Parameterized Complexity Of Representing Models Of MSO Formulas
2026-04-09 • Artificial Intelligence
Artificial IntelligenceComputational Complexity
AI summaryⓘ
The authors expand on Courcelle's theorem, which connects graph properties and a type of logic called MSO2, showing that certain graph problems can be solved efficiently based on treewidth. They prove that solutions to these problems can be represented in compact decision diagrams whose size grows linearly with parameters like treewidth and pathwidth. They also demonstrate limits for one type of diagram (OBDD) under specific conditions, offering insights that link graph theory, logic, and knowledge representation. This work bridges understanding between theoretical computer science and data structure representations.
Monadic second order logic (MSO2)Courcelle's theoremTreewidthPathwidthParameterized complexityDecision diagramsSentential decision diagram (SDD)Ordered binary decision diagram (OBDD)Graph theoryKnowledge representation
Authors
Petr Kučera, Petr Martinek
Abstract
Monadic second order logic (MSO2) plays an important role in parameterized complexity due to the Courcelle's theorem. This theorem states that the problem of checking if a given graph has a property specified by a given MSO2 formula can be solved by a parameterized linear time algorithm with respect to the treewidth of the graph and the size of the formula. We extend this result by showing that models of MSO2 formula with free variables can be represented with a decision diagram whose size is parameterized linear in the above mentioned parameter. In particular, we show a parameterized linear upper bound on the size of a sentential decision diagram (SDD) when treewidth is considered and a parameterized linear upper bound on the size of an ordered binary decision diagram (OBDD) when considering the pathwidth in the parameter. In addition, building on a lower bound on the size of OBDD by Razgon (2014), we show that there is an MSO2 formula and a class of graphs with bounded treewidth which do not admit an OBDD with the size parameterized by the treewidth. Our result offers a new perspective on the Courcelle's theorem and connects it to the area of knowledge representation.