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Pretrain on Small Synthetic Data, Scale Large for Free: Symmetry-Aware Foundation Model for Logic Rule Induction
Yin Jun Phua
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 90%
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Summary
The paper introduces G-NRI, a symmetry-aware foundation model for logical rule induction that extends the Neural Rule Inducer (NRI). By enforcing equivariance under the symmetry group G (comprising example order, atom renaming, polarity flips, and label swaps), the model achieves exact rule-level equivariance without retraining. The authors propose a canonical export mechanism that decodes discrete rules from literal scores, ensuring stability and transferability to larger schemas than those seen during training. Experiments on synthetic and real datasets demonstrate improved accuracy and rule fidelity compared to the baseline NRI.
Entities (8)
Relation Signals (7)
G-NRI → extends → Neural Rule Inducer
confidence 95% · We instantiate it on the Neural Rule Inducer... We restore the remaining symmetries... The full model, G-NRI, uses all five
G-NRI → uses → Canonical Export
confidence 92% · Our central contribution is a canonical export that decodes a discrete rule from literal scores... We instantiate it on the Neural Rule Inducer
G-NRI → enforces → Symmetry Group G
confidence 90% · Enforcing exact symmetry by construction lets one trained inducer scale... The full model, G-NRI, uses all five... restoring all of G
Canonical Export → ensures → Symmetry Group G
confidence 88% · Theorem 2 (Exact rule-level G-equivariance)... yields a signed DNF... exactly equivariant
G-NRI → evaluatedon → UCI Machine Learning Repository
confidence 85% · We evaluate the same checkpoints on 19 Boolean-encoded datasets: 14 tabular benchmarks from the UCI Machine Learning Repository
G-NRI → evaluatedon → MUTAG
confidence 85% · MUTAG (Debnath et al., 1991)... We evaluate the same checkpoints on 19 Boolean-encoded datasets
G-NRI → evaluatedon → CLEVR-Hans3
confidence 85% · CLEVR-Hans3 (Stammer et al., 2021)... We evaluate the same checkpoints on 19 Boolean-encoded datasets
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Abstract
Abstract:Logical rule induction seeks interpretable rules that transfer across propositional schemas. This requires respecting symmetries: atom naming, example order, polarity flips, and label swap. Enforcing exact symmetry by construction lets one trained inducer scale beyond its training schemas. Our central contribution is a canonical export that decodes a discrete rule from literal scores. It needs no retraining and is exactly equivariant whenever those scores respect the symmetries. We instantiate it on the Neural Rule Inducer, a disjunctive-normal-form (DNF) foundation model that natively respects only example order. We restore the remaining symmetries through architecture, inference, and training. On synthetic stress tests, accuracy on the support labels stays stable at much larger schemas, and rule fidelity on fresh inputs remains above the unmodified model. On real data, accuracy improves most on larger schemas. The exported rule is exact on synthetic full-group tests and on schema-valid real-data tests. This is a mathematical property of the export rather than of a specific model, and we validate it empirically only on the NRI. Enforcing symmetry by construction turns this small-data pretrained model into a reusable, interpretable inducer that transfers to larger schemas.
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- Source: https://arxiv.org/abs/2608.00383v1
- Canonical: https://arxiv.org/abs/2608.00383v1
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Symmetry-Aware Foundation Model for Rule Induction Pretrain on Small Synthetic Data, Scale Large for Free: Symmetry-Aware Foundation Model for Logic Rule Induction Yin Jun Phuaphua@comp.isct.ac.jp Institute of Science Tokyo Abstract Logical rule induction seeks interpretable rules that transfer across propositional schemas. This requires respecting symmetries: atom naming, example order, polarity flips, and label swap. Enforcing exact symmetry by construction lets one trained inducer scale beyond its training schemas. Our central contribution is a canonical export that decodes a discrete rule from literal scores. It needs no retraining and is exactly equivariant whenever those scores respect the symmetries. We instantiate it on the Neural Rule Inducer (Phua, 2026), a disjunctive-normal-form (DNF) foundation model that natively respects only example order. We restore the remaining symmetries through architecture, inference, and training. On synthetic stress tests, accuracy on the support labels stays stable at much larger schemas, and rule fidelity on fresh inputs remains above the unmodified model. On real data, accuracy improves most on larger schemas. The exported rule is exact on synthetic full-group tests and on schema-valid real-data tests. This is a mathematical property of the export rather than of a specific model, and we validate it empirically only on the NRI. Enforcing symmetry by construction turns this small-data pretrained model into a reusable, interpretable inducer that transfers to larger schemas. 1. Introduction Inductive Logic Programming (ILP) (Muggleton, 1991; Muggleton and De Raedt, 1994; Quinlan, 1990; Cropper and Morel, 2021) and the Learning from Interpretation Transition (LFIT) setting (Inoue et al., 2014) learn interpretable logical hypotheses, but behave poorly when the data are noisy, partially observed, or contain unseen transitions. Neuro-symbolic work (d’Avila Garcez and Lamb, 2023; De Raedt et al., 2020) aims to keep this interpretability while adding the noise tolerance and zero-shot generalisation of neural networks. A foundation model trained once on synthetic logic episodes should export a compact candidate rule for a new schema in one pass (Figure 1). Logical rules respect structural symmetries by construction, so a zero-shot inducer should respect them as domain properties rather than learning targets. δLFIT2 (Phua and Inoue, 2024) brought a foundation-model framing to the LFIT setting. It recovers transition rules and respects atom permutation by construction. However, it covers only Herbrand bases (sets of all ground atoms) of at most 18 atoms and does not handle noise or missing data. The Neural Rule Inducer (NRI) (Phua, 2026) extended this to generic disjunctive normal form (DNF) rules and to noisy, partially observed data. We adopt its foundation-model framing. Both are trained at small scale, the NRI on only 6–12 variables. Because its weights are not tied to specific atoms, one NRI checkpoint can in principle run on any number of atoms, a variable-schema interface. Its symmetry breaks make this unreliable, however. It does not undo polarity flips or label swaps, and a learned 1 arXiv:2608.00383v1 [cs.LO] 1 Aug 2026 Phua 10001 01100 11011 00110 S N ,Z N 2 Z y 2 S M Permutation- invariant encoder p + p − scores (p + ,p − ), slot 1 Canonical export s j = max(p + j ,p − j ) d j =p + j −p − j ˆ R= (x 1 ∧¬x 3 ) ∨(x 2 ∧ x 4 ) (1) zero literal-sign (2) drop content keys (3) label-swap avg. (4) canonical export (5) symmetrised training original NRI g· (X, Y ) g· ˆ R g g ˆ R(g· (X, Y ))≡ g· ˆ R(X, Y ), all g ∈ G Figure 1:G-NRI, our symmetry-restored NRI. The export commutes with every example, atom, polarity, and label transform in G, defined below. per-example component makes its output depend on atom ordering again. So beyond the training range its accuracy drops and its exported rules stop transforming correctly under label swaps. Enforcing these symmetries by construction removes such shortcuts rather than asking training to approximate them. Our exact guarantees are about commutation. Under each symmetry, the scores and the exported rule transform exactly as they should. The scaling question is whether enforcing this makes the variable-schema interface reliable far beyond the training range. No retraining or per-dataset fitting is added. The construction adds no learned parameters and costs a small constant factor at inference. We make three contributions, the second our central one. (i) We cast binary rule induction under the symmetry groupG=S M × B N × Z y 2 (example order, atom renaming and negation, and label swap), where a map isG-equivariant when it commutes with every transform inG. The NRI is equivariant only toS M , and our construction G-NRI restores all ofG. (i) Our central contribution lifts equivariance from continuous scores to the discrete exported rule, with no retraining. The group is classical, but the guarantee on the discrete output is new. (i) We confirm three claims (RQ1–RQ3, Section 5): exact no-retraining rule equivariance, reliable scaling far beyond training, and improved real zero-shot transfer with compact rules. This guarantee is a mathematical property of the export rather than of any specific model. The canonical export (Theorem 2) applies to any inducer exposingG-equivariant literal scores, but our experiments instantiate and validate it only on the NRI. 2. Background and Symmetry Desiderata A rule-induction problem has many equivalent presentations. Reordering the examples, renaming or reordering the atoms, flipping an atom’s polarity, or swapping the two labels each transform the target rule in a corresponding, reversible way. A map is invariant under such a transform when its output is unchanged and equivariant when its output undergoes the matching transform. Example order asks for invariance and the others for equivariance, so relabelling an episode’s atoms relabels the exported rule the same way. Equivariance also enables scaling. An inducer keyed to the arbitrary names, positions, or polarities of atoms 2 Symmetry-Aware Foundation Model for Rule Induction takes a shortcut that breaks past the training range, whereas an equivariant inducer cannot and transfers from small schemas to large. 2.1. The Neural Rule Inducer (NRI) We adopt the binary NRI as a representative DNF foundation-model inducer (Figure 1). Trained once on many synthetic episodes, it infers a rule for a new episode in one forward pass, with no per-task training. An episode is (X,Y,R), a small labelled dataset ofM examples over a schema ofNatoms, withX ∈ 0,1 M×N andY ∈ 0,1 M . Rows ofX index theMexamples and columns theNatoms, soX[:,j] is itsj-th column andX[i,:] its i-th row. The hidden DNFRsatisfiesR(X) =Y, and the model must recover it from (X,Y) alone. A literal-statistics encoder computes, for each literal (an atomx j or its negation ¬x j ), statistics of how it relates to the label across the examples. A slot decoder then fills a fixed set of clause slots. For each slot it emits soft gates in [0,1], the per-atom inclusion probabilities (p + j ,p − j ) for the literalsx j and¬x j together with a clause-activation gate. Two mechanisms matter for the symmetry analysis below. The model combines information across theMexamples with attention, using a learned key for each example. A pair memory scores pairs of literals (an atom with its own negation is a self-pair ) and keeps the topk by score to initialise the clause slots. Multiplying these gates (a product t-norm) gives the continuous predictionR pred (X), and thresholding them gives the discrete DNF ˆ R. The decoder runs two class rails, one per output class, and no parameter is atom-indexed. The NRI (our baseline) can therefore in principle generalise across schema sizes N . 2.2. Symmetry group of the problem Group-theory recap. A group collects transformations that we can compose and invert and that include an identitye(doing nothing). The symmetries of an episode form one such group, and for a group elementgwe writeg· zfor the result of applyinggto an object z, whether an episode or a rule. The symmetric groupS n is the set of all reorderings ofn items, soS M permutes theMexamples andS N theNatoms. The groupZ N 2 =0,1 N assigns one on/off bit per atom, each flipping that atom’s polarity through exclusive-or⊕, so flipping twice restores it. The single bitZ y 2 =0,1swaps the two labels. A direct product A×Bapplies independent transforms to separate parts at once, whereas a semidirect product A ⋉ Bcouples them. InB N =S N ⋉ Z N 2 , permuting the atoms also relabels which atom each polarity flip acts on. Finally,H ≤ Gdenotes a subgroup, a subset that is itself a group. The natural symmetry group of a binary rule-induction episode is generated by four operations (Figure 1): example-row permutationsS M , atom-column permutationsS N , per-atom polarity flipsZ N 2 =0,1 N (X[:,j]7→ X[:,j]⊕ σ j ), and label swapZ y 2 =0,1 (Y 7→ Y ⊕ τ). We writeB N :=S N ⋉ Z N 2 for the signed atom permutations (an atom permutationπtogether with an independent per-atom polarity flipσ) andG:=S M ×B N ×Z y 2 . Each factor also acts on the rule space, re-indexing literals, rewritingx j 7→¬x j , or takingR to its Boolean complement by De Morgan rewriting. Of these,B N andZ y 2 are symmetries of the underlying Boolean function, while onlyS M is specific to the episode. On real data, only an encoding-preserving subgroupH Σ ≤ B N keeps every example a valid encoding. A rawB N transform can break one-hot exclusivity, so we use it only as a schema-invalid stress test. 3 Phua We check equivariance at two levels. Prediction equivariance compares the prediction on the transformed episode,R pred (g·(X,Y)), with the transformed prediction on the original. Rule equivariance asks whether ˆ R(g·(X,Y)) is logically equivalent (≡) tog· ˆ R(X,Y) after inverse alignment. Neither implies semantic correctness, so we measure rule fidelity to the true generator separately (Figure 2). We reportmaxdev(maximum drift), rule Jaccard (aligned literal overlap), ruleeq (logical-equivalence rate), and validation accuracy (valacc). 3. AG-Equivariant Construction We restoreGwith five components, grouped by the stage at which they act. Architecture, recoveringB N in the encoder: (1) zero theliteralsignfeature and (2) disable the example- attention content keys. Eval-time export, recoveringZ y 2 for the prediction and the fullGfor the rule: (3) a label-swap prediction average and (4) a canonical, tie-complete rule export. Training : (5) symmetrise the loss. Example orderS M is already exact in the NRI. The full model, G-NRI, uses all five, adds no parameters, and keeps the NRI’s zero-shot interface. 3.1. Architectural fixes for B N Three parts of the NRI (Section 2) breakB N . The encoder adds a constantliteralsign feature that separatesx j from¬x j . AZ N 2 flip changes it, so we zero it. The example attention uses a learned key per example, which lets aggregation depend on atom order, so we disable its content keys. The last break is the pair memory’stop-kselection. Training drives each atom and its negation to equal scores, so the self-pairs tie; a fixedtop-kbreaks that tie by atom index, and an atom permutation then changes which pairs are kept, breaking S N . The first two breaks vanish once we zero the feature and disable the content keys. The third is subtler. The trained model keeps a fixed number of top pairs and breaks ties by atom index, so at eval time we instead use a tie-complete selection that keeps every pair whose score ties thek-th best. Because it reads only the set of scores, not atom order, it is permutation-equivariant even at the ties training creates. ForZ N 2 we likewise average each selected pair’s embedding over its polarity flips. Both act at eval time and need no retraining. Proposition 1 (B N equivariance of the soft literal score map). Withliteralsign j =0, example-content keys disabled, tie-complete pair selection, polarity- averaged pair embeddings, and no atom-indexed parameter, the soft per-literal score map (p + ,p − ) : (X,Y)7→[0,1] N ×[0,1] N (the positive- and negative-literal inclusion probabilities) isB N =S N ⋉ Z N 2 equivariant in exact arithmetic (proof, with the measured drift of the unmodified forward pass, in Appendix A). We call the fixed forward the equivariant forward. The tie break it removes is structural, not numerical: it persists in double precision, and the equivariant forward drives the measured atom-permutation drift to the floating-point floor at almost no accuracy cost (Appendix A). 4 Symmetry-Aware Foundation Model for Rule Induction 3.2. Eval-time operations for Z y 2 For an episode (X,Y), the label-swap test-time average, a two-pass test-time augmentation (TTA), is R avg (X,Y ) = 1 2 R pred (X,Y ) + 1− R pred (X, 1−Y ) .(1) A direct substitution givesR avg (X,1−Y) = 1− R avg (X,Y) exactly in real arithmetic, for anyR pred and any (X,Y), so the averaged prediction is label-swap equivariant. This is standard two-pass test-time augmentation (Shanmugam et al., 2021), and it acts at the prediction level only. The average fixes the prediction, not the rule, since the DNF read off one rail need not be the De Morgan complement a label swap requires. Call the two passes of Eq. 1 the rails. A label swap exchanges them and, with them, their predictions and literal scores. The dual-rail export uses this, choosing in a label-swap-invariant way between the positive rail’s rule and the negative rail’s De Morgan complement. The result is a signed DNF, read as that rule or as its complement, and it already gives exact label-swap rule equivalence. The next subsection subsumes it into a full-G, rule-level guarantee. 3.3. Canonical rule export The dual-rail export still uses the NRI’s original per-rail decode, which is notB N -equivariant. It orders equal-scoring atoms by index, which breaksS N , and thresholds each rail on its own, so a near-threshold literal can flip in or out under a polarity swap, which breaksZ N 2 . We replace this readout with a canonical decoder that summarises each atomj’s two literal probabilities (p + j ,p − j ) by two statistics. (The superscripts index literal polarity, and each rail carries its own such pair.) The presence scores j =max(p + j ,p − j ) is invariant under the polarity swap ofx j and¬x j . The signed contrastd j =p + j − p − j has a sign that flips under that swap. Atomjenters a clause whens j ≥ 1 2 andd j ̸= 0, with polarity set by the indicator 1[d j >0] (one whend j >0, zero otherwise), so an exact polarity tie (d j =0) omits the atom rather than choosing a side. A tie-complete rule resolves ties ins j . It admits the whole tie-bucket or abstains, and never breaks a tie by atom index. The export emits literals and clauses in a canonical order. Figure 1 traces one slot. Only atoms 1 and 3 pass the presence threshold, withd 1 >0 andd 3 <0, so the slot decodes tox 1 ∧¬x 3 . A dual-rail selector then chooses between the positive-rail rule and the negative rail’s De Morgan complement with a label-swap-invariant comparator (a symmetrised reference and a sign-flipping key, Appendix B), abstaining on an exact rail tie. This export is a canonicalisation (Kaba et al., 2023; Puny et al., 2022) and needs no retraining. It recovers the dual-rail export as its Z y 2 -only special case. Theorem 2 (Exact rule-level G-equivariance). Suppose the model’s paired literal scores (p + ,p − ) areG-equivariant: invariant underS M , permuted underS N , swapped per atom underZ N 2 , and exchanged between rails underZ y 2 . Then the canonical export, with the polarity-tie and rail-tie abstentions above, tie-complete selection, and canonical ordering, yields a signed DNF ˆ Rwith ˆ R(g·(X,Y))≡ g· ˆ R(X,Y) for every g ∈ G, exactly (up to logical equivalence). 5 Phua Table 1: Which symmetries our construction makes exact, approximate, or empirical. SymmetryObjectGuaranteeBasis S M (example order)predictionexactarchitecture Z y 2 (label swap)prediction, ruleexactEq. 1, Thm. 2 B N (atoms, polarity)literal scoresexact arithmeticProp. 1 G (full)exported ruleexact if below marginThm. 2 H Σ (real, schema-valid)exported ruleexactThm. 3 raw B N (real)exported rulediagnosticstress test The decoder is exact for one reason. It makes no arbitrary choice. It breaks no tie by atom index, assigns no polarity at an exact contrast tie (d j =0), and selects no rail at a key tie, replacing each such choice with a symmetry-respecting abstention or a whole-bucket admission. The theorem constrains only the score interface, not the architecture (proof in Appendix B). The fixed interface takes an episode (X,Y) withX ∈ 0,1 M×N and Y ∈0,1 M , emits per-atom literal scores (p + ,p − ) and a clause gate for each clause slot on two class rails, and returns a DNF over theNatoms. Any inducer with this interface and G-equivariant scores inherits the guarantee (Table 1), with G-NRI the NRI-specific instance. The architectural fixes and tie-complete forward give exactS M score equivariance and, by Proposition 1,B N score equivariance in exact arithmetic. Floating-point arithmetic leaves a tiny residual, but the discrete decode does not change as long as that drift stays below its smallest margin to a decision boundary (the presence threshold, a polarity sign change, a tie-bucket gap, or the rail key), so a small enough residual still yields an exactly equivariant rule. Section 5.1 confirms this. The discrete rule is exactly equivariant at every evaluated synthetic scale and across the schema-valid real-data audits. 3.4. Training-time symmetrisation Symmetrised training folds the same average into the loss (substitutingR avg (X,Y) for R pred (X,Y), two forward passes per step) and turns off an NRI loss term that competes with it. Its effect over the eval-time-only variant (components 1–4) is small but consistently positive, so that variant is a cheaper fallback. 3.5. Scope and sparse-support stability The architectural fixes recover the equivariance of a permutation-invariant set encoder (Zaheer et al., 2017; Lee et al., 2019; Hartford et al., 2018) within the existing architecture. This is also why the construction should scale. WithB N -equivariance the encoder cannot read an atom’s position, the one cue that grows withN, so the decoder’s error tracks the few literals the rule uses rather than the atom countN(a sparse-support stability argument). 4. Experimental Protocol Synthetic generator and conditions. Each episode (X,Y,R) draws a target DNF of at mostk max =6 clauses andℓ max =4 literals,M ∈[24,48] examples, and training schema size Nuniform over [6,12]. We evaluate the frozen model at twelve schema sizes fromN =6 to 6 Symmetry-Aware Foundation Model for Rule Induction N =1024. We compare the unmodified NRI (baseline), EqArch (architectural fixes 1–2), and the full G-NRI (all five components), with two intermediate conditions for the mechanism decomposition (Table 3). All share the NRI’s training recipe and hyperparameters under paired seeds. Real datasets. We evaluate the same checkpoints on 19 Boolean-encoded datasets: 14 tabular benchmarks from the UCI Machine Learning Repository (Kelly et al., 2023), the three MONK’s ILP benchmarks (Thrun et al., 1991), MUTAG (Debnath et al., 1991) (51 features), and CLEVR-Hans3 (Stammer et al., 2021) (3000 scenes as 105 symbolic existential atoms over object attributes; the model never sees the images). Per-datasetNranges 8–116, with 17 of 19 above the training maximum of 12. Evaluation is 5-fold stratified cross-validation with frozen weights. The model conditions on the training fold and exports a DNF, scored on the held-out fold whose labels it never sees. We binarise numeric features at their per-feature median and one-hot encode categoricals. This binarisation is label-free, so it cannot bias the G-NRI-versus-baseline gap (Appendix D). We report three per-dataset-trained supervised anchors (majority class, logistic regression, decision tree), audit rule-level equivariance on all 19 datasets (Section 5.1), and stress-test prediction-level drift on 5 of them. Seeds, metrics, and statistics. Two metrics separate fitting from generalisation. Support- set accuracy (valacc) scores the prediction against the conditioning labels. Fresh-assignment rule fidelity scores the exported rule on freshly sampled assignments (Figure 2). All primary statistics were pre-registered and run overn=8 paired seeds (n=3 for the drift audits). Appendix C details the unit of analysis, the paired nonparametric tests, and corrections. 5. Results We answer the three research questions of Section 1 in turn, reporting the evidence needed to read each result. 5.1. RQ1: Does the canonical export yield exactly G-equivariant rules with no retraining? The output a practitioner uses is the discrete rule, so the guarantee must survive decoding. We therefore test the exported rule directly, over 13 transforms that span theGgenerators and compositions. We measure ruleeq, the fraction of (episode, transform) pairs for which the rule decoded on the transformed episode, mapped back, is logically equivalent to the rule on the original. The variant ruleeq ne excludes empty-rule pairs, so empty (abstaining) exports cannot inflate it (Table 2). On synthetic episodes the full G-NRI is exactlyG- equivariant across the transform set and both audited schema sizes. Both pre-registered criteria, on real-data drift and on synthetic scaling, are also met. Both components are necessary. The original index-based decode is notB N -equivariant, and the canonical decoder on the unmodified forward is inexact (Table 2, upper rows). On real data, we restrict the transforms to those that keep every example a valid encoding (the schema-valid audit). Every dataset with a nontrivial such subgroupH Σ is then exactly equivariant, which ties the guarantee to the actual categorical encodings in the benchmarks. Grouping the audited (episode, transform) trials by their forward prediction drift, an observable proxy for the score-space margin, shows the exported rule stays logically 7 Phua Table 2: Rule-level equivariance of the no-retraining canonical export: worst-case ruleeq (cov.: fraction with a nonempty exported rule). Setting (worst case, 13 transforms)ruleeqruleeq ne cov. Synthetic, full G-NRI1.0001.0000.817 ablation: drop equivariant forward0.9670.966– Real data, 17 of 19 datasets1.0001.0001.000 mushroom, MUTAG (raw, off-manifold)0.7850.7850.867 Table 3: Mechanism decomposition (a) and deployed inference cost (b). (a) Accuracy by added component valacc ConditionN=128N=256 baseline0.772± 0.144 0.613± 0.142 + EqArch0.877± 0.018 0.854± 0.052 + TTA0.913± 0.026 0.910± 0.033 + rule export 0.913± 0.026 0.910± 0.033 G-NRI (all)0.930± 0.015 0.925± 0.026 (b) Inference cost (M =32) N base (ms) G-NRI (ms)ratio 1214.417.9 1.24× 12816.322.0 1.35× 102421.538.2 1.78× equivalent wherever the drift is small and diverges only above a threshold (Figure 2). This matches the margin argument (Appendix B), which bounds the rule change by drift in the decoder’s score inputs rather than in the prediction. 5.2.RQ2: Does restoring symmetry make the existing variable-schema inducer reliable far beyond training? We freeze a checkpoint and evaluate it unchanged out toN =1024 (85×the training maxi- mum). This stress test varies only the schema sizeN. The rule language, the generator family, and the maximum rule complexity (k max ,ℓ max ) stay at their training values, so it probes scaling in atom count rather than transfer to fundamentally more complex logical structures. To isolate the gain from restoring symmetry, we compare against the baseline (which omits the symmetries) and against EqArch (the architectural fixes only). Across schema sizes G- NRI keeps support-set accuracy and equivariance metrics stable, improving over the baseline in every seed. The baseline, by contrast, degrades towards chance byN =1024 (Figure 2). Held-out rule fidelity remains substantially above the baseline, and the lift widens withN. At the largest synthetic schema the model remains above target, and on real data clevr-hans3 is an out-of-distribution (OOD) case where it beats the majority class. By mechanism, the architectural fixes drive most of the gain (Table 3a). They give the largest single increment at both audited schema sizes. Label-swap averaging adds less, the rule export leavesvalacc unchanged because it only affects rule extraction, and symmetrised training adds a marginal residual. The eval-time-only variant, without symmetrised training, reproduces G-NRI’s accuracy to within a couple of points, so it is a cheap fallback. The construction is also cheap at inference (Table 3b): the two-pass operations make the deployed forward-plus-export pipeline 1.2–1.8× the single-pass baseline, with no measurable change in peak memory. 8 Symmetry-Aware Foundation Model for Rule Induction Figure 2: Variable-schema scaling and drift stratification. Table 4: Per-dataset accuracy (%,n=8 seeds): DT = trained decision tree; base/EqA/G = NRI baseline/EqArch/G-NRI. Bold marks the best zero-shot variant. DatasetNDT † baseEqAG DatasetNDT † baseEqAG adult10581.5 65.657.764.4 nursery2798.768.473.1 75.9 breast-cancer993.7 92.792.492.0 spambase5790.771.067.4 79.0 car2196.730.424.8 73.8 tic-tac-toe2793.160.158.6 69.9 credit4680.670.768.6 80.7 vote3294.591.391.6 94.3 diabetes870.072.0 72.671.8 monks-11798.474.174.1 74.6 german6165.358.458.9 60.9 monks-21798.5 61.160.455.9 hepatitis3277.7 80.676.080.0 monks-31796.890.3 96.4 96.4 ionosphere3479.471.968.4 73.1 mutag5184.663.966.9 73.1 kr-vs-kp7399.666.858.3 69.9 clevr-hans3105100.066.465.4 81.5 mushroom116100.078.072.4 81.0 mean (19)89.570.268.6 76.2 † trained on every dataset, so not directly comparable to the zero-shot variants. 5.3. RQ3: Does restoring symmetry improve real zero-shot transfer while keeping rules compact? We next test the same frozen checkpoints, with no per-dataset training, on 19 real datasets of up to 116 atoms. Per-dataset-trained anchors give a reference point. G-NRI stays below them, as expected for a zero-shot model, but closes much of the gap on the larger schemas. The deci- sive comparison is among the zero-shot variants themselves, where the symmetry components are the only difference. Among these zero-shot variants, G-NRI raises mean accuracy over the baseline and wins most datasets under the one-sided sign test (Table 4), with the lift growing with schema size (Table 5a). Within this architecture, restoring symmetry improves the frozen NRI. We read this as evidence about the effect of symmetry on one inducer, not as a claim that G-NRI is a generally competitive rule learner. EqArch sits below the baseline on the mean, so the full construction, not the architectural fixes alone, carries the gain. Against per- dataset-trained CART and RIPPER references, G-NRI induces the sparsest DNFs (Table 5b). 9 Phua Table 5: RQ3 evidence. p = percentage points, Seed Jac. = cross-seed Jaccard, Clause F1 = clause-recovery F1 score. (a) Lift by schema size Schema size N# datasets∆ acc (p) 2–102 −0.40 11–305+3.75 31–1008+4.54 101–10003+5.67 (b) Rule quality MetricG-NRICARTRIPPER Acc. (%)77.085.885.4 Clauses2.048.014.56 Literals5.9637.5816.40 Seed Jac.0.4840.8040.566 Clause F10.3560.4200.865 The exported rules read by inspection. On monks-3 G-NRI returns the ground-truth clause ¬ bodyshapeoctagon∧¬ jacketblue identically across all 8 seeds. 6. Related Work Symbolic and differentiable rule learning. Inductive Logic Programming (Muggle- ton, 1991; Cropper and Morel, 2021), answer-set learners (ILASP) (Law et al., 2014), and LFIT (Inoue et al., 2014) learn interpretable hypotheses by symbolic search. They respect logic’s symmetries but search per task and offer no schema transfer. Differentiable and neuro-symbolic methods (Evans and Grefenstette, 2018; Yang et al., 2017; Manhaeve et al., 2018; Cunnington et al., 2023) are noise-tolerant but also train one model per task. We instead apply one foundation model zero-shot, calibrated by per-dataset-trained anchors. Foundation models and symmetry by construction. The foundation-model inducers we build on,δLFIT2 (Phua and Inoue, 2024) and the NRI (Phua, 2026) (Section 1), recover a rule for each episode in one pass, an instance of amortised in-context inference (Garg et al., 2022; Lake and Baroni, 2023). Group-equivariant and permutation-invariant networks encode symmetry in the architecture (Cohen and Welling, 2016; Zaheer et al., 2017; Maron et al., 2019), including the signed permutations (Agrawal and Ostrowski, 2023) ofB N , but act on continuous representations. We instead enforce symmetry on a discrete symbolic output in a schema-OOD regime (Lake and Baroni, 2018; Stammer et al., 2021), and apply the canonicalisation and frame-averaging view of equivariance (Kaba et al., 2023; Puny et al., 2022; Lim et al., 2023) to rules rather than embeddings. 7. Discussion, Limitations, and Conclusion Limitations and scope. Symmetry sustains support-set accuracy beyond training, and near-equivariant scores export exact G-equivariant rules without retraining. Commutation differs from accuracy, and rule fidelity trails support-set accuracy and per-task learners, worst on multi-class car. RawB N can break encoding and rule equivariance, exact on encoding-preserving mushroom, a symmetry MUTAG lacks. Multi-clause recovery, not atom countN, bottlenecks scaling, and monks-2 ’s two-of-six target lacks small DNFs. Untuned Booleanization (label-free medians, one-hot categorical encoding, one-vs-rest targets) limits generality. 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In Advances in Neural Information Processing Systems (NeurIPS), volume 30, pages 2319–2328, 2017. Manzil Zaheer, Satwik Kottur, Siamak Ravanbakhsh, Barnab ́as P ́oczos, Ruslan Salakhutdi- nov, and Alexander J. Smola. Deep sets. In Advances in Neural Information Processing Systems (NeurIPS), volume 30, pages 3391–3401, 2017. Reproducibility Statement The project distributes its code athttps://github.com/phuayj/g-nri. The repository contains the code, the synthetic data generator, and the scripts that build every table and figure, which is what a reader needs to rerun the experiments end to end. It carries no pre-trained checkpoints, so a reproduction run starts from pre-training. Appendices D and E document the dataset preprocessing and the full per-dataset results. This appendix supports the body and introduces no new headline claims. It contains the proofs of every formal statement, the canonical-export algorithm, the full statistical and per-dataset tables behind the body’s summarised results, and the schema-relative analysis referenced in Section 5.1. Appendix A. Proofs of the equivariance results We prove the score-map equivariance Proposition 1 (including the tie-complete selection it relies on), then verify the label-swap prediction identity of Eq. 1. The rule-level Theorem 2 and the margin argument are in Appendix B. Throughout,R pred : (X,Y)7→[0,1] M denotes the trained NRI’s continuous prediction (positive rail), and we treatX ∈ 0,1 M×N , Y ∈0, 1 M as binary tensors with valid masking suppressed for clarity. Proposition 1 (B N equivariance of the soft literal score map). Proof Letg= (π,σ)∈ B N act onXas (g· X)[:,j] =X[:,π −1 (j)]⊕ σ π −1 (j) . The encoder computes per- literal featuresφ j (X,Y)∈ R D by applying a fixed functionφto thej-th column ofX paired withY. This function does not depend on the indexj, so underπtheφ j permute correspondingly. Underσ,φre-derives the truth-rate, marginal, entropy, and co-occurrence statistics fromX[:,j]⊕ σ j . These are symmetric in the positive/negative-literal swap, so φ j (σ j · X,Y) =swap σ j (φ j (X,Y)), which the shared multilayer perceptron (MLP) and the constant-zeroliteralsignfeature commute with. The cross-attention keys, with example- content keys disabled, are masked example-axis aggregations independent of atom index, henceS N -equivariant andZ N 2 -invariant. The pair memory scores literal pairs by a content function and selects them with a tie-complete rule, admitting every pair whose score ties the k-th best. This consults only the multiset of scores, never an atom index, so it commutes withπ; a fixed-cardinalitytop-kwould instead break a boundary tie by index and fail to commute. The polarity average of each selected pair’s embedding is invariant under σ. Composing these stages, the soft literal scores (p + ,p − ) areB N -equivariant in exact arithmetic. The map is alsoS M -invariant by construction: every example-axis operation is a masked symmetric pool, and the per-literal statistics are counts and rates over examples, so permuting the M examples leaves (p + ,p − ) unchanged. 14 Symmetry-Aware Foundation Model for Rule Induction Native-forward residual and the deployed fix. At initialisation the native forward (fixed-cardinality pair-memorytop-k,k=8) matches this exactness, withB N drift∼4×10 −8 , because random pair scores are generically tie-free. Training, however, drives each atom and its negation to exactly equal pair scores (self-pairs), so the nativetop-know faces a tie and breaks it by atom index. An atom permutation then selects a different pair set and shiftsR pred . A decisive ablation isolates this as the only non-equivariant operation. Disabling pair memory collapses the worst-caseS N drift from 0.079 to 4.8×10 −7 atN =128. The residual is structural, not a precision artefact (float64 reproduces the per-seed worst cases 0.008, 0.031, 0.117 and the average drift to six figures). Replacing the nativetop-kby the tie-complete selector and averaging each pair embedding over its polarity flips (both eval-time, no retraining) restores Proposition 1. Over 3 seeds atN =128 the worst-caseS N /Z N 2 /B N /g combined drift drops from 0.119/0.203/0.143/0.145 to 4×10 −4 /0.002/0.001/0.001 (mean∼10 −6 , the numerical- precision floor), at a cost of−0.75 percentage points (p) in validation accuracy. The example-orderS M and label-swapZ y 2 stay exact (∼3×10 −8 ). At this score equivariance the canonical export is exactly G-equivariant (Theorem 2). The non-tie-complete dual-rail export at the same forward still flips underB N (ruleeq≈0.35 versus 1.000 for the canonical export), so both the tie-complete forward and the canonical decode are necessary. Label-swap prediction identity (Eq. 1). Proof Substitute Y 7→ 1−Y in Eq. 1: R avg (X, 1−Y ) = 1 2 R pred (X, 1−Y ) + 1− R pred (X, 1− (1−Y )) = 1 2 R pred (X, 1−Y ) + 1− R pred (X,Y ) = 1− 1 2 R pred (X,Y ) + 1− R pred (X, 1−Y ) = 1− R avg (X,Y ). The equality is exact in real arithmetic. In float32 the residual is bounded by the rounding error of the two intermediate sums, which we measure as ∼ 3×10 −8 at N =128. Appendix B. Canonical export and proof of Theorem 2 The exporter. Algorithm 1 states the per-rail canonical decoder of Section 3.3. It admits atoms in whole tie-buckets by presence score, assigns polarity by the signed contrast, and removes duplicate and contradictory clauses. The dual-rail selector of Section 3.2 then picks the positive-rail rule or the De Morgan complement of the negative rail by a label- swap-invariant comparator. WithR ± pred the two rails’ predictions, it ranks the rails by their decoded fit to the symmetrised referenceR sym pred = 1 2 (R + pred + 1− R − pred ) and breaks ties by the scalarh= 1 M P i (R + pred − R − pred ) i , which flips sign under a label swap. It abstains on an exact rail tie. Theorem 2 (exact rule-levelG-equivariance). Assume the score map Φ : (X,Y)7→ (p + ,p − ) is G-equivariant in the stated sense. Proof The canonical export factors as Φ followed by the deterministic decode of Algorithm 1 and the dual-rail selection. We check that each generator ofG=S M ×(S N ⋉ Z N 2 )× Z y 2 commutes with this composition. 15 Phua Algorithm 1: Canonical decode (one rail). Input: Literal probabilitiesp∈[0,1] K×2N overKslots on the clean [p + |p − ] axis; clause-activation gates w ∈ [0, 1] K ; budget b; tolerance tieeps. Output: DNF rule ˆ R as a set of clauses of signed literals. ˆ R←∅ for slot k with w k ≥ 1 2 do s j ← max(p k,j ,p k,j+N ),d j ← p k,j −p k,j+N forj= 1,...,N C ←j:s j ≥ 1 2 and |d j | >0 sortCbys j descending; group into buckets tied withintieeps S ←∅; for bucketBin order do if |S| +|B|≤ b then S ←S∪ B; else break; end clause←(j,1[d j >0]) :j ∈Ssorted by (j, polarity) if clause̸=∅then ˆ R← ˆ R∪clause; end return ˆ R with duplicate and contradictory clauses removed S M . Φ is invariant under example permutations by hypothesis, so (s,d) and ˆ Rare unchanged, which matches the trivial action of S M on rule space. S N . Under an atom permutationπ, Φ permutes the score pairs, sos j andd j permute by π. The presence tests j ≥ 1 2 and polarity 1[d j >0] are pointwise and the tie-bucket selection depends only on the multiset of presence scores, taking whole buckets or none. It therefore commutes withπand never consults an atom index. Canonical re-sorting by atom index then yields the same clauses re-indexed by π, i.e. ˆ R(π· X) = π· ˆ R(X). Z N 2 . A polarity flip at atomjswaps (p + j ,p − j ), sos j =maxis unchanged whiled j 7→−d j flips the polarity bit, exchangingx j and¬x j . An exact tied j =0 maps tod j =0, so the atom is omitted in both frames (the inclusion test requiresd j ̸= 0), consistently. Unflipped atoms are untouched, so ˆ R(σ· X) =σ· ˆ R(X). With the previous case this givesB N =S N ⋉ Z N 2 equivariance. Z y 2 . A label swap exchanges the two rails. The referenceR sym pred = 1 2 (R + pred + (1− R − pred )) maps to its complement andh= 1 M P i (R + pred − R − pred ) i negates. Hence the comparator’s leading keys reverse and the selector returns the opposite rail with the flipped polarity flag, which denotes the De Morgan complement. An exact tie in the keys is preserved under the swap, so the selector abstains symmetrically. Hence ˆ R(X, 1−Y )≡ ˆ R(X,Y ). The factors act on independent coordinates of the rule (literal identity, literal polarity, and global complement), so the canonical export commutes with everyg ∈ G, exactly up to logical equivalence. Margin stability. The discrete decode depends on the scores Φ only through finitely many decisions: the comparisonss j ≷ 1 2 , the signs ofd j , the tie-bucket assignments (gaps in the sorted presence scores relative totieeps), and the sign of the rail-selection key. A deviationδ=sup|Φ(g·)− g·Φ(·)|moves each presence scores j =max(p + j ,p − j ) by at most δand each contrastd j =p + j − p − j by at most 2δ. If 2δis below the least gap between any score and the nearest decision boundary, which we call the decode margin, every comparison, sign, and bucket assignment agrees between Φ(g·(X,Y)) andg·Φ(X,Y). The two then decode to the same rule and the exact-arithmetic argument of Theorem 2 applies, so a small enough residual yields an exactly equivariant rule. 16 Symmetry-Aware Foundation Model for Rule Induction The decoder is exactly equivariant by construction. The only path to a violation is a continuous drift in Φ that crosses a decode boundary. This is why the residualS N non-equivariance of Proposition 1 is the sole observed failure mode. Abstention and tie incidence. In the deployed checkpoints the conservative decode surfaces as empty exports rather than tie abstentions. The coverage column of Table 2 reports the nonempty fraction: high on synthetic episodes, complete on the exact real-data row, and lower on the raw off-manifold failures. Rule equivalence restricted to nonempty pairs (ruleeq ne ) is unchanged, so abstention does not inflate the equivariance results. The tie-complete pair selection admits the whole boundary tie bucket, which in the trained checkpoints is the set ofNself-pairs, so the selected set grows linearly withN. Its cost at N =1024 is included in the runtime profile of Table 3. Appendix C. Statistical Analysis We fixed eight statistical hypothesis families in an internal analysis plan before running any audit experiments. All primary tests are one-sided paired Wilcoxon signed-rank (G-NRI > baseline). Table 6 reports the full per-Npaired statistics for each primary metric. The one-sided Wilcoxonp-value floor for ann=8 paired test with all eight signs in the same direction is 1/256≈0.0039. Holm-Bonferroni correction (Holm, 1979) across the twelve-N family raises the floor to 0.0469 (hence the repeated 0.047 entries). We report bootstrap percentilep-values that round to zero as<10 −4 (10,000 resamples). The table reports the group means, the bootstrap 95% confidence intervals on the seed-paired differences, paired effect sizes, and Holm-correctedp-values. Figure 2 visualises the corresponding scaling behaviour. Each invariance profile averagesmaxdevandJaccardoverE=100 episodes and T =20 samples per transform. The eight pre-registered families also cover Brown-Forsythe equality-of-variance tests (Brown and Forsythe, 1974) contrasting the baseline and G-NRI per-seed dispersion. Real-data unit of analysis. On the 19 real datasets the independent unit is the dataset, not the (seed, dataset) cell, because the eight seeds on a dataset share its folds and examples and are correlated. The primary test is a one-sided exact-binomial sign test on the per-dataset win count (14/19,p=0.032; two-sided 0.064), corroborated by a dataset-level one-sided paired Wilcoxon signed-rank test over the 19 per-dataset mean differences (p=0.0014). Excluding the car artefact, these become 13/18 (p=0.048) andp=0.0028. The per-dataset differences are right-skewed by car, so the Wilcoxon symmetry assumption is only approximate and we lead with the sign test. The Wilcoxon pooling all 152 seed×dataset cells (p=1.9×10 −5 ) is descriptive only. It treats correlated seeds as independent and is not a valid test. The subgroup splits (UCI-14,n=14; non-tabular-5,n=5; sign-testp=0.090 andp=0.19) are under-powered and uncorrected for multiplicity, so they are not independent confirmation. Real-data lift by schema size. Table 5a (body) bins the per-dataset G-NRI-minus- baseline accuracy lift by schema sizeN. The lift is negligible on the smallest schemas, which sit inside the training range, and increases monotonically withN. This supports the claim that the gain from restoring symmetry grows with schema size. 17 Phua Table 6:Full primary statistics across all twelve schema sizes,n=8 seeds, paired Wilcoxon signed-rank with Holm-Bonferroni correction. MetricNBaselineG-NRIPaired diff 95% CId / Holm p valacc60.867±0.0190.934±0.0040.067 [0.055, 0.079]3.652 / 0.047* 120.876±0.0130.931±0.0060.055 [0.046, 0.062]4.145 / 0.047* 160.886±0.0120.931±0.0040.044 [0.038, 0.051]4.255 / 0.047* 240.890±0.0060.933±0.0060.043 [0.037, 0.048]5.157 / 0.047* 320.885±0.0160.933±0.0060.047 [0.039, 0.055]3.578 / 0.047* 480.863±0.0420.932±0.0070.069 [0.047, 0.095]1.799 / 0.047* 640.852±0.0590.935±0.0070.083 [0.050, 0.123]1.463 / 0.047* 960.808±0.0970.932±0.0120.124 [0.070, 0.190]1.318 / 0.047* 1280.772±0.1440.930±0.0150.158 [0.079, 0.255]1.146 / 0.047* 2560.613±0.1420.925±0.0260.313 [0.220, 0.402]2.167 / 0.047* 5120.524±0.1040.910±0.0340.386 [0.297, 0.455]3.084 / 0.047* 10240.487±0.0650.894±0.0330.408 [0.350, 0.460]4.768 / 0.047* Z y 2 Jaccard60.556±0.1051.000±0.0000.444 [0.372, 0.506]4.236 / 0.047* 120.502±0.0991.000±0.0000.498 [0.432, 0.559]5.017 / 0.047* 160.458±0.0991.000±0.0000.542 [0.477, 0.604]5.492 / 0.047* 240.399±0.0930.999±0.0020.601 [0.538, 0.658]6.419 / 0.047* 320.356±0.0831.000±0.0000.644 [0.586, 0.693]7.748 / 0.047* 480.329±0.0891.000±0.0000.671 [0.609, 0.723]7.555 / 0.047* 640.300±0.0891.000±0.0000.700 [0.635, 0.751]7.832 / 0.047* 960.255±0.0721.000±0.0000.745 [0.694, 0.788]10.314 / 0.047* 1280.236±0.0760.999±0.0020.763 [0.709, 0.809]10.065 / 0.047* 2560.188±0.0741.000±0.0000.812 [0.759, 0.855]10.994 / 0.047* 5120.197±0.0511.000±0.0000.803 [0.770, 0.834]15.875 / 0.047* 10240.171±0.0421.000±0.0000.829 [0.802, 0.855]19.671 / 0.047* B N max dev60.082±0.0430.028±0.008-0.054 [-0.084, -0.031]-1.290 / 0.047* 120.109±0.0560.019±0.006-0.090 [-0.130, -0.058]-1.584 / 0.047* 160.117±0.0670.016±0.007-0.101 [-0.148, -0.069]-1.576 / 0.047* 240.138±0.0640.009±0.003-0.129 [-0.176, -0.095]-2.008 / 0.047* 320.155±0.0790.007±0.002-0.148 [-0.205, -0.109]-1.891 / 0.047* 480.186±0.1040.005±0.002-0.181 [-0.257, -0.128]-1.745 / 0.047* 640.186±0.0910.003±0.001-0.184 [-0.248, -0.136]-2.022 / 0.047* 960.202±0.1150.002±0.001-0.200 [-0.284, -0.140]-1.740 / 0.047* 1280.193±0.0960.002±0.001-0.190 [-0.260, -0.140]-1.984 / 0.047* 2560.164±0.0650.001±0.001-0.163 [-0.204, -0.121]-2.495 / 0.047* 5120.115±0.0940.001±0.001-0.114 [-0.177, -0.056]-1.212 / 0.047* 10240.088±0.0940.001±0.001-0.087 [-0.151, -0.031]-0.924 / 0.047* g max dev60.576±0.0610.029±0.008-0.547 [-0.586, -0.509]-9.028 / 0.047* 120.561±0.0580.020±0.006-0.541 [-0.575, -0.502]-9.506 / 0.047* 160.560±0.0570.016±0.007-0.545 [-0.579, -0.508]-9.834 / 0.047* 240.585±0.0380.009±0.003-0.576 [-0.604, -0.556]-15.442 / 0.047* 320.603±0.0710.007±0.002-0.596 [-0.646, -0.555]-8.441 / 0.047* 480.660±0.1180.005±0.002-0.655 [-0.737, -0.586]-5.549 / 0.047* 640.674±0.1450.003±0.001-0.671 [-0.767, -0.580]-4.641 / 0.047* 960.760±0.1340.002±0.001-0.758 [-0.845, -0.673]-5.678 / 0.047* 1280.789±0.1540.002±0.001-0.787 [-0.887, -0.686]-5.115 / 0.047* 2560.918±0.0780.001±0.001-0.917 [-0.963, -0.865]-11.828 / 0.047* 5120.968±0.0420.001±0.001-0.967 [-0.989, -0.937]-23.254 / 0.047* 10240.985±0.0180.001±0.001-0.984 [-0.995, -0.972]-53.869 / 0.047* 18 Symmetry-Aware Foundation Model for Rule Induction Real-data symmetry stress. Table 7 reports the real-data prediction-level symmetry- stress suite of Section 4, the mean|∆valacc|under random atom permutations and polarity flips on 5 datasets. It backs the drift criterion of Section 5.1. Table 7:Real-data symmetry stress on 5 datasets, mean|∆valacc|in percentage points (p) across n=125 trials; ratio (b/G) is the baseline-to-G-NRI drift ratio. Transformn baselineEqArchG-NRI ratio (b/G) S N permutation1251.160.000.0264× Z N 2 polarity flip1256.713.353.042.2× Appendix D. Real-data preprocessing We convert each real dataset once to a fixed Boolean schema, shared across all folds. We binarise numeric features at the per-feature median,x 7→1[x > median], for all 19 datasets (the label-free rule of Section 4). We keep already-binary features as is and one-hot encode categorical features (the MONK’s nominal attributes become binary indicators). MUTAG is the standard propositionalisation to 51 binary chemistry and graph-structure features. CLEVR-Hans3 uses 105 existential atoms over object attributes (shape, size, colour, material). Missing values stay masked through binarisation (the threshold uses the non-missing median) and in the episode. Multi-class datasets (car, nursery, clevr-hans3 ) use a One-vs-Rest decomposition. We form one binary episode per class, with the multi-class prediction taken as the arg-max over per-class rule scores. We compute the median thresholds on the full dataset rather than per training fold. Being label-free, this does not bias the zero-shot G-NRI-versus-baseline comparison (Section 4). The per-dataset-trained anchors are calibration references rather than claims, and their absolute accuracies could shift slightly under per-fold thresholds. We flag a strict per-fold binarisation as a confirmatory check (raw continuous features are not retained in the released artefact, so this requires rebuilding the datasets). Appendix E. Real-data per-dataset results and exported-rule quality Table 8 reports per-dataset mean accuracy for the three zero-shot NRI variants on all 19 real datasets, alongside the three per-dataset-trained supervised anchors that calibrate the target-specific gap. G-NRI’s losses to the baseline are marginal (around a point) except monks-2, the largest negative outlier, where it falls below both the baseline and the majority anchor. The trained decision tree solves it, while logistic regression also falls below the majority anchor. The loss is consistent with the multi-clause bottleneck of Section 7, though we have not verified this causally. Separately, on adult all three zero-shot variants sit below even the majority anchor. The model collapses toward the minority class and the exported rule defaults to the majority. Exported-rule quality versus symbolic learners. Table 5b (body) compares the exported G-NRI rule against two per-dataset-trained symbolic learners, the Classification and 19 Phua Table 8:Per-dataset mean accuracy (%,n=8 seeds): trained anchors and zero-shot NRI variants. C is class count; log. reg. is logistic regression. DatasetNC majoritylog. reg.tree baselineEqArchG-NRI 14 UCI tabular benchmarks adult105276.184.681.5 65.657.764.4 breast-cancer9265.595.393.7 92.792.492.0 car21470.087.896.730.424.873.8 credit46255.585.880.670.768.680.7 diabetes8265.172.770.072.072.671.8 german61270.074.365.358.458.960.9 hepatitis32279.383.977.7 80.676.080.0 ionosphere34264.183.579.471.968.473.1 kr-vs-kp73252.296.599.666.858.369.9 mushroom116251.8100.0100.078.072.481.0 nursery27533.391.698.768.473.175.9 spambase57260.693.990.771.067.479.0 tic-tac-toe27265.398.393.160.158.669.9 vote32261.495.694.591.391.694.3 mean over 14 UCI—62.288.887.369.967.276.2 Non-tabular benchmarks (MONKS / MUTAG / CLEVR-Hans3) monks-117249.674.698.474.174.174.6 monks-217265.761.998.5 61.160.455.9 monks-317252.096.496.890.396.496.4 mutag51266.583.084.663.966.973.1 clevr-hans3105333.3100.0100.066.465.481.5 mean over 5 non-tab—71.272.676.3 mean over 19 datasets—70.268.676.2 mean over 16 binary—73.071.376.1 mean over 3 one-vs-rest—55.154.477.1 20 Symmetry-Aware Foundation Model for Rule Induction Regression Trees (CART) learner (Breiman et al., 1984) and the RIPPER rule learner (Cohen, 1995). G-NRI induces by far the sparsest DNFs but, as a single frozen zero-shot model, does not match their per-dataset accuracy. We also attempted Bayesian Rule Lists (BRL) (Letham et al., 2015) as a third symbolic anchor, but it did not converge on any fold under our protocol, so we do not report it. Clause F1 is recovery of ground-truth target clauses. We report cross-seed stability (Seed Jac.), but it is not like-for-like. For G-NRI it varies across independently pretrained checkpoints, whereas for CART and RIPPER it reflects fit-time randomness on fixed data. Example exported rules. Table 9 lists the representative seed-42 exported DNF (fold 0, deduplicated) for the baseline and G-NRI on seven datasets, with the Boolean-schema feature names. For rule quality, the comparison to read is the exported rule’s per-dataset accuracy: by the 8-seed means in Table 4, the G-NRI rule matches or beats the baseline on all but one of these seven, trailing only on diabetes and there only marginally. A check mark flags a clause within a documented ground-truth target, which exists here only for monks-3 ; both variants recover it at seed 42, but only G-NRI exports it on every seed whereas the baseline occasionally collapses to a degenerate rule, so its mean accuracy is the higher. A bold dataset name marks where G-NRI selects substantially different literals from the baseline (literal-set Jaccard below 0.5); the change is in which literals it selects, not in rule size. On mushroom it keys on the absence of a foul odour rather than stalk colour, on diabetes it conjoins plasma glucose with age and body mass, and on spambase and vote it selects an almost disjoint feature set. On tic-tac-toe it reuses the baseline literals and adds the centre-square condition the baseline omits. These are single exports; on real data the rules are otherwise seed-unstable (cross-seed stability in Table 5b). Appendix F. Schema-relative signed-DNF quotient equivariance The rawB N test of the real-data rule-level evaluation applies arbitrary atom permutations and polarity flips, which break the one-hot exclusivity of categorical blocks and so probe the canonical export on inputs that no valid dataset can produce. This appendix models the exported rule’s output space as a signed-DNF quotient space and proves the canonical export equivariant on it under any subgroupH ≤ B N (Theorem 3). Specialised to the on-manifold subgroup that preserves a dataset’s categorical schema, the result explains the rawB N exceptions of Section 5.1 as score-equivariance violations under schema-invalid transforms, not decoder failures. F.1. The signed-DNF quotient space Let [N] =1,...,Nand letL N = [N]×+,−be the set of literals overNatoms. The signed-permutation groupB N =S N ⋉ Z N 2 acts on literals throughh= (π,f) by (i,ε)7→(π(i),ε⊕ f i ), whereπpermutes atom indices andf ∈0,1 N records a per-atom polarity flip. Define the signed-DNF quotient space Q ± N = finite sets of consistent clauses ×0, 1, where a clause is a set of literals. A clause is consistent when it does not contain both (i,+) and (i,−) for any atomi, and the second factor is the output-polarity rail bitbcarried by 21 Phua Table 9: Representative exported DNF (seed 42, fold 0) for the baseline and G-NRI.✓: clause within the ground-truth target; bold: rule differs substantially from the baseline; ¬: negated literal. DatasetMethod Representative exported DNF (seed 42, fold 0) monks-3baseline ¬ bodyshapeoctagon ∧ ¬ jacketblue ✓ G-NRI ¬ body shapeoctagon ∧ ¬ jacketblue ✓ mushroombaseline ¬ stalk-rootc ∧ ¬ stalk-rootr ∧ ¬ stalk-color-above-ringg ∧ ¬ stalk-color-below-ringg G-NRI ¬ odor f ∧ ¬ gill-colorb ∧ ¬ ring-typel ∧ ¬ spore-print-colorh tic-tac-toebaseline top-right-squarex ∧ middle-middle-squarex ∧ bottom-left-squarex ∧ bottom-right-squarex G-NRI(¬ middle-middle-squareo)∨(top-right-squarex ∧ middle-middle-square x ∧ bottom-left-squarex ∧ bottom-right-squarex) spambasebaseline ¬ wordfreqhpgtmedian ∧ ¬ wordfreqhplgtmedian ∧ ¬ word freqgeorgegtmedian ∧ ¬ wordfreqlabsgtmedian G-NRI(¬ wordfreq3dgtmedian ∧ ¬ wordfreqremovegtmedian ∧ ¬ wordfreqfontgtmedian ∧ ¬ wordfreqmoneygtmedian)∨ (¬ wordfreq3dgtmedian ∧ ¬ wordfreqremovegtmedian ∧ ¬ wordfreq000gtmedian ∧ ¬ wordfreqmoneygtmedian) credit-approvalbaseline ¬ A4 l ∧ ¬ A5g ∧ ¬ A6r ∧ ¬ A7z G-NRI ¬ A4l ∧ ¬ A5g ∧ ¬ A6r ∧ ¬ A7o votebaseline(¬ physician-fee-freezen ∧ ¬ el-salvador-aidn ∧ ¬ education-spending n ∧ ¬ crimen)∨(¬ physician-fee-freezen ∧ physician-fee-freezey ∧ ¬ el-salvador-aidn ∧ ¬ crimen) G-NRI(physician-fee-freezen∧¬ physician-fee-freezey ∧¬ export-administration-act-south-african ∧ export-administration-act-south-africay) ∨(¬ adoption-of-the-budget-resolutionn∧ physician-fee-freezen ∧ ¬ physician-fee-freezey ∧ ¬ export-administration-act-south-african) diabetesbaseline plas gtmedian G-NRI(plasgtmedian ∧ agegtmedian)∨(plasgtmedian ∧ massgtmedian ∧ agegtmedian) 22 Symmetry-Aware Foundation Model for Rule Induction the dual-rail selector. The quotient is purely syntactic (literal and clause order, padding, duplicates, and deletion of contradictory clauses), neither the semantic Boolean-equivalence quotient nor the group-action quotient. We writeqfor the quotient map and [·] for a class. For a dataset whose atoms are partitioned into one-hot categorical blocksB 1 ,...,B k , the schema symmetry subgroup is H Σ = Q k b=1 Sym(B b ) ≤ S N , the within-block index permutations, withf= 0 (no polarity flips). Everyh∈ H Σ maps a one-hot input to a one-hot input, soH Σ acts on the data manifold, whereas a generic element of B N does not. F.2. Hypotheses The canonical export is the compositionEof the score map Φ : (X,Y)7→(p + ,p − ) with the per-rail canonical decoder of Algorithm 1 and the dual-rail selector. We require two hypotheses for a subgroup H ≤ B N . (H1) Score-map equivariance. The per-literal soft scores satisfyp ε⊕f i t,π(i) (h· x) =p ε t,i (x) for everyh= (π,f)∈ Hand every slott. ForH Σ the flip vector isf= 0, so the scores only permute within blocks. (H2) Decoder equivariance. Literal selection, the presence scores j =max(p + j ,p − j ), the polarity 1[d j >0], clause gating, and rail-bit selection all commute with the action ofH, with ties resolved by anH-invariant rule or avoided by a strict decode margin (the margin-stability argument of Appendix B). The whole-tie-bucket selection of Algorithm 1 makes this hold. A raw atom-index tie-break would not be H-equivariant. F.3. Quotient action and canonicalisation Two facts underlie the theorem. TheB N action descends to a well-defined action onQ ± N . Hereh= (π,f) maps each complementary pair(i,+),(i,−)to(π(i),+),(π(i),−), so inconsistent clauses map to inconsistent clauses. As a bijection onL N it preserves the remaining syntactic equivalences (duplicate and ordered literals and clauses, padding) and fixes the rail bitb. The sorted-and-padded canonicaliserκ(literals in ascending (atom, polarity) order, clauses in a fixed canonical order, contradictory clauses removed) makesC=κ ◦ qconstant on quotient classes and idempotent (re-applying it changes nothing). F.4. Theorem Theorem 3 (Schema-relative signed-DNF quotient equivariance). LetH ≤ B N and assume (H1) and (H2) hold forH. Then for allh∈ Hand all inputsx, C h· C(E(x)) = C E(h· x) ,equivalently[E(h· x)] = h· [E(x)] in Q ± N . Proof By (H1) the score map commutes withh= (π,f)∈ H: each pair (p + i ,p − i ) is sent to atomπ(i), and the polarity flipf i swapsp + andp − at that atom. By (H2) the canonical 23 Phua Table 10: Schema-relative rule-level equivariance (3 seeds): worst-case ruleeq under schema- valid and raw off-manifold B N transforms (column definitions in the text). DatasetN#blockslog 10 |H Σ |schema ruleeq (all/ne)raw B N ruleeq Nontrivial schema group adult105887.331.000/1.0001.000 mushroom1162158.161.000/1.0000.905 † german-credit611321.251.000/1.0001.000 credit-approval46920.041.000/1.0001.000 kr-vs-kp733611.311.000/1.0001.000 nursery2788.251.000/1.0001.000 tic-tac-toe2797.001.000/1.0001.000 car2166.481.000/1.0001.000 vote32164.821.000/1.0001.000 monks-11764.321.000/1.0001.000 monks-21764.321.000/1.0001.000 monks-31764.321.000/1.0001.000 hepatitis32133.911.000/1.0001.000 Trivial schema group (H Σ =e) breast-cancer-wisconsin900.001.000/1.0001.000 clevr-hans310500.001.000/1.0001.000 diabetes800.001.000/1.0001.000 ionosphere3400.001.000/1.0001.000 mutag5100.001.000/1.0000.875 † spambase5700.001.000/1.0001.000 decoder reads only the presence scores j , the contrast signsign(d j ), the clause gates, and the rail key, each of which commutes with the action ofh. The flipf i leavess i =max(p + i ,p − i ) unchanged and negatesd i , exchanging the literals (i,+) and (i,−), while the permutationπ relabels atoms, and the tie-bucket selection consults only the multiset of presence scores, never an atom index. Hence the multiset of decoded clauses produced fromh· xequals the image underhof the multiset produced fromx, as literal sets. Since the action descends toQ ± N , this image is well defined, so [E(h· x)] =h·[E(x)]. Applying the canonicaliserC, which is constant on classes, gives C(E(h· x)) = C(h· C(E(x))). Specialising toH Σ (wheref= 0) gives exact equivariance on the data manifold: the canonical export commutes with every within-block relabelling of one-hot categorical features. The example-axis groupS M acts trivially on the rule, andZ y 2 toggles only the rail bitb, so the full schema group G Σ = S M × H Σ × Z y 2 is covered by the same argument. Exact versus approximate. Theorem 3 is exact given (H1) and (H2). When the score map is only approximately equivariant, exact rule equality still follows whenever the score drift stays below the decode margins of the margin-stability argument, since every discrete decision then agrees between the two frames. This explains the rawB N exceptions of Section 5.1. They are violations of (H1), a forward-score drift induced by schema-invalid cross-block permutations that crosses the margin, not failures of the decoder (H2), which holds by construction. 24 Symmetry-Aware Foundation Model for Rule Induction F.5. Schema-valid empirical equivariance We instantiateH Σ per dataset from its one-hot block structure and re-run the rule-level equivariance test under the schema-valid transforms only. Of the 19 datasets, 13 have a nontrivial schema groupH Σ ; the other 6 (breast-cancer-wisconsin, clevr-hans3, diabetes, ionosphere, MUTAG, spambase) haveH Σ =e, so their schema ruleeq of 1.000 is vacuous; Table 10 reports them in a separate block. On all 13 nontrivial-schema datasets, across 3 seeds and every schema-valid transform, schema ruleeq (both all pairs and nonempty pairs) equals 1.000 exactly, with an unresolved rail-tie rate of 0, matching the prediction of Theorem 3. In Table 10, “#blocks” counts the one-hot blocks, log 10 |H Σ | gives the schema subgroup’s order, “schema ruleeq (all/ne)” is the worst case over seeds and schema-valid transforms, “rawB N ” the worst case under this audit’s single schema-invalid transform, and † flags the two datasets falling below 1. Per-dataset notes. The main recovery is mushroom, which has a large nontrivial schema group (log 10 |H Σ |= 58.2 over 21 one-hot blocks) and fails this audit’s rawB N transform (ruleeq per seed 0.965/0.925/0.905). UnderH Σ its schema ruleeq is 1.000 with 100% nonempty coverage and non-degenerate rules (≈1.2 clauses, 4.3 literals on average), so the raw failure was a schema-invalidity artefact rather than a decoder defect. MUTAG is the one rawB N failure (worst-case ruleeq 0.875) that schema-relativity does not explain. Its 51 features are non-mutually-exclusive chemical indicators with no one-hot blocks, soH Σ =e. Resolving it would require a domain-specific symmetry beyond categorical schema, which we leave open. Finally, the One-vs-Rest positive-class-2 slice of nursery produces all-empty (abstaining) rules for seeds 42 and 43, so equivariance on that slice is a trivial empty match (its nonempty coverage is 0). The other slices of nursery are genuine. 25