BoolE: Exact Symbolic Reasoning via Boolean Equality Saturation

2025-04-08

AI Summary

This work addresses two key challenges in Boolean symbolic reasoning over gate-level netlists: imprecise functional block identification and high sensitivity of inference results to technology mapping and logic optimization. We propose the first precise Boolean symbolic reasoning framework based on equality saturation. Our method introduces a domain-specific set of Boolean rewriting rules and a novel multi-input multi-output structural extraction algorithm, integrated with e-graph construction, symbolic propagation, and structure-aware hashing to enhance structural insight and inference efficiency. On multiplier architectures, our approach identifies 3.53× more full adders than ABC for carry-save addition (CSA) and 3.01× more for Booth encoding. When integrated into formal verification tasks, it achieves over four orders-of-magnitude speedup in runtime.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Computer Vision: Visual Reasoning & Symbolic Representations

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Boolean symbolic reasoning for gate-level netlists is a critical step in verification, logic and datapath synthesis, and hardware security. Specifically, reasoning datapath and adder tree in bit-blasted Boolean networks is particularly crucial for verification and synthesis, and challenging. Conventional approaches either fail to accurately (exactly) identify the function blocks of the designs in gate-level netlist with structural hashing and symbolic propagation, or their reasoning performance is highly sensitive to structure modifications caused by technology mapping or logic optimization. This paper introduces BoolE, an exact symbolic reasoning framework for Boolean netlists using equality saturation. BoolE optimizes scalability and performance by integrating domain-specific Boolean ruleset for term rewriting. We incorporate a novel extraction algorithm into BoolE to enhance its structural insight and computational efficiency, which adeptly identifies and captures multi-input, multi-output high-level structures (e.g., full adder) in the reconstructed e-graph. Our experiments show that BoolE surpasses state-of-the-art symbolic reasoning baselines, including the conventional functional approach (ABC) and machine learning-based method (Gamora). Specifically, we evaluated its performance on various multiplier architecture with different configurations. Our results show that BoolE identifies $3.53 imes$ and $3.01 imes$ more exact full adders than ABC in carry-save array and Booth-encoded multipliers, respectively. Additionally, we integrated BoolE into multiplier formal verification tasks, where it significantly accelerates the performance of traditional formal verification tools using computer algebra, demonstrated over four orders of magnitude runtime improvements.

Problem

• Exact symbolic reasoning for gate-level netlists in verification and synthesis
• Challenges in identifying function blocks accurately post-structural modifications
• Scalability and performance issues in Boolean symbolic reasoning approaches

Innovation

• Uses equality saturation for Boolean netlists
• Integrates domain-specific Boolean ruleset
• Novel extraction algorithm for high-level structures