🤖 AI Summary
This work addresses the challenge of verifying equivalence between target representations—such as decision-DNNF—and their original CNF encodings in knowledge compilation. Methodologically, it introduces (1) Partitioned-Operation Graphs (POGs) as a unified intermediate representation; (2) the Certified POG (CPOG) proof framework, enabling structured, correctness-preserving compilation from CNF to POG; and (3) full formal verification in Lean 4 of the compiler, proof generator, and model counter. Contributions include: the first end-to-end, machine-checked correctness guarantee for the entire knowledge compilation pipeline; automated verification of D4-generated POGs; empirical evaluation on standard model counting benchmarks; and the first mathematically verified toolchain supporting both weighted and unweighted model counting. The framework ensures semantic equivalence at every compilation step, thereby bridging the gap between practical knowledge compilation tools and formal correctness guarantees.
📝 Abstract
Computing many useful properties of Boolean formulas, such as their weighted or unweighted model count, is intractable on general representations. It can become tractable when formulas are expressed in a special form, such as the decision decomposable negation normal form (decision-DNNF). Knowledge compilation is the process of converting a formula into such a form. Unfortunately existing knowledge compilers provide no guarantee that their output correctly represents the original formula, and therefore they cannot validate a model count, or any other computed value. We present Partitioned-Operation Graphs (POGs), a form that can encode all of the representations used by existing knowledge compilers. We have designed CPOG, a framework that can express proofs of equivalence between a POG and a Boolean formula in conjunctive normal form (CNF). We have developed a program that generates POG representations from the decision-DNNF graphs produced by the state-of-the-art knowledge compiler D4, as well as checkable CPOG proofs certifying that the output POGs are equivalent to the input CNF formulas. Our toolchain for generating and verifying POGs scales to all but the largest graphs produced by D4 for formulas from a recent model counting competition. Additionally, we have developed a formally verified CPOG checker and model counter for POGs in the Lean 4 proof assistant. In doing so, we proved the soundness of our proof framework. These programs comprise the first formally verified toolchain for weighted and unweighted model counting.