Introducing w-Horn and z -Horn: A generalization of Horn and q -Horn formulae Gábor Kusper a , Csaba Biró b , Attila Adamkó c , Imre Baják d a Eszterházy Károly University kusper.gabor@uni-eszterhazy.hu b Eszterházy Károly University and Eötvös Lóránd University biro.csaba@uni-eszterhazy.hu c University of Debrecen adamkoa@inf.unideb.hu d Budapest Business School bajak.imre@uni-bge.hu Submitted: February 2, 2021 Accepted: March 17, 2021 Published online: March 19, 2021 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 Annales Mathematicae et Informaticae Accepted manuscript doi: https://doi.org/10.33039/ami.2021.03.009 url: https://ami.uni-eszterhazy.hu 1