Exhaustive Canonical Skeletons Generation: A Formal Dataset for Decomposable Symbolic Regression

I. INTRODUCTION

• A canonical skeleton subsumes more specific instantiated forms, and explicit closure rules eliminate functional collisions and repeated generation.This targets redundant derivations such as distinct expressions that simplify to the same function.Source: §I · p. 1
• Canonicalized baselines show severe skewness and redundancy, whereas ExCaUS is compact, duplicate-free, and more uniform in unary-operator distribution.The dataset comparison also reports structural duplication rates above 98.7% and 99.9% for the two baselines, versus 0% for ExCaUS.Source: §I · p. 1
• ExCaUS uses a formal recursive decomposition mapped to a deterministic context-free grammar, paired with sampling designed to avoid numerical instability and protected operators.The comparison with existing datasets reports a more uniform distribution of unary operators and reduced bias toward overly simple forms.Source: §I · p. 1
• ExCaUS-Bench contains 150 problems across three tiers, and tested SR methods struggle to recover their underlying functions.Its extended domain ranges make regression more difficult than standard benchmarks with narrower evaluation intervals.Source: §I · p. 1

II. DATASET DEFINITION

• Expressions are rooted trees built from unary and binary operators; canonical atoms and equivalence rules organize constant, linear, and polynomial forms.The paper defines canonical atoms C, L, and P, then specifies closure rules for their combinations.Source: §II · pp. 1–2
• Although univariate, the skeletons are intended as components for decomposable multivariate SR without assuming linear separability.Individual variable-response structures can be progressively merged into a multivariate expression.Source: §I · p. 1, §II · p. 2
• Recursive affine expansion and equivalence rules produce a normal form, with each skeleton subsumed by a unique finite canonical tree.The normal form applies these transformations recursively until no further transformation is possible.Source: §II · p. 2
• The canonical set is finite under bounds on unary-operator count and nesting depth; treating power exponents as external parameters excludes exponents that depend on x.The paper notes that this restriction leaves variable-exponent formulations outside its scope.Source: §II · p. 2

III. GRAMMAR-BASED SKELETON GENERATION

• The target set is finite when total unary-operator count and maximum unary nesting depth are bounded.These bounds are denoted NU and ND.Source: §III · p. 2
• The construction targets exactly-once generation of expressions already in normal form, avoiding redundant evaluation of non-canonical or functionally equivalent forms.The preceding finiteness result applies under bounds on total unary-operator count and maximum unary nesting depth.Source: §III · pp. 2–3
• The grammar produces expressions in their normal form by construction.This is the stated design for constructing the finite canonical set.Source: §III · p. 3

A. Grammar Definition

• The grammar G = (V, Σ, R, σ) defines non-terminals, terminals, production rules, and a start symbol for generating canonical skeletons.Source: A. Grammar Definition · p. 3
• Its non-terminals follow hierarchical levels of the sum-of-products form, with Ω as a sum component, Φ as a product of unary blocks, and Ψ as an atom or unary function.Source: A. Grammar Definition · p. 3
• The atom C is excluded from terminal productions because constants within functional branches are subsumed by additive and multiplicative placeholders higher in the hierarchy.Source: A. Grammar Definition · p. 3
• The unary operator set covers common forms including trigonometric, hyperbolic, exponential, logarithmic, absolute-value, power, and inverse operators.Source: A. Grammar Definition · p. 3

B. Production Constraints for Canonicality

• Three production constraints enforce canonical closure: additive independence, multiplicative irreducibility, and functional irreducibility.They exclude sums subsumed by atoms, products reducible to a single atom, and unary compositions equivalent to linear or polynomial forms.Source: B. Production Constraints for… · p. 3

C. Complexity Constraints

• Global unary-count and compositional-depth bounds restrict functional complexity and ensure the generated skeleton set is finite.The global count cannot exceed NU, while unary nesting is permitted only below ND.Source: C. Complexity Constraints · p. 3

D. Symmetry Breaking and Uniqueness

• Lexicographic ordering canonicalizes sums and products by unary-operator count, nesting depth, then alphabetical operator and atom order.This prevents duplicate representatives caused by commutativity of addition and multiplication.Source: D. Symmetry Breaking and Uniqueness · p. 3

E. Implementation Algorithm

• The generator uses breadth-first state-space expansion, simplifies candidate subtrees, rejects lower-complexity equivalents, and sorts admissible branches before applying global bounds.Completed expressions receive final reduction and sorting before insertion into the unique set.Source: E. Implementation Algorithm · pp. 3–4, B. Production Constraints for… · p. 3

A. Sampling Expression Constants and Support

• Skeleton atoms are instantiated with constants, linear forms, or polynomials whose degree is sampled from U(2, 6), narrowed to U(2, 3) under unary operators.Power exponents are rational values in (0, 7), excluding 0.5, 1, 2, and 3; constants are sampled independently from U(−50, 50).Source: A. Sampling Expression Constants and… · p. 4
• Each instantiated function is evaluated at 10,000 uniformly sampled inputs from U(−10, 10) to produce response values.The input domain stays fixed for evaluation uniformity, while randomized constants naturally scale functions across numerical ranges.Source: A. Sampling Expression Constants and… · p. 4

B. Sampling Data

• Hierarchical parsing handles bounded functions through input shifting and scaling, while vertical singularities receive derivative-driven exclusion boundaries based on local slope.The method avoids fixed spatial windows and coarse-grid detection; safe support points are sampled outside the resulting forbidden subdomains.Source: B. Sampling Data · p. 4

A. Dataset Comparison

• The dataset comparison canonicalizes SeTGAP’s 1.65 million univariate skeletons and NeSymRes’s 10 million skeletons, re-skeletonizing the latter around x1.Table I is identified as a quantitative SR dataset comparison, but its cell values are not supplied here.Source: A. Dataset Comparison · pp. 4–5, Table I · p. 5
• Figure 1 describes dataset frequency rank on log-log axes, nesting-depth and operator-count distributions, and unary-operator proportions.Frequency rank is used to assess whether dataset expansion adds structural variety or repeats high-frequency skeletons.Source: A. Dataset Comparison · p. 5, Figure 1 · p. 5

B. Benchmarking

• ExCaUS-Bench samples 50 skeletons in each of three difficulty tiers and generates one 10,000-point equation dataset per skeleton, split into interpolation and extrapolation sets.Extrapolation uses the bottom and top 5% tails; interpolation uses the middle 90%.Source: B. Benchmarking · p. 5
• Thirteen SR methods are tuned and evaluated over ten runs per equation, with extrapolation performance summarized by MSE and R2 across each tier.The passages describe evaluation procedures and plots but do not provide the resulting method comparisons.Source: B. Benchmarking · p. 5

VI. DISCUSSION

• ExCaUS contains 71,027 unique canonical skeletons with 0% duplication, versus over 98.7% and 99.9% duplication in the canonicalized NeSymRes and SeTGAP datasets.The most frequent NeSymRes skeleton appears over 106 times; ExCaUS's flat frequency-rank curve reduces training memory and computation while providing complete functional-space coverage.Source: §VI · p. 5
• ExCaUS has a more balanced operator distribution across 16 operators, averaging 2.32 operators per expression and 1.58 unary nesting depth, while pow reaches 40% of NeSymRes unary instances.Balance is not perfect because of additional structural filtering rules.Source: §VI · p. 6
• On the Easy tier, PySR, GPGOMEA, and TaylorGP lead in extrapolation R2, while PySR, GPGOMEA, and Operon lead in extrapolation MSE.TaylorGP can score highly on R2 despite shifts or scale mismatches; Operon's constant optimization can reduce residuals despite trend mismatches.Source: §VI · p. 6, Figure 3 · p. 6, Figure 2 · p. 6
• On the 150-problem ExCaUS-Bench, predictive capability degrades from Easy to Medium and Hard, with extrapolation MSE outliers and R2 values near zero or extremely negative.These R2 values indicate failure to capture data variance outside the interpolation range.Source: §VI · p. 6, Figure 2 · p. 6, Figure 3 · p. 6

VII. CONCLUSIONS

• The conclusions present ExCaUS-Bench as a rigorous, non-redundant testbed for future symbolic-regression improvements.Future work will train neural SR models on datasets derived from ExCaUS and its sampling strategy.Source: §VII · p. 6

APPENDIX A RELATED WORK

• Prior tree samplers and templates remain stochastic; distinct derivations can yield functionally redundant expressions and simplicity bias.Grammar-based generation for ODE skeletons also uses exploration, including probabilistic CFGs that favor simpler forms.Source: Appendix A · p. 7

A. Lemma 1: Recursive Canonical Form

• The normalization N maps skeletons to a finite recursive sum-of-products form whose atoms include C, L, and P.The proof proceeds by structural induction; closure rules project products such as P·L onto the atomic leaf P, yielding a finite representative.Source: A. Lemma 1: Recursive Canonical Form · p. 7

B. Theorem 1

• With unary operator count and nesting depth bounded, the set of canonical univariate skeletons is finite.The constraints bound recursive composition, while normalization absorbs additions, multiplications, and placeholders that would otherwise create no new skeleton.Source: B. Theorem 1 · p. 8

APPENDIX C EXPRESSIVE SCOPE AND STRUCTURAL EXTENSIONS

• The appendix presents the recursive structural decomposition as accommodating piecewise-defined functions, complex-valued expressions, and differential equations.Source: Appendix C · p. 8

1) Piecewise-Defined Functions:

• Piecewise boundaries can be represented as unary modulations of functional blocks when the branches satisfy Proposition 1.The framework can use indicator functions or approximate switching smoothly with operators such as tanh.Source: 1) Piecewise-Defined Functions · p. 8

2) Complex-Valued Variables and Fields:

• Complex-valued expressions can be handled by expressing real and imaginary components separately under the same recursive decomposition.Addition combines components; multiplication expands into sums and products of real and imaginary parts.Source: 2) Complex-Valued Variables and Fields · pp. 8–9

3) Differential Equations:

• Differential equations can be recast as decomposable SR by treating precomputed derivatives as additional input variables.The proposed workflow estimates derivatives from data, isolates skeletons for derivative tracks, then merges them with state-variable skeletons.Source: 3) Differential Equations · p. 9

A. Additional Structural Filtering Rules

• Filtering rules restrict operator definitions, nesting, and simplifications to limit redundant or physically unrealistic structures.Source: A. Additional Structural Filtering Rules · p. 9

1) Inverse and Power Operations: •

• The inverse operator is algebraic and adds neither to transcendental-unary count NU nor maximum structural depth ND; generic Pow avoids exponents outside {0.5, 1, 2, 3}, while atom P counts toward NU.Source: 1) Inverse and Power Operations: • · p. 9

2) Structural Closure and Nesting Boundaries: •

• The grammar blocks direct powers of atoms L or P, redundant inverse configurations, and nested power operations that can cause numerical explosion.It also excludes sums or products of independent inverse terms when the grammar already represents an equivalent form.Source: 2) Structural Closure and Nesting… · pp. 9–10

3) Exponential Algebraic Simplifications: •

• Exponential arguments omit an additive constant because it is absorbed into the outer placeholder, and redundant exponential products or inverse nesting are prohibited.Source: 3) Exponential Algebraic… · p. 10

• The pipeline traverses expression trees bottom-up, resolving special operators inside a subtree before modifying its parent.It detects bounded or singular operators, recurses into nested arguments, and then evaluates the parent operator.Source: Appendix D · p. 10
• Nested log and arcsin constraints are resolved in order: enforce the log input domain before scaling its output into arcsin’s [−1, 1] range.Source: Appendix D · p. 10
• Single-bounded operators are adjusted by shifting or negating their inner domain, while double-bounded operators are scaled within stable limits.Source: Appendix D · p. 10
• For singularities, derivative-driven intervals are added to a forbidden-domain list; the example uses narrower exclusions around sharper asymptotes.The example uses α = 0.9995 and contrasts the exclusion windows around x = 4 and x = −4.Source: Appendix D · p. 10

• The E-tier table is titled as a listing of generated equations for the Easy tier.Source: Table E.1 · p. 12
• The M-tier table is titled as a listing of generated equations for the Medium tier.Source: Table E.2 · p. 13
• The H-tier table is titled as a listing of generated equations for the Hard tier.Source: Table E.3 · p. 14
• Figure E.8 plots benchmark equations H21–H40, a subset of the Hard tier.Source: Figure E.8 · p. 22

APPENDIX F SR BENCHMARKING: TUNING AND RESULTS

• Tables F.3–F.8 report per-problem mean and standard deviation for interpolation and extrapolation MSE and R2 across the easy, medium, and hard tiers.Tables F.3–F.5 cover MSE; Tables F.6–F.8 cover R2.Source: Appendix F · p. 24, Table F.3 · p. 25, Table F.4 · p. 26
• Cells marked with a dash indicate invalid expressions or NaN evaluation outputs in all 10 repetitions.Source: Appendix F · p. 24
• Table F.2 lists each SR method’s optimal configuration selected by exhaustive grid search from the hyperparameter grids evaluated for the benchmark.The appendix also reports the evaluated grids in Table F.1.Source: Table F.2 · p. 24, Appendix F · p. 24