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Augmenting Rewrite Rule Sets via Knuth-Bendix Completion
Michael Schifferer, Marcel Ullrich, Sebastian Hack
TL;DR
EqSat explores equivalent programs, but its performance depends on rewrite rules that are usually handcrafted and may miss valuable transformations. This paper evaluates KBC for generating and augmenting those rules, finding improved EqSat simplification speed and effectiveness alongside a speed-versus-effectiveness trade-off for greedy rewriting.
Problem
EqSat depends on rewrite rules that developers handcraft using domain expertise, and manual rule creation can miss valuable transformations.
Source: §1 · p. 1Method
The paper applies Knuth-Bendix Completion to handwritten rules to generate and augment rewrite rules for program optimization.
Source: §1 · p. 1Results
KBC-augmented handwritten rule sets significantly improve EqSat simplification speed and effectiveness, while greedy rewriting is several orders of magnitude faster than EqSat but less effective under tested conditions.
Source: §6 · pp. 5–6Takeaways & Limitations
Critical pairs shorten rewrite chains and can provide novel rewrites, but adding too many can increase e-graph size and decrease performance.
Source: §6 · p. 6Greedy use of a partially completed KBC system rests on the hypothesis that it retains confluent properties; KBC produces a confluent system if it terminates.
Source: §3 · p. 3
Abstract
from arXiv · showhide
Equality Saturation (EqSat) is a powerful technique for program optimization, systematically exploring the search space of candidate programs to overcome the phase ordering problem. However, the feasibility and performance of EqSat rely heavily on the specific rewrite rules used to derive equivalent programs. These rule sets are typically handcrafted, requiring extensive domain expertise and carrying the risk of missing valuable transformations. In this work, we evaluate Knuth-Bendix Completion (KBC) as a method for automatically generating and augmenting rewrite rules for EqSat. Our experiments show faster optimization as well as reaching better terms with previously missed optimizations.
1 Introduction
Equality saturation explores equivalent programs without committing to one rewrite path, but its effectiveness depends on the rewrite rules available. The paper evaluates Knuth-Bendix Completion (KBC) as a way to augment handwritten rules and reports faster, more effective optimization.
1 Introduction
- EqSat represents multiple equivalent programs in an e-graph and explores candidate programs without committing to a single rewrite path.This approach addresses the phase ordering problem, in which optimization order limits final program quality.Source: §1 · p. 1
- EqSat's feasibility and performance depend on rewrite rules that developers handcraft using domain knowledge, intuition, and extensive testing.Manual rule creation can miss valuable transformations, and little is known about the properties of highly effective rule sets.Source: §1 · p. 1
- The paper processes an initial handwritten rule set with KBC, then evaluates generated rules in standard EqSat and a custom greedy rewriting engine.Greedy rewriting trades EqSat's exhaustive exploration for compile-time and memory efficiency, with a risk of inferior result quality.Source: §1 · p. 1
- The paper presents methods for generating and processing rewrite rule sets for program optimization using KBC.Its experimental evaluation compares KBC-augmented rule sets with strictly handwritten rules and examines greedy rewriting as an alternative to EqSat.Source: §1 · pp. 1–2
- KBC-augmented handwritten rules improve EqSat simplification effectiveness and running time compared with strictly handwritten rules.The evaluation also examines greedy rewriting as a resource-efficient alternative, though with less effective simplification than EqSat.Source: §1 · p. 2
2 Term Rewriting Systems
Knuth-Bendix Completion (KBC) uses a simplification order to orient equalities into rewrite rules and uses critical pairs to bridge divergent rewrites. These operations support completion toward a confluent, terminating rewrite system when KBC succeeds.
2 Term Rewriting Systems
- A successful KBC run produces a confluent and terminating term rewriting system (TRS), in which equivalent terms simplify to a unique normal form.The process can instead fail on unorientable equations or fail to terminate.Source: §2 · p. 2
- A simplification order must satisfy compatibility, stability, and subterm ordering; Knuth-Bendix Order weighs terms and breaks ties using precedence and lexicographic checks.Compatibility preserves the order when terms are placed in the same function context, while subterm ordering places a term above each proper subterm.Source: §2 · p. 2
- KBC orients an equality s = t into s → t when s ≻ t, and critical pairs add an equality when a term rewrites along two divergent paths.The added rule bridges the divergence and ensures confluence by creating a shortcut through the proof's peak.Source: §2 · p. 2
- Figure 1 depicts a rewrite proof in which critical pairs tunnel through peaks.The caption identifies the proof's critical-pair shortcut; the accompanying text explains that it bridges divergent rewrite paths.Source: Figure 1 · p. 2, §2 · p. 2
- KBC continuously applies elementary inference rules to equalities E and rewrite rules R.A standard run may succeed with a confluent TRS, fail on unorientable equations, or fail to terminate.Source: §2 · p. 2
3 Approach
The approach uses KBC to generate and process rewrite rules from handwritten axioms, then evaluates the rules in EqSat and greedy rewriting. It varies how unorderable and partially completed rules are handled, including safeguards for partial operations.
3 Approach
- The rule-generation phase uses the twee theorem prover to apply KBC to handwritten arithmetic axioms and derive critical pairs that augment the original rules.The pipeline then deploys generated rule sets across two rewriting environments.Source: §3 · pp. 2–3
- For unorderable rules, the authors compare removing them with keeping both orientations, vary completion limits and replacement versus augmentation, and process partial operations separately.The separate processing is intended to preserve soundness and avoid invalid states such as division by zero.Source: §3 · p. 3
- The authors retain original rules because EqSat uses expanding rules to explore a large term state space, and classify generated rules as normal, shortcut, or novel.Novel rules include rewrites that require inventing a value and are not achieved by EqSat's rule applications.Source: §3 · p. 3
- Rules for partial functions such as division need preconditions: otherwise, 0/x = 0 and x/x = 1 can prove 1 = 0 when x = 0.The approach encodes preconditions as if-then-else constructs, following Claessen and Smallbone [2021].Source: §3 · p. 3
- The EqSat environment exhaustively explores equivalent program representations by applying all possible rules without committing to one rewrite path.This setup measures the maximum simplification power of the KBC-generated rules.Source: §3 · p. 3
- The custom greedy engine applies the same rules sequentially and deterministically until none match, selecting them in Twee's generation order.Unlike EqSat, it does not explore multiple paths; the comparison isolates rule-set effects across the two environments.Source: §3 · p. 3
- The authors test KBC-augmented rule sets in EqSat and report significantly improved simplification speed and effectiveness.KBC produces a confluent rewrite system if it terminates; using a partially completed system greedily is a hypothesis that it retains confluent properties.Source: §3 · p. 3
- Greedy use of a partially completed rewrite system relies on the authors' hypothesis that it exhibits confluent properties.KBC produces a confluent rewrite system if it terminates.Source: §3 · p. 3
4 Experimental Evaluation
The experiments compare KBC-generated rules in EqSat and greedy rewriting, testing rule-set augmentation, size, and postprocessing. Augmentation improves EqSat simplification, while greedy rewriting trades lower effectiveness for much faster execution.
Equality Saturation Performance
- Replacing handwritten rules with KBC-generated rules can leave EqSat in local optima because it needs complexity-increasing rules to bridge equivalent states.The example expands (x^3 - 1)/(x - 1) before simplifying it; KBC retains only reducing rules.Source: §4 · p. 4
- Augmenting handwritten rules with KBC-generated rules yields consistently better EqSat results.Generating up to 150 additional rules maximizes effectiveness; larger sets slow convergence without meaningful term-reduction gains.Source: §4 · p. 4
- 10.46 versus 9.73 mean symbol reduction on large terms with partial functions: E-KBC150 outperforms the E-base handwritten baseline.The study evaluates randomly generated arithmetic terms across both EqSat and greedy rewriting.Source: §4 · pp. 3–4
Greedy Rewriting Performance
- Greedy rewriting with KBC rules improves as the set grows to 1,000 rules, then begins to plateau.Handwritten rules rely on EqSat's ability to increase term size, while KBC rules provide directly simplifying paths for greedy rewriting.Source: §4 · p. 4
Cross-Framework Comparison
- EqSat achieves higher mean symbol reduction, while greedy rewriting reaches its optimum several orders of magnitude faster.On large terms with partial functions, G-KBC1000 takes 0.000081 seconds versus 0.05 seconds for E-KBC150.Source: §4 · p. 4
5 Related Work
Related work includes EqSat applications that rely on manually constructed rewrite rules and approaches to easing rule creation. This paper evaluates KBC for program optimization and notes its potential fit with acyclic e-graphs.
Equality Saturation
- EqSat is used in domains including Herbie and Felix, but practical applications rely almost exclusively on manually constructed rewrite rules.The cited work introduces EqSat to address phase ordering and egg to improve extensibility and performance.Source: §5 · p. 5
Rewrite Rule Synthesis
- Ruler infers rules from empirical behavior, whereas KBC provides formal confluence properties; this paper addresses the limited evaluation of KBC in program optimization.The passage also situates fast relational e-matching and SmoothE as approaches to easing EqSat workflows.Source: §5 · p. 5
Acyclic E-Graphs
- Acyclic e-graphs require directional rewrites, a constraint the authors say aligns with KBC's strictly directed rules.They suggest KBC-generated rules may suit optimization passes in acyclic e-graph frameworks.Source: §5 · p. 5
6 Conclusions
The conclusion reports improved EqSat speed and effectiveness from augmenting handwritten rules with KBC-generated rules, alongside a trade-off for greedy rewriting. Future work proposes expanding the evaluation and exploring ways to identify useful critical pairs.
6 Conclusions
- Augmenting handwritten rules with KBC-generated rules improves EqSat simplification speed and effectiveness.Critical pairs can shortcut rewrite chains and provide novel rewrites, but too many can enlarge the e-graph and reduce performance.Source: §6 · pp. 5–6
- Greedy rewriting is several orders of magnitude faster than EqSat but achieves significantly lower simplification effectiveness under the tested conditions.The authors describe KBC as valuable for enhancing EqSat and reducing the manual burden of rule engineering.Source: §6 · p. 6
6.1 Future Work
- Future work includes evaluating KBC on bitvectors and finding useful critical pairs during e-graph saturation.The authors also propose heuristics for helpful critical pairs and automatic rule generation with properties that could make KBC unnecessary.Source: §6.1 · p. 6