Structure-Guided Automated Reasoning

2023-12-22

arXiv.org

AI Summary

Efficient model checking of Monadic Second-Order (MSO) logic on graphs remains challenging, particularly while preserving structural parameter bounds such as treewidth. Method: We propose the first automated SAT-encoding framework for MSO model checking that strictly preserves the input graph’s treewidth. Our approach compactly reduces arbitrary MSO formulas to equivalent SAT, MaxSAT, #SAT, and ILP instances—each reduction guaranteeing treewidth invariance. Contributions/Results: (1) A novel SAT-based proof of Courcelle’s Theorem; (2) a unified parameterized paradigm for decision, optimization, and counting variants of MSO model checking; (3) tight upper and lower bounds on formula block size in terms of treewidth, conditioned on the Exponential Time Hypothesis (ETH). Experimentally, our solver achieves runtime comparable to domain-specific algorithms—up to polynomial factors. This work establishes the first fully automated, treewidth-preserving translation from MSO to SAT, enabling cross-paradigm solver integration (e.g., SAT, MaxSAT, #SAT, ILP) for MSO reasoning.

Abstract

Algorithmic meta-theorems state that problems definable in a fixed logic can be solved efficiently on structures with certain properties. An example is Courcelle's Theorem, which states that all problems expressible in monadic second-order logic can be solved efficiently on structures of small treewidth. Such theorems are usually proven by algorithms for the model-checking problem of the logic, which is often complex and rarely leads to highly efficient solutions. Alternatively, we can solve the model-checking problem by grounding the given logic to propositional logic, for which dedicated solvers are available. Such encodings will, however, usually not preserve the input's treewidth. This paper investigates whether all problems definable in monadic second-order logic can efficiently be encoded into Lang{sat} such that the input's treewidth bounds the treewidth of the resulting formula. We answer this in the affirmative and, hence, provide an alternative proof of Courcelle's Theorem. Our technique can naturally be extended: There are treewidth-aware reductions from the optimization version of Courcelle's Theorem to MaxSAT and from the counting version of the theorem to #SAT. By using encodings to SAT, we obtain, ignoring polynomial factors, the same running time for the model-checking problem as we would with dedicated algorithms. Another immediate consequence is a treewidth-preserving reduction from the model-checking problem of monadic second-order logic to integer linear programming (ILP). We complement our upper bounds with new lower bounds based on ETH; and we show that the block size of the input's formula and the treewidth of the input's structure are tightly linked.

Problem

• High-level Logic Transformation
• Lang{sat} Formulation
• Tree-width Preservation

Innovation

• Tree-width Preservation
• Integer Linear Programming
• Efficient Transformation