Introducing w-Horn and z-Horn: A generalization of Horn and q-Horn formulae

Abstract

In this paper we generalize the well-known notions of Horn and 𝑞 -Horn formulae. A Horn clause, by definition, contains at most one positive literal. A Horn formula contains only Horn clauses. We generalize these notions as follows. A clause is a 𝑤 -Horn clause if and only if it contains at least one negative literal or it is a unit or it is the empty clause. A formula is a 𝑤 -Horn formula if it contains only 𝑤 -Horn clauses after exhaustive unit propagation, i.e., after a Boolean Constraint Propagation (BCP) step. We show that the set of 𝑤 -Horn formulae properly includes the set of Horn formulae. A function 𝛽 ( 𝑥 ) is a valuation function if 𝛽 ( 𝑥 ) + 𝛽 ( ¬ 𝑥 ) = 1 and 𝛽 ( 𝑥 ) ∈ { 0 , 0 . 5 , 1 } , where 𝑥 is a Boolean variable. A formula ℱ is a 𝑞 -Horn formula if and only if there is a valuation function 𝛽 ( 𝑥 ) such that for each clause 𝐶 in ℱ we have that ∑︀ 𝑥 ∈ 𝐶 𝛽 ( 𝑥 ) ≤ 1 . In this case we call 𝛽 ( 𝑥 ) a 𝑞 -feasible valuation for ℱ . In other words, a formula is 𝑞 -Horn if and only if each clause in it contains at most one “positive” literal (where 𝛽 ( 𝑥 ) = 1 ) or at most two half ones (where 𝛽 ( 𝑥 ) = 0 . 5 ). We generalize these notions as follows. A formula ℱ is a 𝑧 -Horn formula if and only if ℱ ′ = BCP ( ℱ ) and either ℱ ′ is trivially satisfiable or trivially unsatisfiable or there is a valuation function 𝛾 ( 𝑥 ) such that for each clause 𝒞 in ℱ ′ we have that ( 1 or ∑︀ 𝑥 ∈ 𝐶 ∧ 𝛾 ( 𝑥 )=0 . 5 𝛾 ( 𝑥 ) = 1 . In this case we call 𝛾 ( 𝑥 ) to be a 𝑧 -feasible valuation for 𝐹 ′ . In other words, a formula is 𝑧 -Horn if and only if each clause in it after a BCP step contains at least one “negative” literal (where 𝛾 ( 𝑥 ) = 0 ) or exactly two half ones (where 𝛾 ( 𝑥 ) = 0 . 5 ). We show that the set of 𝑧 -Horn formulae properly includes the set of 𝑞 -Horn formulae. We also show that the 𝑤 -Horn SAT problem can be decided in polynomial time. We also show that each satisfiable formula is 𝑧