E-Graph Retargetable Approach

• The paper introduces an e-graph based retargetable approach that employs equality saturation and ILP-based extraction to achieve global optimization across diverse EDA tasks.
• It details a systematic methodology involving e-graph construction, rewrite system design, and cost modeling to drive significant improvements in area and delay.
• The approach demonstrates robust retargetability, allowing seamless adaptation to varying backend constraints and synthesis objectives with formal correctness guarantees.

An E-graph based retargetable approach is a formal methodology for design-space exploration, optimization, and synthesis in electronic design automation and compilation, centering on the e-graph data structure. An e-graph compactly encodes large exponential spaces of equivalent expressions under structural, algebraic, or semantic rewriting rules, enabling global optimization through equality saturation and decoupled extraction of canonical representations tailored to particular hardware technologies or optimization objectives. Recent advances demonstrate its use for logic synthesis, high-level synthesis, register-transfer level (RTL) optimization, multiplier design, automatic code proof, and even hardware-software co-design for processor architectures, with retargetability referring to the framework's ability to adapt to new metrics or backend constraints by substituting rewrite sets or cost models. This approach has yielded significant Area/Delay improvements, robust retargeting across operand widths or libraries, and proven correctness guarantees.

1. E-Graph Structure: E-Nodes, E-Classes, and Congruence Closure

The e-graph is a directed acyclic graph (DAG)-like structure comprised of e-classes (equivalence classes) and e-nodes (operator applications plus child links) (Chen et al., 21 Mar 2024, Coward et al., 18 Jun 2024, Wanna et al., 2023). Each e-class encodes a set of semantically equivalent Boolean, bit-vector, or IR expressions. An e-node is a tuple $\langle$op, c₁, …, cₖ$\rangle$ specifying the operator and pointers to operand e-classes.

Key properties:

• Hash-consing is employed for operator/child tuples—structurally identical instantiations merge.
• Union-find algorithms manage e-class merges as new equivalences are discovered by rewrite application.
• Congruence closure is maintained: if two child classes $c_i ≡ d_i$, then nodes with identical operators and child links reside in the same class.
• Annotated operators preserve arithmetic, bitwise, control-flow, or semantic tags, including operand bitwidth or characteristics for RTL/HDL contexts (Coward et al., 2022).

Initialization involves parsing the circuit, IR, or expression into e-node fragments, recursively mapping their structure into e-classes.

2. Equality Saturation and Rewrite System Design

Equality saturation is the central search strategy: rather than incrementally applying heuristically ordered transformations, all rewrite rules fire in parallel over the e-graph, growing the set of represented implementations without discarding prior forms (Chen et al., 21 Mar 2024, Kourta et al., 2021, Cheng et al., 2023). This is phase-order-insensitive and avoids local minima by postponing selection until saturation.

Components:

• Rewrite rules are defined as pairs or triples (cond, LHS, RHS), parameterized by attributes (bitwidth, signage).
• Pattern-matching applies LHS patterns to all e-classes; each match under cond(m) triggers insertion of RHS into the same class.
• Conditional rewrites enable dynamic legality checks (e.g., loop fusion, width constraints) by invoking external transformations or checking predicates.
• Saturation is bounded by time or node-count limits (e.g., 2.5 million nodes or 300 s).

Acceleration techniques:

• Iteration Level Check halts rewrites on reaching a proving goal (Kourta et al., 2021).
• Pulsing and Non-Provable Patterns Detection balance exploration/exploitation and early termination in code-proving contexts.
• Divide-and-conquer and hierarchical instantiation enable scalability for large multipliers, datapaths, or parameterized designs (Wanna et al., 2023).

3. Extraction Algorithms and Cost Modeling

After saturation, exponentially many equivalent implementation candidates exist in the e-graph. Extraction selects one per output e-class to build the physical circuit or IR (Chen et al., 21 Mar 2024, Coward et al., 18 Jun 2024, Wanna et al., 2023, Coward et al., 2022).

Common extraction strategies:

• Pool extraction: sample candidates by minimum depth, minimum size, and randomized selection to cover diverse cost optima (Chen et al., 21 Mar 2024).
• ILP-based extraction: formulate a 0–1 integer linear program over e-node variables $x_n$ to minimize $\sum_n cost(n) x_n$, with constraints for consistency, acyclicity, and selection (one node per e-class). This enables global reuse of sub-expressions, crucial for area/delay optimization in RTL synthesis (Coward et al., 18 Jun 2024, Coward et al., 2022).
• Hierarchical/phase-based extraction: some flows (e.g., multipliers) perform distinct reduction and Boolean gate-phase extractions (Wanna et al., 2023).

Cost functions:

• Technology-aware: regression models (e.g., XGBoost over AST features) predict true area or delay post-mapping for each candidate (Chen et al., 21 Mar 2024).
• Theoretical area: gate count, delay per operator type, or domain-specific metrics (e.g., prefix adders, Booth multipliers) (Coward et al., 18 Jun 2024, Coward et al., 2022).
• Control/data-flow hybrid models: latency for loops, area for arithmetic/data nodes, with cost constraints reflecting backend scheduling (Cheng et al., 2023).

Interchangeability of cost models enables retargeting: substituting the regression, theoretical, or ILP function steers the engine toward new QoR or technology libraries.

4. Retargetability Across Technologies, Parameters, and Synthesis Goals

Retargetability denotes the capacity to swap out rewrite sets or cost models and thereby adapt the engine to new backend standards, operand widths, synthesis objectives, or domain-specific architectures.

Mechanisms:

• Parametric rewrites: symbolic bitwidth/signage parameters in patterns and conditions allow a single pass to build implementations for all sizes and operand types (Coward et al., 18 Jun 2024, Coward et al., 2022).
• Pluggable cost models: as synthesis metrics or target libraries change, swapping the evaluation model is all that is required for the engine to optimize for area, delay, power, or timing (Chen et al., 21 Mar 2024, Coward et al., 18 Jun 2024).
• Automated library generation: for parameterized RTL or HLS designs, the approach yields entire libraries specialized for each width, architecture, or instruction-set extension (Coward et al., 18 Jun 2024, Coward et al., 2022).
• Backend constraints: direct incorporation of extracted loop parameters, latency bounds, or area caps into the e-graph extraction and ILP optimization (Cheng et al., 2023).

For code-proving, the core engine is parameterized by frontend IR, ruleset, and extraction metric, making it universally retargetable within any compiler or DSL (Kourta et al., 2021).

5. Key Experimental Results and Impact

Across logic, RTL, HLS, and compiler contexts, e-graph based retargetable approaches have demonstrated:

• Logic synthesis: E-Syn achieves on average 15.29% delay savings (delay-mode) and 6.42% area savings (area-mode) compared to state-of-the-art AIG-based ABC flows, with a wider Pareto frontier and an additional 21% delay reduction via pool extraction + ML cost evaluation (Chen et al., 21 Mar 2024).
• RTL optimization: ROVER achieves up to 63% area reduction at equal/better timing, auto-adapting among ripple/CSA architectures across widths, and outputs equivalence certificates for back-end verification (Coward et al., 18 Jun 2024).
• Code proof: Caviar+ proves 90% of Halide expressions (vs. 81% for Halide’s greedy TRS), running 18.9× faster than vanilla saturation and retargetable to arbitrary DSLs (Kourta et al., 2021).
• HLS and control/data-flow optimization: SEER yields up to 38× speedup with only 1.4× area overhead, with a unified MLIR↔e-graph pipeline, dynamic MLIR passes, and robust retargetability (Cheng et al., 2023).
• Multiplier circuits: OptiMult delivers up to 46% latency reduction for squarers and 9% for multipliers, discovering non-standard architectures missed by manual design, scalable up to 10 bits (Wanna et al., 2023).
• Parameterized datapath optimization: e-graph approaches generalize carry-save clustering and MCM solutions, up to 71% area and 77% delay reductions over hand-tuned RTL, and yield piecewise-optimal architectures across bitwidths (Coward et al., 2022).

6. Formal Guarantees, Scalability, and Limitations

Formal properties:

• Congruence and soundness: e-class merges retain all equivalences; every output drawn via extraction is functionally faithful under the rewrite set (Coward et al., 18 Jun 2024).
• Certificate production: chains of rewrites yield formal equivalence proofs, compatible with industrial regression flows (Coward et al., 18 Jun 2024).

Scalability:

• Equality saturation, though exponential in general, is tractable in practice with heuristics (node/time caps, divide-and-conquer, phase separation) and fast union-find structures.
• Accelerated variants (pulsing, goal detection, dynamic rewrites) mitigate runtime overhead in compilers (Kourta et al., 2021, Cheng et al., 2023).

Limitations:

• Saturation and extraction complexity grow with larger rulesets or operand sizes; maintaining tractable e-graph sizes requires judicious application ordering and dynamic rewriting (Wanna et al., 2023, Cheng et al., 2023).
• Some extraction modes may neglect subexpression sharing (fan-out=1), trading area for simplicity (Wanna et al., 2023).
• ILP formulations are NP-hard but solved efficiently in practice for reasonable graph sizes (Coward et al., 18 Jun 2024, Coward et al., 2022).

7. Future Prospects and Cross-Domain Extensions

The e-graph based retargetable methodology is extending into:

• Holistic hardware-software co-design: for ASIP and domain-specialized processor landscapes, e-graph pattern matching enables automated exploitation of new custom instructions (ISAX), skeleton-component matching, and robust integration with MLIR and downstream LLVM flows (Zou et al., 27 Nov 2025).
• Context-aware AR scene matching: “E-Graph” scene graphs enable semantic retargeting of AR layouts across diverse indoor environments, abstracting spatial relations for content arrangement (Tahara et al., 2020).
• Ongoing work involves hybrid rewrite models, integration with machine learning-driven cost prediction, and further scalability optimizations.

The approach represents a unified paradigm for logic, RTL, compiler, and software/hardware co-design, with proven gains in area, delay, flexibility, and correctness, underpinned by the e-graph structure and retargetable extraction.