Testing Expressibility Is Hard

We study the expressibility problem: given a finite constraint language Γ on a finite domain and another relation R, can Γ express R? We prove, by an explicit family of examples, that the standard witnesses to expressibility and inexpressibility (gadgets/formulas/conjunctive queries and polymorphisms respectively) may be required to be exponentially larger than the instances. We also show that the full expressibility problem is co-NEXPTIME-hard. Our proofs hinge on a novel interpretation of a tiling problem into the expressibility problem. Key words: constraint, relation, expressive power, inverse satisfiability, structure identification, conjunctive query, primitive positive formula, polymorphism, domino system, nondeterministic exponential time

• Type: Journal
• Year: 2010
• Where: CP
• Authors: Ross Willard
• Keywords: Artificial Intelligence, CP 2010, Explicit Family, Expressibility Problem, finite Constraint Language