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Clause (logic)

${\displaystyle l_{1}\vee \cdots \vee l_{n}}$

Empty clauses[edit]

A clause can be empty (defined from an empty set of literals). The empty clause is denoted by various symbols such as ${\displaystyle \emptyset }$, ${\displaystyle \bot }$, or ${\displaystyle \Box }$. The truth evaluation of an empty disjunctive clause is always ${\displaystyle false}$. This is justified by considering that ${\displaystyle false}$ is the neutral element of the monoid ${\displaystyle (\{false,true\},\vee )}$.

The truth evaluation of an empty conjunctive clause is always ${\displaystyle true}$. This is related to the concept of a vacuous truth.

Implicative form[edit]

Every nonempty (disjunctive) clause is logically equivalent to an implication of a head from a body, where the head is an arbitrary literal of the clause and the body is the conjunction of the complements of the other literals. That is, if a truth assignment causes a clause to be true, and all of the literals of the body satisfy the clause, then the head must also be true.

This equivalence is commonly used in logic programming, where clauses are usually written as an implication in this form. More generally, the head may be a disjunction of literals. If ${\displaystyle b_{1},\ldots ,b_{m}}$ are the literals in the body of a clause and ${\displaystyle h_{1},\ldots ,h_{n}}$ are those of its head, the clause is usually written as follows:

${\displaystyle h_{1},\ldots ,h_{n}\leftarrow b_{1},\ldots ,b_{m}.}$

• Ifn = 1 and m = 0, the clause is called a (Prolog) fact.
• Ifn = 1 and m > 0, the clause is called a (Prolog) rule.
• Ifn = 0 and m > 0, the clause is called a (Prolog) query.
• Ifn > 1, the clause is no longer Horn.

See also[edit]

• Conjunctive normal form
• Disjunctive normal form
• Horn clause

References[edit]

External links[edit]

• Clause logic related terminology

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