Admissible Explanation Space

• Admissible explanation space is a defined set of valid, semantically grounded inputs or hypotheses that yield interpretable model explanations.
• These spaces are constructed via feature mappings, hypothesis enumeration, or SMT-based region interpolation to ensure mathematical and semantic consistency.
• Empirical results demonstrate their role in enhancing attribution sparsity, hypothesis coverage, and sound region-based explanations in diverse applications.

An admissible explanation space defines a set of inputs, hypotheses, or features over which explanations of model predictions or scientific phenomena are valid, meaningful, and grounded in problem-specific semantics. This notion arises in multiple settings—including post-hoc XAI for time series, sampling-based hypothesis discovery, and logic-based neural network classification—with context-dependent definitions and requirements. Across applications, admissible explanation spaces delimit the universe of candidate explanations or input variations that are both interpretable to humans and mathematically consistent with a model, data, or task.

1. Formal Definitions Across Domains

The concept of admissible explanation space is instantiated differently depending on domain and goal:

• Attribution for time series models: An admissible explanation space $(E, \varphi, b, d_E)$ comprises a reversible, almost everywhere differentiable map $\varphi: X \rightarrow E \subset \mathbb{R}^m$ (with time series $X \subset \mathbb{R}^n$), an interpretable coordinate system in $E$, a baseline $b \in E$ so that $\varphi^{-1}(b)$ represents signal absence, and a metric $d_E$ such that small perturbations in $E$ yield bounded changes in $X$ (Rezaei et al., 2024). Requirements include invertibility ($\forall x \in X,\, \varphi^{-1}(\varphi(x)) = x$), (local) differentiability, baseline validity, interpretability, and metric regularity.
• Set-valued scientific hypothesis generation: In model-driven discovery, the admissible explanation space $\varphi: X \rightarrow E \subset \mathbb{R}^m$0 is the enumerated set of all hypotheses from some syntactic universe $\varphi: X \rightarrow E \subset \mathbb{R}^m$1 which are exactly consistent with given observations $\varphi: X \rightarrow E \subset \mathbb{R}^m$2; formally, $\varphi: X \rightarrow E \subset \mathbb{R}^m$3 with a deterministic validator $\varphi: X \rightarrow E \subset \mathbb{R}^m$4 (Chen et al., 17 Oct 2025).
• Neural network region explanations: For classifiers $\varphi: X \rightarrow E \subset \mathbb{R}^m$5, an admissible explanation space $\varphi: X \rightarrow E \subset \mathbb{R}^m$6 relative to a decision $\varphi: X \rightarrow E \subset \mathbb{R}^m$7 is any region such that $\varphi: X \rightarrow E \subset \mathbb{R}^m$8 and $\varphi: X \rightarrow E \subset \mathbb{R}^m$9 contains at least one $X \subset \mathbb{R}^n$0 classified as $X \subset \mathbb{R}^n$1 (Labbaf et al., 27 Nov 2025). In this context, $X \subset \mathbb{R}^n$2 is typically represented as a quantifier-free linear real arithmetic formula $X \subset \mathbb{R}^n$3.

2. Construction Methodologies

The strategies for constructing or operationalizing admissible explanation spaces vary by setting:

• Feature-space mappings for attribution: For time series, admissible spaces are built using transformations $X \subset \mathbb{R}^n$4 (e.g., FFT, STFT, time-difference), each with a corresponding inverse and interpretable basis. Standard XAI methods (IG, SHAP) are applied post hoc in $X \subset \mathbb{R}^n$5 by “wrapping” the model with $X \subset \mathbb{R}^n$6. All candidate spaces must have an invertible and differentiable mapping $X \subset \mathbb{R}^n$7 to enable gradient-based attribution (Rezaei et al., 2024).
• Explicit enumeration and validation: In set-valued explanation, $X \subset \mathbb{R}^n$8 is constructed by generating all candidate hypotheses and filtering with the validator $X \subset \mathbb{R}^n$9 (by domain: graph constraints, projection and gravity, Boolean equivalence). Canonicalization routines collapse task-specific symmetries to ensure distinctness (Chen et al., 17 Oct 2025).
• Craig interpolation and SMT: For neural network region explanations, the admissible set $E$0 is generated by encoding the network and class decision as a constraint system, then deriving an interpolant $E$1 from the unsatisfiable conjunction of a seed point $E$2 and the negated decision region. $E$3 then provably defines a region where $E$4 for all $E$5; optional refinement using unsatisfiable core minimization yields tighter regions (Labbaf et al., 27 Nov 2025).

3. Metrics and Evaluation

Different metrics are deployed to measure explanation quality:

• Attribution metrics: For time series, sparsity $E$6 and faithfulness flip rate $E$7 quantitatively assess the compactness and faithfulness of the attribution in various explanation spaces (Rezaei et al., 2024).
• Hypothesis space coverage: In model-based scientific workflows, three central metrics are used: Validity (precision), Uniqueness (non-redundancy), and Recovery (coverage). Explicit formulas:

$E$8

These metrics disentangle correct constraint adherence from diversity and completeness; high Validity alone does not guarantee coverage of $E$9 (Chen et al., 17 Oct 2025).

• Soundness and completeness: For neural network explanations, the soundness guarantee is paramount: $b \in E$0. Completeness—comprising maximality of $b \in E$1—is not required, though interpolation strategies can tune the region size (Labbaf et al., 27 Nov 2025).

4. Concrete Instantiations

Admissible explanation spaces manifest concretely as:

Domain	Space Construction	Format
Time Series Attribution	$b \in E$2: FFT, STFT, diff, ...	$b \in E$3
Hypothesis Set (HypoSpace)	$b \in E$4 via enumeration/validation	Finite set
Neural Net Regions (Space Exp)	Interpolant $b \in E$5 via SMT	Region $b \in E$6

Time Series: Examples include time domain ($b \in E$7), frequency ($b \in E$8), time-frequency ($b \in E$9), min-zero ($\varphi^{-1}(b)$0), and first-difference ($\varphi^{-1}(b)$1) spaces, each with tailored baselines and semantics (Rezaei et al., 2024).

HypoSpace: Case studies span:

• Causal graphs (DAGs) under intervention constraints.
• 3D voxel grids under projection and gravity constraints.
• Boolean expression trees with observed I/O constraints (Chen et al., 17 Oct 2025).

Neural Networks: Regions defined as half-spaces or polyhedra (e.g. $\varphi^{-1}(b)$2) produced by interpolation provably cover points where the model’s output is invariant (Labbaf et al., 27 Nov 2025).

5. Empirical Findings and Practical Prescriptions

Empirical results highlight the impact and utility of admissible explanation spaces:

• Time series: Explanation domains other than time (notably STFT, difference, min-zero) achieve higher sparsity and faithfulness on benchmarks (e.g., STFT on AudioMNIST achieves $\varphi^{-1}(b)$3, 100% faithfulness; frequency-domain explanations reach $\varphi^{-1}(b)$4 faithfulness on FordA) (Rezaei et al., 2024). Choice of admissible space is guided by signal modality, interpretability, computational overhead, and baseline semantics.
• Hypothesis generation: As $\varphi^{-1}(b)$5 scales, strong LLMs sustain high Validity but exhibit declining Uniqueness and Recovery, exposing mode collapse; comprehensive exploration of $\varphi^{-1}(b)$6 requires diversity-focused decoding or explicit memory (Chen et al., 17 Oct 2025).
• Neural network explanations: Interpolant-based admissible spaces offer tighter, more expressive and scalable regions than axis-aligned intervals or abductive (per-sample) explanations. In practical benchmarks, region-based explanations relax more features, require fewer solver calls, and run faster than alternative methods (Labbaf et al., 27 Nov 2025).

6. Theoretical Guarantees and Limitations

Admissible explanation spaces come with distinct theoretical properties:

• Soundness: For region explanations via interpolation, every $\varphi^{-1}(b)$7 in $\varphi^{-1}(b)$8 maintains output $\varphi^{-1}(b)$9.
• Maximality: Not required by definition; possible to adjust via different interpolation schemes to trade off between region volume and shape (Labbaf et al., 27 Nov 2025).
• Composability: Admissible spaces can be constructed without retraining the model or modifying XAI methods, fostering compatibility and agnosticity (Rezaei et al., 2024).
• Challenges: In hypothesis settings, full enumeration is feasible only in structured domains of moderate combinatorial size; for large-scale domains, efficient approximation remains an open problem. For attribution, the choice of mapping $d_E$0 and baseline $d_E$1 is critical for interpretability and faithfulness.

7. Significance and Future Directions

The admissible explanation space concept enables interpretable, theoretically grounded, and context-sensitive explanations across machine learning and computational science. Its utility has been demonstrated for increasing post-hoc explanation fidelity, rigorously evaluating LLM hypothesis diversity, and establishing sound region-based guarantees in neural classification. Open questions include maximally efficient enumeration and proposal strategies for large $d_E$2; formal definitions of interpretability for complex explanation domains; and scale-out of logic-based region explanation to deep or high-dimensional networks. Across settings, admissible explanation spaces serve as crucial infrastructure for systematic, high-coverage, and semantically meaningful model understanding (Rezaei et al., 2024, Chen et al., 17 Oct 2025, Labbaf et al., 27 Nov 2025).