Logic for Computer Scientists/Propositional Logic/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.
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 :
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.
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
- , 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: