Adaptive Hierarchical Symbolic Abstraction

• Adaptive Hierarchical Symbolic Abstraction Mechanism (AHSAM) is a trigger-on-stagnation system in DIGSP that converts specialized genetic programming individuals into shared symbolic constructs.
• It uses ANOVA-based filtering to select statistically significant expressions, compressing and validating them before reinjection across populations.
• Empirical studies in SFRC beam modeling show that AHSAM improves convergence, reduces model complexity, and enhances interpretability compared to baseline models.

Searching arXiv for the cited AHSAM paper and closely related hierarchical symbolic abstraction work. Adaptive Hierarchical Symbolic Abstraction Mechanism (AHSAM) is a trigger-on-stagnation component introduced within Domain-Informed Genetic Superposition Programming (DIGSP), a symbolic regression framework for engineering systems governed by separable physical mechanisms. In DIGSP, the input space is partitioned into domain-specific feature subsets, independent genetic programming populations evolve to model material-specific effects, and ensemble fitness promotes inter-population cooperation. AHSAM intervenes only after stagnation across all populations: it performs analysis of variance- (ANOVA) based filtering to identify statistically significant individuals, compresses them into symbolic constructs, and injects them into all populations through a validation-guided pruning cycle. In the reported SFRC beam case study, this mechanism is presented as the means by which local specialization is converted into shared, higher-level symbolic structure, with the stated goals of improving convergence, generalization, and interpretability (Khorshidi et al., 20 Sep 2025).

1. Conceptual role within domain-informed symbolic regression

AHSAM is defined in the context of DIGSP rather than as a standalone general-purpose abstraction algorithm. DIGSP addresses engineering systems in which multiple, largely independent physical mechanisms are assumed to superpose in an interpretable way. Its basic organization is domain-informed partitioning, early isolation of GP populations, and later cooperation through ensemble fitness. Within that organization, AHSAM is the mechanism that converts evolved symbolic expressions into reusable abstractions and redistributes them across the full system (Khorshidi et al., 20 Sep 2025).

The motivation for AHSAM is explicitly tied to limitations of monolithic symbolic regression over heterogeneous, high-dimensional feature spaces. The reported problems are poor convergence, overfitting/redundancy, and reduced interpretability. AHSAM is therefore positioned as a hierarchical and modular knowledge-sharing cycle: populations first specialize on their assigned mechanisms, then selected symbolic sub-expressions are elevated into abstractions that can participate in symbolic superposition across domains. The intended effect is not merely search acceleration, but a restructuring of the symbolic vocabulary so that higher-level constructs can be reused by populations that did not originally evolve them (Khorshidi et al., 20 Sep 2025).

A common misconception is to treat AHSAM as the entire DIGSP framework. The reported architecture is narrower. DIGSP supplies the domain-informed decomposition and multi-population evolutionary scaffold; AHSAM is the adaptive abstraction-and-injection procedure that is activated when isolated evolution and ensemble fitness are no longer producing improvement. Another misconception is to view AHSAM as a continuously active abstraction layer. The mechanism is specifically described as trigger-on-stagnation rather than always-on (Khorshidi et al., 20 Sep 2025).

2. Triggering logic and operating cycle

AHSAM is not active from initialization. The trigger rule is that, for each population, if no improvement in fitness is observed for $T$ consecutive generations, and this stagnation occurs concurrently across all populations, AHSAM is activated. In the SFRC study, $T = 25$ generations (Khorshidi et al., 20 Sep 2025).

The reported operating cycle has a fixed internal order. First, system-wide stagnation is detected. Second, outputs from leading individuals are collected within each population. Third, ANOVA-based filtering is used to determine which symbolic expressions qualify as abstraction candidates. Fourth, each selected expression is compressed into a named symbolic construct. Fifth, each construct is evaluated for predictive utility and pruned if necessary. Sixth, validated constructs are injected into all populations as new terminal symbols. Seventh, evolutionary search resumes with an enriched symbolic vocabulary. The procedure may repeat upon future stagnations, producing an evolving hierarchical library of discovered mechanisms (Khorshidi et al., 20 Sep 2025).

This sequencing is important for interpreting the “adaptive” qualifier in AHSAM. Adaptation does not mean continuous online reorganization after every generation. Rather, the mechanism adapts to a particular failure mode of multi-population evolution: concurrent stagnation. The hierarchy likewise does not arise from a pre-specified ontology. It is induced incrementally by repeated cycles of abstraction, validation, and reinjection, so that newly created constructs can later serve as building blocks for additional constructs (Khorshidi et al., 20 Sep 2025).

3. Statistical filtering, symbolic compression, and reinjection

The first internal phase of AHSAM is ANOVA-based symbolic filtering. For each population, outputs of candidate individuals over the training samples are compared using one-way ANOVA, and all individuals with $p \leq 0.05$ are selected as abstraction candidates. The stated purpose is to retain symbolic expressions whose outputs are statistically significant and meaningfully distinct across the data (Khorshidi et al., 20 Sep 2025).

The second phase is symbolic compression. Each selected individual $g^{(i)}(x)$ is compressed into a named abstract feature,

$z_i(x) = g^{(i)}(x).$

These $z_i$ constructs become new terminal symbols. Reinjection is cross-population: every validated abstraction is added to the terminal set of all populations, including populations associated with other physical domains. This is the step that enables symbolic superposition, because children in later generations can recombine original features with cross-mechanism abstractions (Khorshidi et al., 20 Sep 2025).

Selection by ANOVA is not the final gate. AHSAM includes a validation-guided pruning cycle. For each $z_i$, its standalone RMSE as a regressor is computed and compared to the mean population RMSE. If the abstraction’s RMSE is no worse than the baseline, reinjection is allowed. If not, subtrees are recursively pruned from $z_i$ and the abstraction is retested; the process repeats until predictive utility becomes non-inferior or the abstraction is discarded. This reported design directly addresses a second misconception: statistical significance alone is not treated as sufficient for admission into the shared symbolic library. Structural compactness and predictive utility are both enforced before reinjection (Khorshidi et al., 20 Sep 2025).

The hierarchy in AHSAM is therefore operational rather than merely taxonomic. Lower-level GP individuals are compressed into higher-level symbolic terminals, and those terminals subsequently participate in future expression growth. The abstraction process also remains tightly coupled to validation, which distinguishes it from purely compressive macro-construction strategies that do not explicitly test whether the introduced abstractions degrade predictive performance (Khorshidi et al., 20 Sep 2025).

4. Reported empirical behavior in the SFRC beam case study

DIGSP with AHSAM was benchmarked against a baseline multi-gene genetic programming (BGP) model on a dataset of steel fiber-reinforced concrete (SFRC) beams. The evaluation used 30 independent trials with 65% training, 10% validation, and 25% testing splits. Across these trials, DIGSP consistently outperformed BGP in training and test root mean squared error (RMSE), while no significant difference was observed in validation RMSE due to limited sample size (Khorshidi et al., 20 Sep 2025).

The reported quantitative outcomes are summarized below.

Quantity	DIGSP	BGP
Train RMSE	$1.22 \pm 0.17$	$1.61 \pm 0.13$
Test RMSE	$T = 25$0	$T = 25$1
Median tree size	41 nodes	204 nodes

The Wilcoxon rank-sum test confirmed statistical significance with $T = 25$2, and a more specific test-set value is reported as $T = 25$3. The error distributions are described as tighter with fewer outliers for DIGSP. These observations are attributed to the combination of domain-informed structural decomposition and symbolic abstraction, rather than to symbolic abstraction alone (Khorshidi et al., 20 Sep 2025).

Interpretability claims are also concrete in the SFRC setting. The model is reported to decompose predictions into contributions from concrete, steel, and fibers, consistent with engineering intuition and physical reality. A superpositional form is explicitly given as

$T = 25$4

The reported mechanism contributions align with physical expectations: concrete dominates, steel is secondary, and fibers assist post-crack resilience. Parameter sensitivities for DIGSP are reported as positive for $T = 25$5 and $T = 25$6, and negative for $T = 25$7 (Khorshidi et al., 20 Sep 2025).

These results establish AHSAM’s empirical role within DIGSP as a search-space restructuring device that also modifies model semantics. The reported reduction in median tree size from 204 to 41 nodes suggests that abstraction-and-pruning is functioning not only as a knowledge-transfer mechanism but also as a bloat-control mechanism. The absence of a significant validation-RMSE difference due to limited sample size also provides an internal qualification: the evidence for generalization is strongest on the reported test results, not on the validation split (Khorshidi et al., 20 Sep 2025).

5. Position within the broader literature on hierarchical symbolic abstraction

The term “AHSAM” is specific to the DIGSP case study, but the mechanism belongs to a broader research trajectory concerned with adaptive, hierarchical, and task-dependent symbolic abstraction. The literature provided spans robotic perception, hierarchical reinforcement learning, mathematical reasoning, symbolic control, and post-hoc model explanation.

Paper	Abstraction target	Relation to AHSAM
"Language-guided Adaptive Perception with Hierarchical Symbolic Representations for Mobile Manipulators" (Fahnestock et al., 2019)	Object detectors and part-whole relations	Hierarchical symbolic modeling is activated by language and yields task-minimal world models
"Hierarchical State Abstraction Based on Structural Information Principles" (Zeng et al., 2023)	RL states via an optimal encoding tree	Data-driven hierarchy and abstraction at any tree level suggest principled multi-granular compression
"Goal Space Abstraction in Hierarchical Reinforcement Learning via Reachability Analysis" (Zadem et al., 2023)	Goal symbols defined by reachability-equivalent state sets	Dynamic refinement preserves environment dynamics during symbolic goal discovery
"Constructing Abstraction Hierarchies Using a Skill-Symbol Loop" (Konidaris, 2015)	MDP states and actions induced by skills	Skills induce the appropriate symbolic representation in an alternating acquisition loop
"LEMMA: Bootstrapping High-Level Mathematical Reasoning with Learned Symbolic Abstractions" (Li et al., 2022)	Higher-level mathematical actions	Solved trajectories are rewritten in terms of reusable abstractions that later serve as new primitives
"Hierarchical Symbolic Reasoning in Hyperbolic Space for Deep Discriminative Models" (Santhirasekaram et al., 2022)	Multi-level symbolic explanations	A learned abstraction tree organizes symbols from fine-grained to fully abstract levels

Several contrasts are technically informative. In language-guided adaptive perception, hierarchical symbols such as “handle-of-door” are introduced so that the world model includes only the objects and relations necessary for a command, for example distinguishing “drive to the door” from “open the door” (Fahnestock et al., 2019). In SISA, abstraction is organized by an optimal encoding tree and can be performed at any tree level, with conditional structural entropy used to compensate for sampling-induced essential information loss (Zeng et al., 2023). In GARA, symbolic goals emerge by splitting state partitions whenever reachability relations violate the desired abstraction property, so that groups of states remain functionally equivalent for hierarchical control (Zadem et al., 2023). In the skill-symbol loop, abstraction hierarchies are built by alternating skill acquisition and representation acquisition, with the abstract state space determined by the available skills (Konidaris, 2015). In LEMMA, learned abstractions are mined from successful solution trajectories, rewritten as higher-level actions, and then reused hierarchically in later reasoning (Li et al., 2022). In hyperbolic symbolic reasoning, an abstraction tree is induced from a quantized latent space so that explanations can be extracted at multiple abstraction levels (Santhirasekaram et al., 2022).

This suggests that AHSAM’s distinctiveness lies less in the general idea of hierarchy than in its specific combination of stagnation-triggered activation, ANOVA-based statistical gating, validation-guided pruning, and cross-population reinjection within domain-partitioned symbolic regression. The broader literature supports the general value of adaptive hierarchy, but AHSAM’s reported novelty is the use of these elements to support symbolic superposition in an engineering regression setting (Khorshidi et al., 20 Sep 2025).

6. Interpretation, limitations, and plausible extensions

The reported evidence supports a specific interpretation of AHSAM. It is a mechanism for converting locally evolved symbolic expressions into a shared abstraction library under a conservative admission policy: stagnation must be system-wide, ANOVA must identify significance, validation must show non-inferior utility, and pruning must remove destabilizing structure before reinjection if necessary. This makes AHSAM neither a purely statistical filter nor a purely evolutionary operator. It is a gated bridge between local symbolic discovery and global symbolic reuse (Khorshidi et al., 20 Sep 2025).

Its limitations are also explicit. The empirical demonstration is a case study on SFRC beams within DIGSP, and the abstract states that DIGSP offers “a principled and interpretable modeling strategy for systems where symbolic superposition aligns with the underlying physical structure” (Khorshidi et al., 20 Sep 2025). A plausible implication is that transfer to domains lacking separable physical mechanisms, or lacking meaningful superpositional structure, may be less direct. Likewise, the reported absence of a significant difference in validation RMSE due to limited sample size indicates that some aspects of the empirical picture remain sensitive to evaluation protocol and dataset scale (Khorshidi et al., 20 Sep 2025).

The broader literature indicates several directions that could, in principle, be combined with AHSAM-like mechanisms. Reachability-based symbolic refinement preserves environment dynamics in hierarchical control (Zadem et al., 2023); structural-information methods provide a fully data-driven hierarchy without manual tuning of cluster number or depth (Zeng et al., 2023); and skill-induced symbolic state construction links available actions to the correct abstract representation (Konidaris, 2015). This suggests, though it does not establish, that future variants of AHSAM could be paired with data-driven tree construction, dynamics-preserving refinement criteria, or skill-conditioned symbol induction.

In its present reported form, however, AHSAM should be understood narrowly and precisely: an adaptive hierarchical symbolic abstraction mechanism inside DIGSP that is triggered by concurrent evolutionary stagnation, identifies statistically significant symbolic individuals, compresses them into reusable constructs, subjects them to validation-guided pruning, and injects them across populations to support symbolic superposition, improved convergence, and interpretable decomposition in SFRC beam modeling (Khorshidi et al., 20 Sep 2025).