Instruction Set and Language for Symbolic Regression
Symbolic regression search spaces suffer from structural redundancy: expression DAGs with $k$ internal nodes admit $\Theta(k!)$ distinct node-numberings that encode the same mathematical expression. This paper proposes IsalSR, a representation framework that computes a pruned canonical string—a complete labeled-DAG isomorphism invariant—to collapse all equivalent forms into a single canonical representation. The approach promises to reduce effective search space size by $O(k!)$ and can be integrated into any existing SR algorithm as a preprocessing step.
The paper presents a novel and technically rigorous solution to the structural redundancy problem in symbolic regression. The two-tier instruction alphabet and pruned backtracking canonicalization algorithm are well-conceived, and the empirical validation across 961 DAGs is comprehensive. However, the central theoretical guarantees remain unproven conjectures, and the absence of end-to-end integration experiments with actual SR solvers limits the demonstration of practical utility.
The representation design is elegant: the alphabet ensures every string decodes to a valid DAG without rejection filters, and the commutative encoding (replacing Sub/Div with unary Neg/Inv) simplifies isomorphism to handling only the Pow operator. The search space reduction claim is empirically validated through exhaustive permutation testing showing that for $k \leq 8$ nodes, all $k!$ equivalent representations collapse to one canonical string, with 100% invariance verified across all 961 test DAGs.
The paper relies on two central conjectures (round-trip fidelity and canonical invariance) whose proofs are explicitly deferred as "future work," leaving the theoretical foundations incomplete. The canonicalization algorithm fails on degenerate structures containing "edges directed into a Var node," causing three P3 failures in validation. Most critically, the scalability study CPU measurements are "under analysis and will be reported in a subsequent revision," and integration experiments demonstrating actual fitness evaluation savings in SR pipelines are entirely absent—only property validation is shown.
The evidence strongly supports the isomorphism invariance property through exhaustive testing of up to $k! = 40,320$ permutations per DAG for $k \leq 8$, confirming the Orbit-Stabilizer theorem predictions. However, the paper lacks empirical comparison with alternative canonicalization schemes or baseline SR methods—while related work like GraphSR is cited, no comparative metrics (e.g., canonical string length, computation time) are provided. The benchmarks (Nguyen, AI Feynman) validate representation properties but are not used to demonstrate superior search efficiency against non-canonicalized baselines.
The algorithms are documented with detailed pseudocode (Tables 2–4) and deterministic random seeds (base seed 42), facilitating reproducibility. However, the promised "public code repository" is not linked in the text, and the scalability experiment data (CPU time vs. DAG size) is explicitly withheld as "under analysis." Without the code repository URL or the missing scalability results, independent verification of the computational claims and reproduction of the canonicalization timings is currently impossible.
A fundamental but largely unaddressed obstacle in Symbolic regression (SR) is structural redundancy: every expression DAG with admits many distinct node-numbering schemes that all encode the same expression, each occupying a separate point in the search space and consuming fitness evaluations without adding diversity. We present IsalSR (Instruction Set and Language for Symbolic Regression), a representation framework that encodes expression DAGs as strings over a compact two-tier alphabet and computes a pruned canonical string -- a complete labeled-DAG isomorphism invariant -- that collapses all the equivalent representations into a single canonical form.
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