Full text of the second edition of Artificial Intelligence: foundations of computational agents, Cambridge University Press, 2017 is now available.

### 5.4.1 Horn Clauses

The definite clause language does not allow a contradiction to be stated. However, a simple expansion of the language can allow proof by contradiction.

An **integrity constraint** is a clause of the form

*false←a*

_{1}∧...∧a_{k}.where the *a _{i}* are atoms and

*false*is a special atom that is

*false*in all interpretations.

A **Horn
clause** is either a definite
clause or an integrity
constraint. That is, a Horn clause has either *false*
or a normal atom as its head.

Integrity constraints allow the system to prove that some conjunction of atoms is
*false* in all models of a knowledge base - that is, to prove disjunctions of negations
of atoms. Recall that *¬p* is the **negation**
of *p*, which is true in an interpretation when *p* is false in that
interpretation, and *p∨q* is the
**disjunction** of *p* and *q*, which is true in an interpretation if *p* is true or *q* is true
or both are true in the interpretation. The integrity
constraint *false←a _{1}∧...∧a_{k}* is logically
equivalent to

*¬a*.

_{1}∨...∨¬a_{k}A Horn clause knowledge base can imply negations of atoms, as shown in Example 5.16.

**Example 5.16:**Consider the knowledge base

*KB*:

_{1}*false←a∧b.*

*a←c.*

*b←c.*

The atom *c* is *false* in all models of *KB _{1}*. If

*c*were

*true*in model

*I*of

*KB*, then

_{1}*a*and

*b*would both be

*true*in

*I*(otherwise

*I*would not be a model of

*KB*). Because

_{1}*false*is

*false*in

*I*and

*a*and

*b*are

*true*in

*I*, the first clause is

*false*in

*I*, a contradiction to

*I*being a model of

*KB*. Thus,

_{1}*c*is

*false*in all models of

*KB*. This is expressed as

_{1}*KB*

_{1}¬cwhich means that *¬c* is *true* in
all models of *KB _{1}*, and so

*c*is

*false*in all models of

*KB*.

_{1}Although the language of Horn clauses does not allow disjunctions and negations to be input, disjunctions of negations of atoms can be derived, as Example 5.17 shows.

**Example 5.17:**Consider the knowledge base

*KB*:

_{2}*false←a∧b.*

*a←c.*

*b←d.*

*b←e.*

Either *c* is *false* or *d* is *false* in every model of *KB _{2}*. If
they were both

*true*in some model

*I*of

*KB*, both

_{2}*a*and

*b*would be true in

*I*, so the first clause would be false in

*I*, a contradiction to

*I*being a model of

*KB*. Similarly, either

_{2}*c*is

*false*or

*e*is

*false*in every model of

*KB*. Thus,

_{2}

KB_2 ¬c∨ ¬d. KB_2 ¬c∨ ¬e.

A set of clauses is **unsatisfiable** if it
has no models. A set of clauses is provably **inconsistent** with respect to a proof procedure
if *false* can be derived from the clauses using that proof
procedure. If a proof procedure is sound and complete, a set of
clauses is provably inconsistent if and only if it is unsatisfiable.

It is
always possible to find a model for a set of definite clauses. The
interpretation with all atoms *true* is a model of any set of definite
clauses. Thus, a definite-clause knowledge base is always
satisfiable. However, a set of Horn clauses can be unsatisfiable.

**Example 5.18:**The set of clauses

*{a, false←a}*is unsatisfiable. There is no interpretation that satisfies both clauses. Both

*a*and

*false←a*cannot be

*true*in any interpretation.

Both
the top-down and the bottom-up proof procedures can be used to prove
inconsistency, by using *false* as the query. That is, a Horn clause
knowledge base is inconsistent if and only if *false* can be derived.