Dynamic Boolean Synthesis with Zero-suppressed Decision Diagrams

2025-12-07

AI Summary

Boolean synthesis is a fundamental problem in formal methods, yet existing ZDD-based approaches lack a systematic dynamic programming (DP) framework. This paper introduces the first ZDD-oriented DP framework for Boolean synthesis, unifying functional synthesis and QBF-solving requirements. By introducing the “magic number,” we establish a unified characterization of both time complexity and treewidth upper bounds, thereby revealing, for the first time, the theoretical exploration–exploitation trade-off inherent in the planning–execution paradigm. The framework synergistically integrates ZDDs’ compact representation, symbolic computation, and DP, achieving significant efficiency gains in settings suffering from state-space explosion. Experimental evaluation demonstrates that our implemented tool outperforms baseline methods in both generality and performance. It establishes a novel paradigm for industrial-strength synthesizers and catalyzes systematic investigation of the magic number as a new analytical dimension.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Motivated by functional synthesis in sequential circuit construction and quantified boolean formulas (QBF), boolean synthesis serves as one of the core problems in Formal Methods. Recent advances show that decision diagrams (DD) are particularly competitive in symbolic approaches for boolean synthesis, among which zero-suppressed decision diagram (ZDD) is a relatively new algorithmic approach, but is complementary to the industrial portfolio, where binary decision diagrams (BDDs) are more often applied. We propose a new dynamic-programming ZDD-based framework in the context of boolean synthesis, show solutions to theoretical challenges, develop a tool, and investigate the experimental performance. We also propose an idea of magic number that functions as the upper bound of planning-phase time and treewidth, showing how to interpret the exploration-exploitation dilemma in planning-execution synthesis framework. The algorithm we propose shows its strengths in general, gives inspiration for future needs to determine industrial magic numbers, and justifies that the framework we propose is an appropriate addition to the industrial synthesis solvers portfolio.

Problem

• Develops a dynamic ZDD framework for boolean synthesis.
• Addresses theoretical challenges in synthesis with decision diagrams.
• Proposes magic number concept for planning-execution trade-offs.

Innovation

• Dynamic ZDD framework for boolean synthesis
• Magic number concept for planning time bound
• Integration into industrial synthesis solvers portfolio