Abstract

We study the semantics of propositional interference-based proof systems such as DRAT and DPR. These are characterized by modifying a CNF formula in ways that preserve satisfiability but not necessarily logical truth. We propose an extension of propositional logic called overwrite logic with a new construct which captures the meta-level reasoning behind interferences. We analyze this new logic from the point of view of expressivity and complexity, showing that while greater expressivity is achieved, the satisfiability problem for overwrite logic is essentially as hard as SAT, and can be reduced in a way that is well-behaved for modern SAT solvers. We also show that DRAT and DPR proofs can be seen as overwrite logic proofs which preserve logical truth. This much stronger invariant than the mere satisfiability preservation maintained by the traditional view gives us better understanding on these practically important proof systems. Finally, we showcase this better understanding by finding intrinsic limitations in interference-based proof systems.

Highlights

  • The story of SAT solving is one of great success, which has made SAT solvers widely used in practical applications due to their ability to routinely solve instances with millions of variables

  • Expressivity and complexity of overwrite formulas Having extended propositional logic PL with the overwrite connective to obtain the new logic overwrite propositional logic (OPL) and its restrictions Overwrite clausal normal form (OCNF) and uniformly overwrite clausal normal forms (UOCNF), it is natural to ask how these logics compare with respect to expressivity and what is the complexity of deciding their satisfiability

  • The inclusion arrow shows that clausal normal form (CNF) can be embedded in PL; the coiled arrow shows that transforming PL formulas into equivalent CNF formulas is worst-case exponential; and the dashed arrow shows that the satisfiability problem for PL can be polynomially reduced to the satisfiability problem for CNF

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Summary

Introduction

The story of SAT solving is one of great success, which has made SAT solvers widely used in practical applications due to their ability to routinely solve instances with millions of variables. Two recent interrelated breakthroughs in SAT solving are the extension of conflict-driven clause learning (CDCL) solvers [34] with inprocessing techniques that allow efficient solving in fragments where CDCL has exponential behavior [1, 24, 19] and the introduction of increasingly expressive and easy to check proof systems to certify the correctness of unsatisfiability results [7, 11, 41, 17]. We argue that for any practical application overwrite logic is no more different from CNF formulas than propositional logic, and Tseitin-like procedures [30] exist with similar complexity This new perspective enables inferences that cannot be performed with interference reasoning. Since many model checking approaches are interpolation-based [27], or can work as interpolant generators [4], there are compelling reasons to study the interaction between interpolation and interference reasoning

Preliminaries
Redundancy notions on CNF logic
Interference-based proofs
Overwrite propositional logic
Qualitative expressivity
Quantitative expressivity
Complexity of the satisfiability problem
Understanding DPR
Satisfiability preservation as a proof invariant
Proof-dependent semantic invariants
DPR proofs as truth-preserving proofs on UOCNFs
New insights with overwrite logic
Conclusion
Full Text
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