The Next Whisky Bar

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 − ).…”