in technical computing environments (Flatt et al., 2021).
• Universal set-approximation in deep learning verification (Wang et al., 2020).
• Explicit quantum error bars with minimal measurement overhead (Weber et al., 2021).

7. Limitations and Open Directions

Despite broad applicability, key limitations are apparent:

• Precision vs. Efficiency: There is an inherent trade-off between tightness of interval enclosures (precision) and algorithmic tractability. For expressive models (e.g., deep nets) or high-uncertainty settings, intervals may become excessively wide or difficult to compute tightly (Wang et al., 2020).
• Completeness: In nonconvex or nonlinear domains, interval abstractions can be too conservative for fine-grained verification.
• Domain-Specific Extensions: Extensions such as error intervals, movability flags, three-valued validity, and tailored boundary-lifting are required to address specific challenges in modeling, error handling, or optimization (Flatt et al., 2021).

Future directions include refinement to richer abstract domains (polyhedra, zonotopes), integration with probabilistic or set-valued analysis, more scalable optimization under interval uncertainty, and the development of domain-specific interval semantics for emerging areas such as stochastic hybrid systems, large-scale quantum computation, and safe AI.

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References

(0810.4196, Flatt et al., 2021, Hahn et al., 2017, Huynh et al., 3 Oct 2025, Hladík, 2014, Baird et al., 2023, Deshmukh et al., 2015, Wang et al., 2020, Lechner et al., 2022, Weber et al., 2021)