1. Clausal form :
In this form, the formula is made up of a number of clauses, where each clause is composed of a number of literals connected by OR logical connectives only.
A clausal form formula must be transformed into another formula with the following characteristics :
- All variables in the formula are universally quantified. Hence, it is not necessary to include the universal quantifiers explicitly for all. The quantifiers are removed, and all variables in the formula are implicitly quantified by the universal quantifier.
- As the formula is made up of a number of clauses, and each clause is composed of a number of literals connected by OR logical connectives only. Hence, each clause is a dis-junction of literals.
- To form a formula, the clauses themselves are connected by AND logical connectives only. Hence, clausal form of a formula is a conjunction of clauses.
2. Horn Clausal form :
A Horn clause is a clause (a disjunction of literals) with at most one positive, i.e. unnegated, literal. A clause with at most one positive (unnegated) literal is called a Horn Clause.
Types of Horn Clauses :
- Definite clause / Strict Horn clause : Has exactly one positive literal.
- Unit clause : Definite clause with no negative literals.
- Goal clause : Horn clause without a positive literal.
Examples :
Clausal form :
Literals can be positive literals or negative literals. For the forms of the individual clauses where each of is a disjunction of literals. For the clause form :
NOT(P1) OR NOT(P2) OR ..... OR NOT(Pn) OR Q1 OR Q2 OR ..... OR Qm
The above clause has n negative literals and m positive literals. This clause can be transformed into the following equivalent logical formula :
P1 AND P2 AND ..... AND Pn => Q1 OR Q2 OR ..... OR Qm
where ‘=>’ is the implies symbol.
Horn Clausal form :
(i) The Horn clause,
NOT(P1) OR NOT(P2) OR ... OR NOT(Pn) OR Q
Can be transformed into the clause :
P1 AND P2 AND … AND Pn => Q
which is written in Datalog as the following rules.
Q :- P1, P2, ..., Pn.
The above Datalog Rule, is hence a Horn clause.
Meaning : If the predicates P1 AND P2 AND … AND Pn, are all true for a particular binding to their variable arguments, then Q is also true and can hence be inferred.
(ii) The Horn clause,
NOT(P1) OR NOT(P2) OR ... OR NOT (Pn)
Can be transformed into
P1 AND P2 AND … AND Pn =>
which is written in Datalog as follows :
P1, P2, ..., Pn.
The above Datalog expression can be considered as an integrity constraint, where all the predicates must be true to satisfy the query.