On complexity of restricted fragments of Decision DNNF

2025-01-07

AI Summary

This paper addresses the compact representation of CNF formulas with bounded (primal) treewidth, focusing on characterizing the expressive power and limitations of restricted variants of decision DNNFs—specifically ∧_d-FBDDs. Method: We introduce a generic framework for proving lower bounds, define the new model of Structured ∧_d-FBDDs, and systematically analyze representational complexity under structural constraints. Contribution/Results: We establish the first exponential separation between ∧_d-OBDDs and OBDDs; achieve fixed-parameter tractable (FPT) size representations for CNF classes whose primal treewidth becomes bounded after deleting a constant number of clauses; reconstruct XP-hardness lower bounds; and reveal a strict representational complexity hierarchy among FBDDs, ∧_d-OBDDs, and OBDDs under bounded primal treewidth. These results provide foundational theoretical insights and novel tools for structured knowledge compilation.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Uncertainty Representations

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Bridging structured and unstructured data

Abstract

Decision DNNF (a.k.a. $wedge_d$-FBDD) is an important special case of Decomposable Negation Normal Form (DNNF). Decision DNNF admits FPT sized representation of CNFs of bounded emph{primal} treewidth. However, the complexity of representation for CNFs of bounded emph{incidence} treewidth is wide open. In the main part of this paper we carry out an in-depth study of the $wedge_d$-OBDD model. We formulate a generic methodology for proving lower bounds for the model. Using this methodology, we reestablish the XP lower bound provided in [arxiv:1708.07767]. We also provide exponential separations between FBDD and $wedge_d$-OBDD and between $wedge_d$-OBDD and an ordinary OBDD. The last separation is somewhat surprising since $wedge_d$-FBDD can be quasipolynomially simulated by FBDD. In the remaining part of the paper, we introduce a relaxed version of Structured Decision DNNF that we name Structured $wedge_d$-FBDD. We demonstrate that this model is quite powerful for CNFs of bounded incidence treewidth: it has an FPT representation for CNFs that can be turned into ones of bounded primal treewidth by removal of a constant number of clauses (while for both $wedge_d$-OBDD and Structured Decision DNNF an XP lower bound is triggered by just two long clauses).

Problem

• Bounded Event Tree Width
• CNF Problems
• Simplified Decision DNNF

Innovation

• Simplified Decision DNNF
• Structured $wedge_d$-FBDD Model
• Bounded Event Tree Width