🤖 AI Summary
This study addresses the problem of correctly and efficiently encoding compressed natural deduction proofs—represented as directed acyclic graphs, known as DLDSs—into Boolean circuits. Focusing on level-compressed DLDSs in pure implicational minimal logic, the work proposes a path-assignment-based dependency propagation semantics that guarantees logical equivalence between the original proof and its circuit encoding. Innovatively, it presents the first formalization in the Lean theorem prover of a verifiable mapping from DLDSs to Boolean circuits, and establishes a rigorous correspondence between path extraction in uncompressed tree-like fragments and circuit acceptance conditions. This contribution not only enables machine-verified translation but also provides a formally checkable circuit evaluator, thereby strengthening the foundation for certified proof-to-circuit transformations.
📝 Abstract
In this article, we present a method for encoding Dag-Like Derivability Structures (DLDS), obtained via horizontal compression of Natural Deduction proofs in purely implicational minimal logic, as Boolean circuits. These DLDS compress Natural Deduction tree-like proofs into directed acyclic graphs, preserving logical correctness while reducing redundancy. We formally define the circuit construction process and establish its pointwise correctness, showing that, for any fixed path assignment, the resulting Boolean circuit agrees with the intended dependency-propagation semantics. A Lean formalization establishes machine-checked guarantees for the circuit evaluator and includes a restricted bridge for the uncompressed simple-tree fragment, connecting valid DLDS instances in that fragment to genuine circuit acceptance of their extracted paths under the route and discharge conditions formalized in Lean. This approach opens new perspectives for automated theorem proving and formal certification.