A characterization of efficiently compilable constraint languages
Christoph Berkholz, Stefan Mengel, Hermann Wilhelm
TL;DR
The paper delivers a complete dichotomy for when CSP constraint languages Γ admit compact representations in DNNF and its structured variants, based on strong blockwise decomposability and its uniform counterpart. It proves that strongly uniformly blockwise decomposable languages yield polynomial-time compilation to ODDs, while languages failing this property force exponential lower bounds for structured DNNF; a parallel dichotomy holds for unstructured representations via FDDs. A key methodological contribution is linking blockwise decomposability to rectangle covers and strong balance, enabling constructive algorithms (ODD/FDD) and unconditional lower bounds that do not rely on unproven assumptions. The Boolean case collapses to a tight dichotomy where only equalities and disequalities are tractable, and the paper also provides decidability results for the central notions, plus a separating example demonstrating the distinctness of the two decomposability notions. These results advance the understanding of the knowledge compilation landscape by precisely delineating the boundary between tractable and intractable constraint languages and by offering practical criteria to classify languages.
Abstract
A central task in knowledge compilation is to compile a CNF-SAT instance into a succinct representation format that allows efficient operations such as testing satisfiability, counting, or enumerating all solutions. Useful representation formats studied in this area range from ordered binary decision diagrams (OBDDs) to circuits in decomposable negation normal form (DNNFs). While it is known that there exist CNF formulas that require exponential size representations, the situation is less well studied for other types of constraints than Boolean disjunctive clauses. The constraint satisfaction problem (CSP) is a powerful framework that generalizes CNF-SAT by allowing arbitrary sets of constraints over any finite domain. The main goal of our work is to understand for which type of constraints (also called the constraint language) it is possible to efficiently compute representations of polynomial size. We answer this question completely and prove two tight characterizations of efficiently compilable constraint languages, depending on whether target format is structured. We first identify the combinatorial property of ``strong blockwise decomposability'' and show that if a constraint language has this property, we can compute DNNF representations of linear size. For all other constraint languages we construct families of CSP-instances that provably require DNNFs of exponential size. For a subclass of ``strong uniformly blockwise decomposable'' constraint languages we obtain a similar dichotomy for structured DNNFs. In fact, strong (uniform) blockwise decomposability even allows efficient compilation into multi-valued analogs of OBDDs and FBDDs, respectively. Thus, we get complete characterizations for all knowledge compilation classes between O(B)DDs and DNNFs.