Boolean Constraint Satisfaction Problems: When Does Post's Lattice Help?

The propositional satisfiability problem SAT, i.e., the problemto decide, given a propositional formula φ (without loss ofgenerality in conjunctive normal form CNF), if there is anassignment to the variables in φ that satisfies φ, is thehistorically first and standard NP-complete problem [Coo71].However, there are well-known syntactic restrictions for whichsatisfiability is efficiently decidable, for example if everyclause in the CNF formula has at most two literals (2CNF formulas)or if every clause has at most one positive literal (Horn formulas)or at most one negative literal (dual Horn formulas), see [KL99].To study this phenomenon more generally, we study formulas with“clauses” of arbitrary shapes, i.e., consisting ofapplying arbitrary relations R ⊆ {0,1} k to (notnecessarily distinct) variables x 1,...,x k . A constraint languageΓ is a finite set of such relations. In the rest of thischapter, Γ and Γ′ will always denote Booleanconstraint languages. A Γ-formula is a conjunction of clausesR(x 1,...,x k ) as above using only relations R from Γ. Thefor us central family of algorithmic problems, parameterized by aconstraint language Γ, now is the problem to determinesatisfiability of a given Γ-formula, denoted byCsp(Γ).