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Formalization of Logical Calculi in Isabelle/HOL


Fleury,  Mathias
Automation of Logic, MPI for Informatics, Max Planck Society;
International Max Planck Research School, MPI for Informatics, Max Planck Society;

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Fleury, M. (2020). Formalization of Logical Calculi in Isabelle/HOL. PhD Thesis, Universität des Saarlandes, Saarbrücken. doi:10.22028/D291-30179.

Cite as: https://hdl.handle.net/21.11116/0000-0005-AE07-0
I develop a formal framework for propositional satifisfiability with the conflict-driven clause learning (CDCL) procedure using the Isabelle/HOL proof assistant. The framework offers a convenient way to prove metatheorems and experiment with variants, including the Davis-Putnam-Logemann-Loveland procedure. The most noteworthy aspects of my work are the inclusion of rules for forget and restart and the refinement approach. I use the formalization to develop three extensions: First, an incremental solving extension of CDCL. Second, I verify an optimizing CDCL (OCDCL): Given a cost function on literals, OCDCL derives an optimal model with minimum cost. Finally, I work on model covering. Thanks to the CDCL framework I can reuse, these extensions are easier to develop. Through a chain of refinements, I connect the abstract CDCL calculus first to a more concrete calculus, then to a SAT solver expressed in a simple functional programming language, and finally to a SAT solver in an imperative language, with total correctness guarantees. The imperative version relies on the two-watched-literal data structure and other optimizations found in modern solvers. I used the Isabelle Refinement Framework to automate the most tedious refinement steps. After that, I extend this work with further optimizations like blocking literals and the use of machine words as long as possible, before switching to unbounded integers to keep completeness.