Horn clause

In mathematical logic and logic programming, a Horn clause is a logical formula of a particular rule-like form which gives it useful properties for use in logic programming, formal specification, and model theory. Horn clauses are named for the logician Alfred Horn, who first pointed out their significance in 1951.

Definition

A Horn clause is a clause (a disjunction of literals) with at most one positive, i.e. unnegated, literal.

Conversely, a disjunction of literals with at most one negated literal is called a dual-Horn clause.

A Horn clause with exactly one positive literal is a definite clause; a definite clause with no negative literals is sometimes called a fact; and a Horn clause without a positive literal is sometimes called a goal clause (note that the empty clause consisting of no literals is a goal clause). These three kinds of Horn clauses are illustrated in the following propositional example:

In the non-propositional case, all variables in a clause are implicitly universally quantified with scope the entire clause. Thus, for example: