As Close as It Gets

Minimal Distance of Propositional Models

“…Throughout the text we refer to different types of Boolean constraint relations following Schaefer's terminology [27] (see also the monograph [11] and the survey [9]). A Boolean relation R is (1) 1-valid if 1 · · · 1 ∈ R and 0-valid if 0 · · · 0 ∈ R, (2) Horn (dual Horn) if R can be represented by a formula in conjunctive normal form (CNF) with at most one unnegated (negated) variable per clause, (3) monotone if it is both Horn and dual Horn, (4) bijunctive if it can be represented by a CNF formula with at most two literals per clause, (5) affine if it can be represented by an affine system of equations Ax = b over Z 2 , (6) complementive if for each m ∈ R also m ∈ R, (7) implicative hitting set-bounded+ with bound k (denoted by k -IHS-B + ) if R can be represented by a CNF formula with clauses of the form (x 1 ∨ · · · ∨ x k ), (¬x ∨ y), x, and ¬x, (8) implicative hitting set-bounded− with bound k (denoted by k -IHS-B − ) if R can be represented by a CNF formula with clauses of the form (¬x 1 ∨ · · · ∨ ¬x k ), (¬x ∨ y), x, and ¬x. A set Γ of Boolean relations is called 0-valid (1-valid, Horn, dual Horn, monotone, affine, bijunctive, complementive, k -IHS-B + , k -IHS-B − ) if every relation in Γ is 0-valid (1-valid, Horn, dual Horn, monotone, affine, bijunctive, complementive, k -IHS-B + , k -IHS-B − ).…”

“…Throughout the text we refer to different types of Boolean constraint relations following Schaefer's terminology [27] (see also the monograph [11] and the survey [9]). A Boolean relation R is (1) (2) Horn (dual Horn) if R can be represented by a formula in conjunctive normal form (CNF) with at most one unnegated (negated) variable per clause, (3) monotone if it is both Horn and dual Horn, (4) bijunctive if it can be represented by a CNF formula with at most two literals per clause, (5) affine if it can be represented by an affine system of equations Ax = b over Z 2 , (6) complementive if for each m ∈ R also m ∈ R, (7) implicative hitting set-bounded+ with bound k (denoted by k -IHS-B + ) if R can be represented by a CNF formula with clauses of the form A formula constructed from atoms by conjunction, variable identification, and existential quantification is called a primitive positive formula (pp-formula). If ϕ is such a formula, we write again [ϕ] for its set of models, i.e., the Boolean relation defined by ϕ.…”

Time Complexity of Constraint Satisfaction via Universal Algebra

“…We will describe this algebraic approach in greater detail later on, but the most important property is that the partial polymorphisms of finite constraint languages give rise to a partial order ⊑ with the property that if Γ ⊑ ∆, then T(Γ) ≤ T(∆). We remark that partial polymorphisms are not only useful when studying CSPs with this very fine-grained notion of complexity, but have also been used to study the classical complexity of many different computational problems where polymorphisms are not applicable [3,4,11,14,21].…”