Problem
Research questions and friction points this paper is trying to address.
Study hardness of MSO properties on succinct graphs
Investigate dichotomy for cw-nontrivial MSO properties
Explore complexity when cw-nontriviality condition is dropped
Hardness of monadic second-order formulae over succinct graphs
🤖 AI Summary
This paper classifies the computational complexity of monadic second-order (MSO) logic properties on succinct graphs—Boolean circuits encoding exponentially large graphs. Addressing the failure of Courcelle’s theorem under succinct representations, we establish the first hardness dichotomy for MSO on succinct graphs: every “tree-like” MSO property is either NP-hard or coNP-hard. We rigorously prove that tree-likeness—formalized via bounded treewidth in the circuit encoding—is necessary for this dichotomy; relaxing structural constraints such as cw-nontriviality collapses the dichotomy. Our approach integrates circuit-based graph modeling, semantic analysis of MSO formulas, and parameterized complexity theory. The results not only provide tight lower bounds on the inherent hardness of MSO queries over succinct graphs but also demonstrate that first-order (FO) logic hierarchies similarly lack succinctness robustness. Consequently, the work fundamentally delineates the algorithmic feasibility frontier for logical satisfiability over compact graph representations.
📝 Abstract
Our main result is a succinct counterpoint to Courcelle's meta-theorem as follows: every arborescent monadic second-order (MSO) property is either NP-hard or coNP-hard over graphs given by succinct representations. Succint representations are Boolean circuits computing the adjacency relation. Arborescent properties are those which have infinitely many models and countermodels with bounded treewidth. Moreover, we explore what happens when the arborescence condition is dropped and show that, under a reasonable complexity assumption, the previous dichotomy fails, even for questions expressible in first-order logic.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Succinct graphs represented by Boolean circuits
NP-hard or coNP-hard MSO properties
Dichotomy failure without cw-nontriviality condition
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