Logic for Computer Scientists/Propositional Logic/Horn clauses

Horn clauses In this subsection we introduce a special class of formulae which are of particular interest for logic programming. Furthermore it turns out that these formulae admit an efficient test for satisfiability. Definition 9 A formula is a Horn formula if it is in CNF and every disjunction contains at most one positive literal. Horn clauses are clauses, which contain at most one positive literal.                                             where is a tautology and is an unsatisfiable formula. In clause form this can be written as and in the context of logic programming this is written as:              For Horn formula there is an efficient algorithm to test satisfiability of a formula : Deciding Satisfiability of Horn Formulae If there is a subformula of the kind label every occurrence of in . Apply the following rules until none of them is applicable: If is a subformula and are all labeled and is not labeled then label every occurrence of in . If is a subformula and are all labeled then Stop: Unsatisfiable Stop: Satisfiable The assignment iff is labeled is a model. Theorem 6 The above algorithm is correct and stops after steps, where is the number of atoms in the formula. As an immediate consequence we see, that a Horn formula is satisfiable if there is no subformula of the form . Horn formulae admit unique least models, i.e. is a unique least model for if for every model and for every atom B in F holds: if then . Note, that this unique least model property does not hold for non Horn formulae: as an example take which is obviously non Horn. is a least model and as well, hence we have two least models. Problems Problem 20 (Propositional) Let be a propositional logical formulae and a subset atomic formula occurring in . Let be the formula which results from by replacing all occurrences of an atomic formulae by . Example: . Prove or disprove: There exists an for or , so that is equivalent to a Horn formula (i.e. ). Problem 21 (Propositional) Apply the marking algorithm to the following formula F. Which is a least model? Problem 22 (Propositional) Decide which one of the indicated CNFs are Horn formulae and transform then into a conjunction of implications: