JuliaSymbolics and Hash Consing

• JuliaSymbolics and Hash Consing is a framework that ensures unique symbolic subexpressions via canonicalization to avoid redundancy.
• It uses a global weak-reference hash table and efficient alpha-equivalence hashing, yielding O(1) amortized performance in deduplication.
• Benchmark results demonstrate substantial improvements in memory usage and computation speed for large-scale symbolic computations and code generation.

JuliaSymbolics, a high-performance symbolic computation ecosystem for the Julia programming language, has incorporated hash consing as a central mechanism for canonicalizing and deduplicating expression trees. This integration targets the memory and performance bottlenecks inherent in symbolic manipulation, especially expression swell due to repeated subterms. Combined with recent advances in hashing modulo alpha-equivalence, JuliaSymbolics now enables highly efficient structure sharing and canonicalization, which benefit large-scale symbolic computations and model-based code generation pipelines.

1. Hash Consing Foundations in JuliaSymbolics

Hash consing is a technique for maintaining a unique, canonical instance of every distinct symbolic subexpression (up to structural or alpha-equivalence). In JuliaSymbolics, each Expr node’s unique identity is determined at construction time by a hash function, and the system maintains a global hash table:

$$\mathcal{T} : \mathbb{H} \longrightarrow \{\text{weak references to canonical } \mathit{Expr}\}$$

where $\mathbb{H}$ is the 64- or 128-bit hash universe. The global table is implemented as a WeakValueDict, so entries are purged when no strong references remain. The initialized table:

const GLOBAL_CONS_TABLE = WeakValueDict{UInt64, Expr}()

ensures that for any strongly-reachable expression $e$, $\mathcal{T}[e.\text{hash}]$ either yields the canonical pointer for $e$ or nothing if the entry expired. Both lookup and insertion amortize to $O(1)$ on average due to the hash-table’s design (Zhu et al., 24 Sep 2025).

2. Hash-Consing Procedure and Complexity

Every new node is constructed by routing through the hash cons function:

function hash_cons(e::Expr) h = e.cached_hash === nothing ? compute_hash(e) : e.cached_hash e.cached_hash = h existing = GLOBAL_CONS_TABLE[h] if existing !== nothing && structurally_equivalent(existing, e) return existing end GLOBAL_CONS_TABLE[h] = e return e end

This procedure ensures structural uniqueness except for unlikely hash collisions. In LaTeX pseudocode:

$$\begin{array}{l} \textbf{Procedure } \mathsf{CONS}(e):\ 1. h \leftarrow \text{cached or fresh hash of } e\ 2. \text{if } \mathcal{T}[h] \equiv_{\rm struct} e \text{ then return } \mathcal{T}[h]\ 3. \mathcal{T}[h] \leftarrow e\ 4. \text{return } e \end{array}$$

The structural check $\equiv_{\rm struct}$ is executed only on hash collisions, which are extremely rare for 64/128-bit hashes. The amortized computational cost per operator is $O(1)$.

A direct implication for symbolic manipulation is that expression formation transitions from tree construction ($O(N_{\text{tot}})$ nodes) to DAG formation ($O(N_{\text{uniq}})$ distinct subterms), fundamentally reducing both time and space when repeated subexpressions are abundant (Zhu et al., 24 Sep 2025).

3. Integration with Julia’s Macro, IR, and Code Generation Pipeline

Hash consing is interposed at several layers in JuliaSymbolics:

• AST Construction: Macros such as @variables and overloaded symbolic constructors in SymbolicUtils.jl invoke hash consing for each node, yielding DAGs rather than trees.
• IR Lowering: After macro expansion, the symbolic AST (now a DAG) is lowered to Julia’s intermediate representation, with maximal term sharing.
• Code Generation: The code generator performs a topological traversal of the hash-consed DAG, emitting each subterm exactly once. The resulting IR is more compact, and Julia’s JIT can perform aggressive inlining and CSE.
• Memory Management: Weak references ensure that unused subterms are eligible for garbage collection, preventing memory leaks.

This pipeline realizes cross-stage optimization: symbolic differentiation and simplification benefit from subterm sharing, and downstream tasks (such as code generation or evaluation) traverse a much smaller, uniquely-shared graph (Zhu et al., 24 Sep 2025).

4. Memory and Computational Complexity Analysis

For a symbolic computation with $N_{\rm tot}$ raw subterms (as in a tree) and $N_{\rm uniq}$ unique subterms (as in a DAG after consing):

• Memory usage:

$$\text{Pre-consing:} \ M_0 = \Theta(N_{\rm tot}) \ \text{Post-consing:} \ M = \Theta(N_{\rm uniq}) \ \text{Reduction factor:} \ \frac{M_0}{M} = \frac{N_{\rm tot}}{N_{\rm uniq}}$$

• Time complexity:

$$\text{Differentiation pre-consing:} \ T_0 = O(N_{\rm tot}); \quad \text{post-consing:} \ T = O(N_{\rm uniq})$$

For operations such as Jacobian computation in sparse models, $N_{\rm tot}/N_{\rm uniq}$ can be $O(n)$ or higher, directly translating to a similar factor reduction in memory and computation (Zhu et al., 24 Sep 2025). Code generation and downstream compilation stages realize similar multiplicative speedups due to replacing $O(N_{\rm tot})$ tree traversals with $O(N_{\rm uniq})$ DAG traversals.

5. Practical Benchmark Results

The implementation yields varying improvements based on the degree of subterm duplication. In benchmark models:

Stage	Time Speedup (BCR)	Memory Reduction (BCR)	Time Speedup (XSteam)	Memory Change (XSteam)
Jacobian	3.2×	2.0×	0.8× (slower)	0.4× (↑2.5×)
Code Generation	1.5×	1.8×	5.0×	1.0×
Compilation	2.0×	–	10.0×	–
Evaluation	2.0×	–	20–100×	–

Results indicate that workloads with pervasive subterm overlap (e.g., large biochemical models) scale especially well. For less redundant workloads (XSteam Jacobian), hash consing introduces modest overhead (e.g., 0.8× time, 2.5× memory) in the initial symbolic stage, but downstream benefits (codegen, evaluation) remain substantial. Figures in the source study illustrate that for increasing model size, downstream speedups consistently dominate any upfront consing cost (Zhu et al., 24 Sep 2025).

6. Hashing Modulo Alpha-Equivalence

Standard hash consing only deduplicates subterms up to structural identity. To recognize terms identical under variable renaming, integration of an efficient alpha-equivalence hash function is required (Maziarz et al., 2021). The method, as implemented for JuliaSymbolics, maintains per-node summaries consisting of a “structure hash” and a “VarMap” (tracking variable occurrence patterns):

struct ESumm struct_hash::UInt varmap::Dict{Symbol,UInt} end

The main hash function per node $e$ is:

• $H_S$ (“structure hash”): a strong random-compositional summarization of the node type and children.
• $H_V$ (“varmap hash”): uses XOR as a commutative aggregator over variable–position hashes.

Overall hash: $\text{hashESumm} = F'(H_S, H_V)$, with $F'$ a strong binary combiner. This procedure:

1. Guarantees that alpha-equivalent expressions hash identically.
2. Achieves $O(n (\log n)^2)$ time complexity for $n$-node ASTs, with near-optimal empirical performance compared to alternatives (e.g., locally-nameless, De Bruijn indexing).
3. Collision probability is well-controlled: for $b$-bit hashes and trees up to $10^9$ nodes, the probability $\ll 10^{-10}$ with $b=128$.

Empirical measurements in machine learning ASTs demonstrate $>10\times$ speedups over locally-nameless approaches for large, unbalanced trees. Integration into JuliaSymbolics simply requires a pre-pass to uniquify binders and a per-node hash attachment on construction (Maziarz et al., 2021).

7. Limitations, Trade-Offs, and Future Directions

While hash consing in JuliaSymbolics delivers substantial improvements when duplicate subexpressions are frequent, there are inherent trade-offs:

• Low-duplication workloads: Overhead from hashing and weak-ref management can result in small slowdowns and increased memory (up to 2.5× as observed). This derives from the cost of recursive hash calculation without subsequent sharing.
• Parallel symbolic construction: The current global table is not shared cross-thread; introducing cross-thread tables could further amortize work.
• Equivalence-aware sharing: Canonicalization is structural or alpha-based only; for higher-level algebraic equivalence, integration with e-graph frameworks is identified as a key extension.

Adaptive hash consing—dynamically toggling consing in regions with low expected redundancy—could mitigate some overhead. The anticipated integration of e-graphs would enable sharing among algebraically equivalent subterms, addressing higher-level expression duplication encountered in AI-driven symbolic reasoning (Zhu et al., 24 Sep 2025).

• "Efficient Symbolic Computation vis Hash Consing" (Zhu et al., 24 Sep 2025)
• "Hashing Modulo Alpha-Equivalence" (Maziarz et al., 2021)